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(George) Yeh1ă  N \6Susan SharpHansen2ă  NX 9Barry Lester3ă  N0 8Robert Strobl1ă  N 6Jeffrey Scarbrough4ă  N I.The Pennsylvania State University1ă 2University Park, PA 16802  N@ s4AQUA TERRA Consultants2ă 4Mountain View, CA 94043  N j8GeoTrans, Inc.3ă 6Sterling, VA 22170  NP 7AScI Corporation4ă 7Athens, GA 30613 2EPA Contract No. 68CO0019 ~8Project Monitor 9Robert Carsel &/Environmental Research Laboratory -U.S. Environmental Protection Agency 7Athens, GA 30613 - /ENVIRONMENTAL RESEARCH LABORATORY 2 .OFFICE OF RESEARCH AND DEVELOPMENT -U.S. ENVIRONMENTAL PROTECTION AGENCY Y4ATHENS, GEORGIA 30613%0*0*0*      :DISCLAIMER׃ The work presented in this document has been funded by the United States Environmental Protection Agency. It has been subject to the Agency's peer and administrative review, and has been approved as an EPA document. Mention of trade names or commercial products does not constitute endorsement or recommendation for use by the U.S. Environmental Protection Agency.  0*(( ;FOREWORD׃ As environmental controls become more costly to implement and the penalties of judgement errors become more severe, environmental quality management requires more efficient analytical tools based on greater knowledge of the environmental phenomena to be managed. As part of this Laboratory's research on the occurrence, movement, transformation, impact, and control of environmental contaminants, the Assessment Branch is developing management or engineering tools that can be used by States to protect public drinking water wells from possible contamination. The 1986 Amendments to the Safe Drinking Water Act require each State to develop and submit to the U.S. EPA a wellhead protection program. As part of the program, States must establish procedures for delineating wellhead protection areas around each water well or well field which supplies a public water system. In order to delineate wellhead protection areas in agricultural regions using the assimilative capacity criterion, the 3DFEMWATER/3DLEWASTE model has been developed. These finite element numerical codes simulate 1) flow and transport in threedimensional variablysaturated porous media under transient conditions, 2) multiple distributed and point sources/sinks, and 3) processes which retard the transport of contaminants. Rosemarie C. Russo, Ph.D. Director Environmental Research Laboratory Athens, Georgia (0*(( ;ABSTRACT׃ The 1986 Amendments to the Safe Drinking Water Act require each State to develop and submit to the U.S. EPA a wellhead protection program. As part of the program, States must establish procedures for delineating wellhead protection areas around each water well or well field which supplies a public water system. Of the five criteria that have been suggested by the U.S. EPA for delineating wellhead protection areas, the assimilative capacity criterion is potentially the most accurate. It takes into account the reduction in concentration of contaminants being transported toward a well caused by chemical and environmental processes at the land surface and in the vadose and saturated zones. Nationwide, agricultural areas are located in many diverse hydrogeologic environments. Recharge and pumping rates can vary widely within an area because of irrigation practices and/or climate. In addition, contamination scenarios must consider multiple point and nonpoint source loadings of pesticides which vary both spatially and temporally. In order to delineate wellhead protection areas in agricultural regions using the assimilative capacity criterion, the use of a numerical model which accounts for 1) flow and transport in threedimensional variablysaturated porous media under transient conditions, 2) multiple distributed and point sources/sinks, and 3) processes which retard the transport of contaminants, is needed. This document describes two related numerical codes, 3DFEMWATER and 3DLEWASTE, which can be used to delineate wellhead protection areas in agricultural regions using the assimilative capacity criterion. 3DFEMWATER (A Threedimensional Finite Element Model of WATER Flow through SaturatedUnsaturated Media) simulates subsurface flows, whereas 3DLEWASTE (A Hybrid ThreeĩDimensional LagrangianEulerian Finite Element Model of WASTE Transport through SaturatedUnsaturated Media) models contaminant transport. Both codes 1) treat heterogeneous and anisotropic media consisting of as many geologic formations as desired, 2) consider both distributed and point sources/sinks that are spatially and temporally dependent, and 3) accept four types of boundary conditions (i.e., Dirichlet (fixedhead or concentration), specifiedflux, Neumann (specifiedpressurehead gradient or specifieddispersive flux), and variable). The variable boundary condition in 3DFEMWATER simulates evaporation/infiltration/ seepage on the soilair interface and in 3DLEWASTE, simulates mass infiltration into or advection out of the system. 3DLEWASTE contains options to model adsorption using a linear, Freundlich, or Langmuir isotherm, dispersion, and firstorder decay. This report was submitted in partial fulfillment of Work Assignment Number 1, Contract Number 68CO0019 by AQUA TERRA Consultants, under the sponsorship of the U.S. Environmental Protection Agency. This report covers the period May 1991 to July 1992, and work was completed as of August 1992.X` hp x (#%'0*,.8135@8: key. Then follow instructions and respond to prompts presented on the monitor screen by the interactive installation program. Complete installation instructions are also printed on each external diskette label. The 3DFEMWATER/3DLEWASTE distribution diskette sets implement software product installation standards to insure the most errorfree, maintainable, and useracceptable distribution of CEAM products. It has a unique menu option, command, fullscreen (interactive), diagnostic, errorrecovery, help, and selective installation capabilities using stateoftheart humanfactors engineering practices and principles. NOTE: The contents of the distribution diskettes can be copied to another set of "backup" diskettes using the DOS DISKCOPY command. Refer to the DOS Reference Manual for command application and use. The "backup" diskettes must be the same size and storage density as the original, source diskettes. 2.6 INSTALLATION VERIFICATION AND ROUTINE EXECUTION Refer to the following sections in the READ.ME file for complete instructions concerning installation verification and routine execution of the 3DFEMWATER/3DLEWASTE model: 44File name and content.(#4 44Routine execution.(#4 44Run time and performance.(#4 44Minimum file configuration.(#4 2.7 CODE MODIFICATION Included in the diskette set are: 44An executable task image file for the 3DFEMWATER/3DLEWASTE model.(#4 44FORTRAN source code files.(#4 44Command and/or "make" files to compile, link, and run the task image file (*.EXE).(#4 If the user wishes to modify the model or any other program, it will be up to the user to supply and/or obtain: 44An appropriate text editor that saves files in ASCII (nonbinary) text format.(#4 '0*((Ԍ44FORTRAN development tools to recompile and link edit any portion of the model.(#4 CEAM cannot support, maintain, and/or be responsible for modifications that change the function of any executable task image (*.EXE), DOS batch command (*.BAT), and/or "make" utility file(s) supplied with this model package. 2.8 TECHNICAL HELP For questions and/or information concerning: 44Installation and/or testing of the 3DFEMWATER/3DLEWASTE model and/or support programs or files, call 706/5463590, 3548 for assistance.(#4 443DFEMWATER/3DLEWASTE model and/or program content, application, and/or theory, call 706/5463210 for assistance.(#4 44Use of the CEAM electronic bulletin board system (BBS), contact the BBS system operator (SYSOP) at 706/5463590.(#4 44Other environmental software and documentation distributed through CEAM, contact the Model Distribution Coordinator at 706/5463549.(#4 44Other support available through CEAM, contact Mr. Dermont Bouchard, CEAM Manager:(#4 44` ` By mail at the following address:(#` 44 Center for Exposure Assessment Modeling (CEAM) 44 Environmental Research Laboratory 44 U.S. Environmental Protection Agency 44 960 College Station Road 44 Athens, Georgia 306052720 44` ` By telephone at 706/5463130.(#` 44` ` By fax at 706/5463340.(#` 44` ` Through the CEAM BBS message menu and commands. The CEAM BBS communication parameters and telephone number are listed above.(#` 2.8.1 Electronic Bulletin Board System (BBS) To help technical staff provide better assistance, write down a response to the following topics before calling or writing. If calling, be at the computer, with the computer on, and in the proper subdirectory when the call is placed. XProgram information:(# '0*((Ԍ44Describe the problem, including the exact wording of any error and/or warning message(s).(#4 44List the exact steps, command(s), and/or keyboard key sequence that will reproduce the problem.(#4 Machine information: 44List computer brand and model.(#4 44List available RAM (as reported by DOS CHKDSK command).(#4 44List available extended memory (XMS).(#4 44List name and version of extended memory (XMS) manager (i.e., HIMEM, VDISK, RAMDRIVE, etc.).(#4 44List available hard disk space (as reported by DOS CHKDSK command).(#4 44List the brand and version of DOS (as reported by DOS VER command).(#4 44List the name of any memory resident program(s) installed.(#4 44Printer brand and model.(#4 44Monitor brand and model.(#4 NOTE: If contacting CEAM by mail, fax, or BBS, include responses to the above information in your correspondence. 2.9 DISCLAIMER Mention of trade names or use of commercial products does not constitute endorsement or recommendation for use by the United States Environmental Protection Agency. Execution of the 3DFEMWATER/3DLEWASTE model, and modifications to the DOS system configuration files (i.e., /CONFIG.SYS and /AUTOEXEC.BAT) must be made at the user's own risk. Neither the U.S. EPA nor the program authors can assume responsibility for model and/or program modification, content, output, interpretation, or usage. CEAM software products are built using FORTRAN77, assembler, and operating system interface command languages. The code structure and logic of these products is designed for singleuser, singletasking, nonLAN environment and operating platform for microcomputer installations (i.e., single user on a dedicated system). A user will be on their own if he/she attempts to install a CEAM product on a multiuser, multitasking, and/or LAN based system (i.e., Windows, DESQview, any LAN). CEAM cannot provide installation, operation, and/or general user support under any combination of these'0*(( configurations. Instructions and conditions for proper installation and testing are provided with the product in a READ.ME file. While multiuser/multitasking/LAN installations could work, none of the CEAM products have been thoroughly tested under all possible conditions. CEAM can provide scientific and/or application support for selected products if the user proves that a given product is installed and working correctly. 2.10 TRADEMARKS 44IBM, Personal Computer/XT (PC/XT), Personal Computer/AT (PC/AT), PC DOS, VDISK, and Personal System/2 (PS/2) are registered trademarks of International Business Machines Corporation.(#4 44DESQview is a trademark of Quarterdeck Office Systems, Inc.(#4 44Sun and SunOS are registered trademarks of Sun Microsystems, Inc.(#4 44SPARC is a registered trademark of SPARC International, Inc.(#4 44UNIX is a registered trademark of American Telephone and Telegraph.(#4 44SVS FORTRAN77 is a trademark of Silicon Valley Software.(#4 44PRIME and PRIMOS are trademarks of Prime Computers, Inc.(#4 44Microsoft, RAMDRIVE, HIMEM, MS, and MSDOS are registered trademarks of Microsoft Corporation.(#4 44Windows is a trademark of Microsoft Corporation.(#4 44RM/FORTRAN is a trademark of Language Processors, Inc.(#4 44DEC, VAX, VMS, and DCL are trademarks of Digital Equipment Corporation.(#4 44386 and 486 are trademarks of Intel Corporation.(#4 44U.S. Robotics is a registered trademark and Courier HST is a trademark of U.S. Robotics, Inc.(#4 80*((   X4` hp x (#%'0*,.8135@8:=h_a} RIGHT .} ddsenr[6r1rr(rXr rh rrhddHa&r r)dd*r]ddddrfor rhq r< rhdd; HaK1Kfor Khh K Khdd2 !a$(#(#(#(#8!'#$ P!0*((1'#A'#j a`!'#H%P    Figure 1 ! Figure 1 aA2x0*xddWELL2.WPG<<'OJFigure 3.2 Variable pore spacing in soil under saturated flow conditions.$(#(#(#(#A'#$where #xcPdddddjdd9LO` ? !(33c)+HFUNC {theta _e~=~{ theta _w~~ theta _{wr}} OVER { phi ~~ theta _{wr}}} ddsedd7widdwrL-LKLdd"wr+߸$(#(#(#(#`!'#$-#xdddoodd9LO` u !(33d)FUNC {gamma ~=~1~~1/ beta } ..i.1..1'./.-$(#(#(#(#!'#$ and  N8 w ` ` = moisture content (dimensionless) - ` ` = porosity (dimensionless)  N wr ` ` = residual moisture content (dimensionless) , ` ` = soilspecific exponents (dimensionless)  ` ` = soilspecific coefficient (1/L)  Np# ha ` ` = air entry pressure head (L)  NH$ e ` ` = effective moisture content (dimensionless) Note that the soilmoisture content is defined as the porosity times the degree of saturation. Typical soilmoisture curves generated from Equations 33a and 33b are presented in Figures 3.3a and 3.3b. P'0*((1'#VAP'#'#!PԌ#d6X@K0@#љ Figure 1 A Figure 1 aya2x0*xddFIG3-3A.WPG <<y$(#(#(#(#.a'#$# i2PkCY&P# yx0hh0*xddWELL3-3B.WPG <<yFigure  3.3.` ` Logarithmic plot of constitutive relations for sand, clay, silty loam, and sandy ` ` loam: (a) moisture content vs. pressure head and (b) relative permeability vs. ` ` moisture content  (based on data presented in Carsel and Parrish, 1988).0' 0*(('#Wa 0ԌThe water capacity term or storage term used in 3DFEMWATER can be written in the form: _#xddddddd9!L` !(34) FUNC {F(h)~=~d theta sub w /dh}  LFsL(Lh;L)LLdLdd]"wL/Ldh_$(#(#(#(#!'#$It should be noted that due to the relatively small influence of compressibility on water capacity in the unsaturated zone (with respect to the drainage potential), soil and water compressibility have been ignored in the storage term. When analytical functions are used to describe the nonlinearity of the relative conductivity, the derivative with respect to pressure head of the water content versus pressure head function must also be analytically defined. The equation governing saturated flow represents a limiting case of Richard's equation where the relative permeability is a constant of 1.0 and the water capacity is a constant equal to the specific yield for an unconfined aquifer or specific storage for a confined aquifer. 3.1.2 Boundary Conditions and Transient Source/Sink Terms  Unique solutions to variablysaturated flow problems are generated by solving Richard's equation in conjunction with 1) a set of boundary conditions defined at the physical edges of the modeled system and 2) where appropriate, source/sink terms applied within the system (Figure 3.4). Boundary conditions available in the 3DFEMWATER model include fixedhead (Dirichlet) boundaries, specifiedflux (Cauchy) boundaries, specifiedpressurehead gradient  N (Neumann) boundaries and variable (headdependent flow) boundaries.Fixedhead or Dirichlet boundaries are boundaries defined by prescribing pressure heads at specified boundary nodes so that: #xdddddUdd9!L`VJ (35)j(FUNC {h~=~h_d (x_b ,y_b ,z_b ,t)~on~B_d} LhLiLhdd"d L(oLxdd"bL,uLydd"bL,{Lzdd"bL,LtL)LonLBdd "dj$(#(#(#(#!'#$where  Nx hd` `  %= specified pressure head (L)  NP Bd` `  %= portion of the system boundary subject to a Dirichlet ` `   % boundary condition  N xb,yb,zb %= spatial coordinates on the boundary (L) Dirichlet boundaries are typically used to define the perimeters of bodies of water, the water table location, and leaking surface impoundments or other waste disposal facilities containing specified levels of water. Specified pressure heads may be constant or allowed to vary with time reflecting physical processes such as water level fluctuations associated with seasonal changes in rainfall and evapotranspiration rates.  N! The specifiedflux (Cauchy) boundary represents the portions of the system boundary where infiltration or evapotranspiration rates can be quantified. The specifiedflux boundary condition can be written: @H$!0*((!'#|!'#!@ y0*xddWELL3-4A.WPG"y$(#(#"$y0*xddWELL3-4B.WPG"y" " " " " " " " " " " " " " " " " " " " " (#(# Figure  3.4.` ` Conceptual model and mathematical approximation for variablysaturated flow ` ` system. Within the modeled system, transient source/sink terms may be ` ` applied as point sources/sinks or as distributed sources/sinks.  0'"0*(("W"0 !#xddddddd9#L`wk(36)wOFUNC {n CDOT k_r K_s CDOT ( GRAD h~+~ GRAD z)~=~q_c(x_b ,y_b ,z_b ,t)~on~B_c } LsLnL;Lkdd"rLKddC"sLL(GL+LhXLL+iLzL)zL'Lqdd"cL(-Lxdd"bL,3 Lydd "b L,9 Lzdd "b L,? Lt L)P Lona LBdd "cw$(#(#(#(#!!'#$where n ` ` = outward unit vector normal to the boundary +h ` ` = pressure head gradient +z ` ` = gravity gradient  N qc ` ` = specified flux rate (L/T)  N Bc` ` = portion of the system boundary subject to a specifiedflux ` `  boundary condition  NH kr ` ` = relative permeability  N Ks ` ` = saturated hydraulic conductivity (L/T)  N The specifiedflux condition is analogous to a Neumann boundary condition for saturated flow problems differing only in the nonlinear nature of the effective hydraulic conductivity. The specified boundary is simulated by assigning water flux rates along specified element sides. Flux rate versus time profiles can be input to account for seasonal or other timevariant changes in rainfall and evapotranspiration rates. The default boundary condition for  N 3DFEMWATER is a zero specifiedflux boundary condition, qc=0.  N Also available in 3DFEMWATER is a specifiedpressurehead gradient (Neumann) boundary condition of the form:  N@ A#xddddd dd9#L`(37)BFUNC {n CDOT k_r K_s CDOT GRAD h~=~q_n (x_b ,y_b ,z_b ,t)~on~B_n} LsLnL;Lkdd"rLKddC"sLL+GLhLLqdd"nCL(Lxdd "bIL,Lydd"bOL,Lzdd"bUL,Lt L) Lon LBddA "nߤ$(#(#@(#(#!A'#$where qn (L/T) is the portion of the boundary flux attributable to the pressurehead gradient  N and Bn is the portion of the system boundary subject to a specifiedpressurehead gradient boundary condition. For unsaturated flow problems, the presence of this option provides the user an efficient way of evaluating systems with vertically extensive vadose zones. As long as the area of interest in a study is above the capillary fringe, the specifiedpressurehead gradient boundary condition allows the user to truncate the system above the water table without knowing fluxes or pressure heads a priori (Figure 3.5). By choosing the specifiedpressurehead gradient boundary condition option for element faces defining the bottom  N boundary of the system, and setting the flux qn equal to zero, the bottom boundary becomes a gravity drainage boundary. This is equivalent to the code allowing the user to specify a flux  N8 along a horizontal bottom boundary of qc=krKs. This assumption of zero vertical change in pressure head near the bottom boundary is a reasonable assumption for slowly varying flow conditions and represents the outflow boundary condition that is usually assumed for field drainage experiments. This boundary condition is not appropriate for use in modeling the saturated zone.  NH$ The variable composite boundary condition represents a combined Dirichlet/ specifiedflux boundary. It allows for timevariant infiltration/ evapotranspiration rates with limits set on the maximum and minimum pressure heads which the boundary nodes may attain. The variable boundary conditions during periods of precipitation are: @'#0*((!'#|!#'#A#@ yx!hx0*ddWELL3-5.WPG$x!y$(#(#(#(#"'#$ Figure 3.5.` ` Use of a pressurehead gradient boundary condition to simulate a portion of the ` ` unsaturated zone. 0'$0*(('#7'$0 a#xdddddUdd9%L`J (38a)i'FUNC {h~=~h_p(x_b ,y_b ,z_b ,t)~on~B_v} LhLiLhdd"p L(oLxdd"bL,uLydd"bL,{Lzdd"bL,LtL)LonLBdd "vi$(#(#(#(#!a'#$or #xddddddd9%L`k(38b)xPFUNC {n CDOT k_r K_s CDOT ( GRAD h~+~ GRAD z)~=~q_p (x_b ,y_b ,z_b ,t)~on~B_v } LsLnL;Lkdd"rLKddC"sLL(GL+LhXLL+iLzL)zL'Lqdd"pL(-Lxdd"bL,3 Lydd "b L,9 Lzdd "b L,? Lt L)P Lona LBdd "vx$(#(#8(#(#!'#$and during periods of nonprecipitation are: #x dddddUdd9%L`J (38c)l*FUNC {h~=~h_p (x_b ,y_b ,z_b ,t)~on~B_v }  LhLiLhdd"p L(oLxdd"bL,uLydd"bL,{Lzdd"bL,LtL)LonLBdd "vl$(#(#p(#(#!'#$or #xdddddUdd9%L`J (38d)k)FUNC {h~=~h_m (x_b ,y_b ,z_b ,t)~on~B_v } LhLiLhdd"m L(oLxdd"bL,uLydd"bL,{Lzdd"bL,LtL)LonLBdd "vk$(#(# (#(#!'#$or #xddddddd9%L`k(38e)xPFUNC {n CDOT k_r K_s CDOT ( GRAD h~+~ GRAD z)~=~q_e (x_b ,y_b ,z_b ,t)~on~B_v } LsLnL;Lkdd"rLKddC"sLL(GL+LhXLL+iLzL)zL'Lqdd"eL(-Lxdd"bL,3 Lydd "b L,9 Lzdd "b L,? Lt L)P Lona LBdd "vx$(#(#0(#(#!'#$where  N hp ` ` = maximum pressure head (L)  Nh qp ` ` = maximum infiltration rate (L/T)  N@ hm ` ` = minimum pressure head (L)  N qe ` ` = maximum evapotranspiration rate (L/T)  N Bv` ` = portion of the system boundary subject to a variable ` `  boundary condition Physically, the maximum pressure head limit on the boundary prevents the generation of inappropriate surface water mounding. The minimum pressure headrestraint keeps the evaporation process from drying the soil near the boundary to moisture levels lower than residual saturation levels. The variable boundary condition can be used to approximate seepage faces within the studied area.  N Internal source/sink terms, as represented by the term q (L3/T/L3) in Equation 31 are also accounted for in 3DFEMWATER. As with the boundary conditions, the source/sink terms can be constant or allowed to vary with time. Two source/sink options are available in the code. The first is a distributed source/sink option and the second is a point source/sink option. The distributed source option is a source intensity that is integrated over the volume of an  N" element. The user prescribes a source intensity, q2 (L3/T/L3), or flux rate per unit volume for each distributed source element. This option allows a user modeling a large area to approximate the influence of several wells within an element. The point source/sink option is generally used to represent production or injection wells.  N& Wells are represented as volumetric water fluxes, q1 (L3/T), applied at a nodal point or to better represent a screening interval, a column of nodal points (Figure 3.6). If vertically p'%0*((Q'#|a%'# % '#%'#t%'#%p y%rx0*ddWELL3-6.WPG&%y$(#(#'#$ '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# '# &(#(#Figure  3.6. Using a series of nodes to represent a screened well interval. 0&&0*(('#o+&0 *adjacent nodes are used to represent the screened interval of a well, the volumetric flux must be distributed among the nodes. The most appropriate distribution of the total flux is in  N proportion to the effective conductances, Ce, of the individual nodes where the effective conductance of each node is defined as: #xddddd dd9'L` (39)BFUNC {C_e ~=~0.5 LEFT [(K_s )_{n1}L_{n1}~+~(K_s )_n L_n RIGHT ]} LCddu"eLL0 L.oL5sg zL(WLKdd"sL)dd]"ndd"dd"1LLddu"ndd"dd"1rLL(LKdd"s%L)dd"nLLdd+ "nߔ$(#(#8(#(#!'#$where n1 and n are indices referring to the element below the node and the element above the node respectively and 0.5L is half the thickness of an element.  NH Timevariant boundary conditions and source/sink flux or flux intensity rates are defined by a series of paired time and value points. This paired data is used to assemble a lookup table from which appropriate values are obtained using linear interpolation at specified times of analysis. Constant values can be specified by assigning the same value to a set of two time/data point pairs, making sure that the simulation time is fully spanned. 3.1.3 Initial Conditions The solution of Richard's equation also requires the initialization of pressure head values such that:  N !#x@ddddd%dd9'L` b (310)C FUNC {h~=~h_i (x,y,z,t=0)~\in~R} LhLiLhdd"i L(oLxL,7LyL,LzcL,Lt+LL0L)LinLRC$(#(#(#(#!!'#$where hi is the initial pressure head distribution (L), and R is the region of interest (Figure 3.7). Besides providing a frame of reference for transient analyses, the initial conditions are used to set the nonlinear parameters at the beginning of a simulation. For transient problems, an appropriate set of initial pressure head values may either be input directly or derived from a steadystate simulation. For more information on these options see Section 4.1.11. y! `0*xddWELL3-7.WPG'y Figure  3.7. Pressure head versus time at a nodal point on the finite element grid. @''0*((!'# '@'#4!'@ 3.1.4 SteadyState When analyzing the influence of transient stresses, such as well production schemes and drought conditions, on the flow system, a starting point must be assumed. The user defines boundary conditions and flow parameters as best he/she can, then does an initial simulation to allow the system to reach an equilibrium or steadystate (Figure 3.8). The steadystate simulation then defines the pressure head at all points in the system and it is from this initial condition that a transient simulation is started. Although the actual system is never really in steadystate, by using averaged conditions (i.e., rainfall, etc.) a reasonable starting point is generated. If the steadystate simulation fails to converge or the results poorly match field data, flow parameters and/or boundary conditions should be adjusted to improve the starting conditions. The steadystate or equilibrium condition is generated by removing the temporal term from Equation 31. The system is then defined as the equilibrium reached under the average conditions. Besides being used for initial conditions for a transient simulation, the steadystate flow option can also be used in conjunction with a transient transport simulation. Since the flow system will generally reach equilibrium under nonchanging stresses faster than an associated solute transport problem, using a steadystate flow field and average conditions to define the advective portion of solute transport will often give a good approximation of the change of solute distribution over time. The savings in computational effort can be considerable and, given the uncertainty of parameters in the system, an acceptable approximation may be reached. yAx0*xddWELL3-8.WPG( y Figure  3.8.` ` Pressure head versus time at nodal point where steadystate solution is being ` ` approached. '(0*((  3.2 NUMERICAL APPROXIMATION IN 3DFEMWATER The 3DFEMWATER model was developed to solve the variablysaturated flow equation described in Section 3.1. In the model, Richard's equation (Equation 31) is approximated using the Galerkin finite element technique. The time integral term in Equation 31 is approximated using backwards or central (CrankNicholson) difference in time. The nonlinearity of the system is treated using Picard iteration and the generated set of linearized equations is solved using a block iterative method. 3.2.1 Galerkin Formulation In 3DFEMWATER, Richard's equation is approximated using the Galerkin finite element method (Pinder and Gray, 1977) where the dependent variable, pressure head, is approximated by a trial function of the form:  N UA#x0dddddV dd9)L`q(311)5FUNC {h~=~N_j (x_i ,t)h_j (t)~~~~j~=~1,2, DOTSLOW ,n} LhLiLNdd"j L(oLxdd"iL,uLtL)=Lhdd"jL(CLtL)0LjLL1L,R L2 L, L~ L, LnU$(#(# (#(#!A'#$where Nj(xi,t) are the threedimensional shape functions and hj(t) are nodal values of pressure head at time t for the n nodes of which the finite element grid is comprised (Figure 3.9). #d6X@K0@#yaxh0*xddFIG3-9.WPG) y# i2PkCY&P# Figure  3.9.` ` Finite element grid for production from a single well in a variablysaturated ` ` porous medium. 0')0*((0'#$A)0 Substituting the trial functions into Equation 31 and applying the Galerkin criterion, a set of weighted residual minimization equations are generated of the form:  N  a#x`ddddddd9*L`(312)FUNC { INT from R sub s W_i LEFT [F(h)~{ PARTIAL h hat } OVER { PARTIAL t} ~~ GRAD CDOT [k_r K_s CDOT ( GRAD h hat ~+~ GRAD z)]q RIGHT ]dR~=~0} dd4R<<LspWddincvLxn]wn^ c}L^ n^ ]~6F(hb)(=0,I0h=|,|tc+t[<kddrKddDs(H+hY  +j z )2 ]  q dR L0 $(#(#(#(#`!a'#$where Wi are the weighting functions and Rs is the volume being simulated. For the Galerkin method, the weighting functions are the same as the shape functions.  N Substituting Wi = Ni and Equation 31 into Equation 312 results in: #x` ddddddd92-13*L`8(313)FUNC {STACK {INT from R sub s ~N_i LEFT [F(h)~{ PARTIAL (N_j h_j )} OVER { PARTIAL t}~- ~ GRAD CDOT LEFT [k_r K_s CDOT LEFT (h_j GRAD N_j~+~ GRAD z RIGHT ) RIGHT ]~-~q RIGHT]~dR~=~0, ~~~i~=~1,2,...n~~# ALIGNR j~=~1,2,...n~~}}  ddR<<LswNddMin[vL[sxn[wn}Lsn~wFw(Gwhw)q,(NNddjhddVj)Z,tTww+ew:tK :{wkddPMrwKddMs. w :e :l whdd MjW w+ wNdd! Mj wS w+ wz wnwqKwdR\w w0mw,wiZww1kw,w23w,w.w._w.wn-jZ--1k-,-23-,-.-._-.-n$(#(#H (#(#8!'#$where n is the number of nodes. Integration by parts can be used to rid Equation 313 of all second order derivatives, leaving a set of equations of the form:  N '#x$ddddd(\dd9*x``(314)efunc { int from R sub s F (h) N sub i } func { { partial (N sub j h sub j )} over { partial t} dR } `+` func { int from R sub s k sub r K sub s grad N sub i cdot ( h sub j grad N sub j + grad z ) dR } # func { int from B sub s N sub i bold n cdot }func { k sub r K sub s cdot ( h sub j grad N sub j + grad z ) dB int from R sub s N sub i q dR ~=~ 0 }  ddR<<LspuFu(8uhu)uNddfKi,4(Nddj:hddj) , tUudR/uddR<<sukddlKruKddKsJu+uNdd KiP u u( uhdd~ Kj u+ uNdd Kj u$ u+ uz u)P udRddddc4B<<sNdd*ifn.kddrKdd6sr(:hddj+@NddjF+z)rdB:   dd 4R<< s Ndde i q dR  0'$(#(#(#(#!'#$where Bs is the region boundary. The integrals given in Equation 314, which are taken over the entire region being modeled, can be replaced by the summation of integrals taken over the individual elements of which the finite element grid (Figure 3.9) consists. This finite element approximation generates a set of n nodal equations of the form: `#xddddddd9*L`E(315a)FUNC {STACK {A_{ij}~{dh_j} OVER { PARTIAL t}~+~B_{ij} h_j~=~C_i~,~ ALIGNR ~~~~i~=~1,2,...n~~~~~~# ALIGNR ~~~~j~=~1,2,...n~~~~~~}} )AddijYKndhdd8ij,t))Bddij[)hddjF))CddYi),)i] ) )1n ), )26 ), ). ).b ). )n-jQ - -1b -, -2* -, -. -.V -. -n`$(#(#P(#(#`!'#$where a#x8"dddddLdd9*L` (315b)LFUNC{ A_{ij}~=~ SUM FROM {k=1} TO {m}~ INT from R sub e F(h)N_i^e N_j^e dR} Addsij4IddmddVkddVddZV1dd 4R<<JenF(6h)Ndd}.edddiNdd8.eddjtdRa$(#(#(#(#`!'#$ #xp&dddddIdd9*x`P (315c)I~func { B sub ij ~=~ sum from {k=1} to m int from R sub e grad N sub i sup e cdot k sub r K sub s cdot grad N sub j sup e dR }  Bdduij6Idd mddVkdd Vdd\V1dd4R<<e'+Ndd .eddiFkddrLKddsR+Ndd5.eddjqdRI$(#(#!(#(#`!'#$ %*0*((a`'# a* '#*'#?*'#;!*8"'#K%*p&'#)* and  N d!#xddddddd9+x`{(315d)Xfunc { C sub i ~=~ sum from {k=1} to m } left [ func { int from R sub e k sub r K sub s cdot } func { grad N sub i sup e cdot grad z dR } `+` func { int from R sub e N sub i sup e q dR } `+` func { int from B sub s N sub i sup e bold n } cdot func { k sub r K sub s cdot }func { ( h sub j grad N sub j sup e `+` grad z ) } right ]func { dB }  CdduiIddmddVkddVdd V1nvLxnwn}Ln~dd4R<<OeskddrKdd{s+Ndd.eddi:+zfdR@dd4R<<e Ndd .edd} i q6 dR   dd 4B<< s Nddf .eddM i n j kdd r Kddrs(vhddj+|Ndd.eddjI+#z)dBd$(#(#(#(#`!!'#$where m is the number of elements into which the system is discretized and Ne denotes elemental shape functions. 3.2.2 Solution Techniques To solve the series of linearized ordinary differential equations presented in Equation 315a, the time differential is replaced by a finite difference formulation, resulting in working equations for 3DFEMWATER of the form: )A#xcdddddwdd9+L`a(316)FUNC {{A_{ij}} OVER { DELTA t_k}~ LEFT (h_j^{k+1}~~h_j^k RIGHT )~+~wB_{ij}^{k+w} h_j^{k+1}~+~(1w)B_{ij}^{k+w}h_j^k ~=~C_i^{k+w}} (RAddij=LLtdd"kwewlhdd9kddudd1dd*j6hddbkddIjwBddkddddwddijQhddkdddd>1ddjp ( 18  w )d Bdd kdd ddQ wdd ij hdd kdd j >Cddkdddd+wddi)$(#(# (#(#`!A'#$where k+1 represents the current time level, k the previous time level, t the length of the current time step and w the time weighting function (1.0=backwards in time; 0.5=Crank Nicholson or centered in time). Note that the associated transport code, 3DLEWASTE, utilizes a backwardsintime scheme. Therefore, when using 3DFEMWATER to generate a flow field for a 3DLEWASTE simulation, the backwardsintime option should be used. This prevents the possibility of a mismatch in the interpolation of timevariant boundary condition and source/sink flux values. For each time step, the solution method involves an outer and inner iterative scheme (Figure 3.10) where the outer iterations control convergence of the nonlinear terms in the equations and the inner iterative scheme controls the blockiterative method of solving the linearized set of equations. For each nonlinear iteration, the linearized set of equations is solved using relative permeability and storage terms updated using pressure head values generated during the previous nonlinear (outer) iteration. Relative permeability and storage terms for the first iteration in a time step are based on pressure head values from the previous time step, or for the first time step, from the initial conditions. Because of the strong nonlinear nature of the soil moisture curves, the outer iterative scheme may become unstable. To help circumvent this problem it is often helpful to damp the iterative changes in the pressure head. One method of damping the iterative changes is through the use of an underrelaxation factor. Implementation of the underrelaxation factor for the outer iterations in 3DFEMWATER is as follows:  a#xH'ddddddd9+L`s (317).FUNC {h_i^{r+1}~=~(1~~u )h_i^r~+~u h_i^{r+1}} Lhddrdddd1ddu4iL.L(L1?LLuPL)Lhdd3rdd4iLeLuLhdd>rddzdd1dd/4i $(#(#"(#(#!a'#$where u is the outer underrelaxation factor and r is the iteration number. If damping is needed, values below one should be used. Acceleration or overrelaxation (1.0sddzdd1dd/4i$(#(#(#(#!'#$where s denotes the inner iteration number and o is the overrelaxation factor. The optimal value of the overrelaxation factor usually falls between 1.5 and 1.9. A good starting point is o = 1.72. 3.3 3DLEWASTE 3DLEWASTE is designed to simulate the movement of dissolved species through a variablysaturated porous medium. Typical applications for 3DLEWASTE include the examination of: 1) leachate migration from landfills and surface impoundments, 2) the influence on water quality of pesticide and fertilizer applications, and 3) the environmental impact of leaky containment structures such as underground and above ground storage tanks (Figure 3.12). Velocity fields needed to define the advective pathways of water bearing the chemicals are provided by associated 3DFEMWATER simulations. 0 %-0*((P'#h-0 #d6X@K0@# 3'3'StandardDILN0SCR.PRSdp'3'3StandardDILN0SCR.PRSdp. 30*0*xddWELL3-11.WPG.) mFigure 3.11 Use of vertical or horizontal nodal slices in the block interative method.$.....$0.x.!.0Ԓ# i2PkCY&P# 30*ph0*xddFIG3-12.WPG/0* hJ̒Figure 3.12 Migration of dissolved contaminants through the unsaturated zone into an unconfined aquifer @system. $.....$0/x.!/0Ԓ '3'3StandardDILN0SCR.PRSdp3'3'StandardDILN0SCR.PRSdp.0 3.3.1 Governing Equations The governing equation for advectivedispersive solute transport through variablysaturated porous media, based on the laws of conservation of mass and flux, can be written in the form: #x'dddddu_dd90x{`b (319)!func { theta { partial C} over {partial t} } `+` func { rho sub b { partial S} over { partial t} } ~=~func { grad cdot ( theta bold D cdot grad C ) } `` func { bold V }func { cdot grad C } `` func { lambda ( theta C } `+` func { rho sub b S ) } `+` func { Q C sub \in } `` func { Q C }  ,C.,.t#ddjlb,8S.,8.t+ o(7D+cC)=V { + CU  / (  Cm  #ddG lb S )]Q7Cddlin'QCߨ$(#(#8(#(#`!'#$where  Np ` ` = moisture content (L3/L3)  NH #b` ` = bulk density of the porous medium (M/L3)  N C` ` = concentration of the dissolved species (M/L3) S` ` = species concentration in the adsorbed phase (M/M) t` ` = time (T)  N  V ` ` = Darcy velocity (i.e., specific discharge) (L/T) +` ` = del operator indicating divergence  NZ +` ` = del operator indicating gradient   N2  D ` ` = dispersion coefficient tensor (L2/T) ` ` = material decay constant (T) Q` ` = water source/sink rate (M/T)  N Cin` ` = dissolved species concentration of the source/sink  N ` `  fluid (M/L3)  ND Note that for a fluid sink, such as a production well, Cin = C.  N The dispersion coefficient tensor, D , which defines the spreading of the dissolved species as it is advected through the system, is defined as: 6#x.ddddd dd90X{`*(320)FUNC { theta BOLD D ~=~ \ FUNC { alpha _T} FUNC {| BOLD V | delta ~+~( FUNC {alpha _L ~~\ alpha _T} ) FUNC {BOLD {VV} /FUNC {| BOLD {V}|~+~ theta FUNC {a_m} tau delta }}}} LsLD LL 1Ldd"TL|5LVL|L LWL(Ldd"LLQL Ldd"TUL)LVV L/ L|I LV L|Z L Lk Ladd "m L)q L 6$(#(#~(#(#!'#$where  N | V | ` ` = magnitude of the Darcy velocity vector (L/T)   ` ` = Kronecker delta tensor  N L ` ` = longitudinal dispersivity (L)  Nh T ` ` = transverse dispersivity (L)  N@ am ` ` = molecular diffusion coefficient (L2/T) ) ` ` = tortuosity coefficient () The dispersivity parameters quantify the magnitude of longitudinal and lateral spreading of the dissolved species as it is advected through the system. This spreading, called hydrodynamic dispersion, is due to the combined influence of 1) water movement through the complex pathways, and 2) associated mixing patterns (Figure 3.13) that occur over spatial scales not accounted for by the flow field approximated in the advective term of Equation 319. These complex pathways may vary in scale from pore spaces and microfractures to larger scale features such as joints and fracture zones. Since dispersion is a function of features of various scales, the appropriate value of dispersivity must take into@'00*((!'#r 0.'#"0@ yx%0*xddFIG3-13.WPG1<%y$(#(#(#(#&'#$Figure  3.13.` ` Diagram showing the effect of scale on hydrodynamic dispersion processes. 0&10*(('#o+10 consideration the distance the species travels. The molecular diffusion coefficient in Equation 320 quantifies the spreading due to molecular diffusion. In order to solve Equation 319 for a single dependent variable, the constitutive relationship between the species concentrations in the dissolved and adsorbed phases must be defined. The 3DLEWASTE code allows the user to choose from three relationships: 1) a linear isotherm, 2) Freundlich isotherm, or 3) Langmuir isotherm. The isotherms, as determined in laboratory partitioning experiments, can be plotted in loglog form to derive: V#xH dddooCdd92X`S (321)!FUNC {\log~S~=~\n~log~C~+~\log~K} .log.S1..n.log.C.Z.log.KV$(#(#(#(#!'#$or #xdddggdd92X` ( $(322)bFUNC {S~=~KC^n } .S.i.KCddLcnb$(#(# (#(#!'#$where n is slope of the plot of log S versus log C and K is the Saxis intercept (Freeze and Cherry, 1979). Equation 322 defines the Freundlich isotherm, which is often used to describe the partitioning between the dissolved and absorbed phases. When the isotherm has a slope n=1, the isotherm is linear and the relationship can be defined as:  N !#x'@ddddd_dd92X{`  #(323)FUNC {dS OVER dC~=~K_d} (=dS=.dCcKddvld$(#(#(#(#`!!'#$where Kd is called the distribution coefficient (L3/M). Linear isotherms are often used to describe the adsorption of hydrophobic organic compounds to organic matter in soils. The distribution coefficient is described as a function of the organic carbon content of the soil as:  NP )A#xdddddodd92X{`e "(324)FUNC {K_d~=~f_{oc}K_{oc}} LKddu"dLLfdd "ocLKdd"oc)$(#(#P(#(#!A'#$where foc is the fractional organic carbon content and Koc is the normalized distribution  N coefficient. There are many published lists of values for Koc (e.g., Lyman et al., 1982; U.S. EPA, 1986; Verschueren, 1983). Data are available primarily for pesticides and, to a lesser  N` degree, aromatic and polycyclic aromatic compounds. If data on Koc are not available for a  N8 particular chemical, a value can be estimated from empirical relationships between Koc and some other property of the chemical such as the water solubility, S, the octanolwater partition  N coefficient, Kow, or the bioconcentration factor for aquatic life, BCF. Lyman et al. (1982) tabulate 12 such regression equations obtained from data sets of different classes of chemicals. One commonlyused relationship (Karickhoff et al., 1979) takes the form: ;a#x(ddddd]dd92X{` F !(325)FUNC {K_{oc}~=~0.41K_{ow}} LKdds"oc4LL0EL.L41qLKdd"ow;$(#(#H$(#(#!a'#$p&20*((QH '#42'#2@'#!2'#A2('#*a2p The Langmuir isotherm takes the form:  N P#xE`ddddd}dd93X{`- "(326)$FUNC {S~=~{S_{max}KC} OVER {1~+~KC}} S3SddmaxKC.1D..KCP$(#(#(#(#`!'#$where Smax is the maximum concentration allowed in the medium. The effective decay constant, , is a degradation constant that can be used to quantify the effects of radioactive decay, or the composite effects of hydrolysis and biodegradation. When used to quantify the effects of hydrolysis and biodegradation, the effective decay coefficient (for a linear isotherm) takes the form:  N i#xcddddd.dd93X` (327)gFUNC {lambda ~=~{ lambda _1 theta ~+~ lambda _2 K_d rho _b } OVER { theta ~+~K_d rho _b }~+~ lambda _b} dd17dd21Kddd#dd7b7LLLKdd"d3L#dd"b~ddbi$(#(# (#(#`!'#$where 1 is the firstorder hydrolysis rate constant for the dissolved species, 2 is the first NX order hydrolysis rate constant for the sorbed species, and b is the firstorder biodegradation rate constant. The dissolved species firstorder hydrolysis rate can be written in terms of the  N acidcatalyzed (Ka), basecatalyzed (Kb), and neutral (Kn) hydrolysis rate constants as:  N #xhdddddo dd93X{`e (328)x1FUNC {lambda _1~=~K_a [H^+~]~+~K_n~+~K_b [OH^~]} Ldds"1LLKdd "aGL[LHdd L]RLLKdde"nLLKdd"b9L[LOHddvL]x$(#(#(#(#!'#$where [H+] is the hydrogen ion concentration and [OHé] is the hydroxyl ion concentration. The sorbed phase firstorder hydrolysis rate is considered to be a function of the acid and neutral hydrolysis rates and is usually written in the form: #xxddddddd93X{` j (329)+FUNC {lambda _2~=~ alpha K_a [H^+~]~+~K_n } Ldds"2LL LKddo"aL[LHdd L]LcLKdd"nߝ$(#(#(#(#!'#$where  is the acidcatalyst hydrolysis rate enhancement factor for the sorbed phase with a typical value of 10.0. Note that for a nonlinear isotherm the formulation in 3DLEWASTE is  N valid only if 1 = 2. The governing equation for advectivedispersive solute transport in a porous medium, as presented in Equation 319, describes the transport from an Eulerian or fixed framework. The numerical algorithm may begin to oscillate and fail to converge to a solution of this equation when the advective term starts to dominate over the dispersive term and the equation takes on a hyperbolic nature. Dominance of the advective term over the dispersive term is reflected in the nondimensional Peclet number, which is defined as the ratio of the product of the velocity magnitude and distance advected to the dispersion coefficient. In finite element analysis the critical Peclet number is the local Peclet number of an element, where the local Peclet number is defined as: #x(ddddddd93X{`  #(330)FUNC {P ~=~L/ alpha sub L } LPLiLLL/1Ldd"L$(#(#H$(#(#!'#$where L is the element length. p'30*((Q`'# 3'#3h'#k3x'#{3('#*3pԌOne method of circumventing the numerical problems (i.e., oscillation and failure to converge) associated with Peclet numbers greater than 2 is to address the system through a moving (i.e., Lagrangian) coordinate system. In the Lagrangian formulation for solute transport in a porous medium, the temporal term is defined as a material derivative of the form: I!#x=ddddd udd94X{`(331)FUNC {LEFT ( theta ~+~ rho _b~{dS} OVER {dC} RIGHT ) DC OVER Dt~=~ theta ~{ PARTIAL C} OVER { PARTIAL t}~+~ rho _b~{ PARTIAL S} OVER { PARTIAL t}~+~ BOLD {V} CDOT }FUNC { GRAD {C}} gn5#ddlbdS.dCDC.Dt },C.,.te#ddlbg | , S| ., .t O V  +{ CI$(#(#8(#(#`!!'#$where D denotes the material derivative.  NH The advective term, V +C, written in index notation becomes: A#xcddddd7dd94xB` (332)U\func { bold V cdot }func { grad C ~=~ dx sub i over dt {partial C} over {partial x sub i} }  Vs+;CdxddiLdt+,C L,pLxdd"iU$(#(# (#(#`!A'#$where the repeated indices indicate summation. Substituting Equation 331 into Equation 319, the governing equation for a Lagrangian framework becomes:  N a#x=dddddudd9EQ33A4x{`(333a)gleft ( theta `+` func { rho sub b dS over dC } right ) func { DC over Dt ~=~ grad cdot } ( func { theta bold D cdot } func { grad C } ) `-` func { lambda ( theta C } `+` func { rho sub b S } ) `+` func { Q C sub \in } `-` func { QC }  gn5!#ddlbdS.dC-DC-.DtS+d(,DX+C ) p ( 8 C $ #dd lb S( )  Qx Cdd linhQCg$(#(#(#(#`!a'#$for a linear isotherm. The average linear velocity, V *, for a linear isotherm becomes: #x=ddddd$udd94X{`b (333b)TbFUNC { BOLD {V} ^* } ~=~ FUNC { BOLD {V} } /LEFT ( FUNC { theta ~+~ rho _b~ dS OVER dC } RIGHT )  VddsV /mgn@#ddQlbdS.dCT$(#(#(#(#`!'#$For a nonlinear isotherm, the Lagrangian equation becomes: #x'dddddh_dd9EQ344x{`@ (334a)func { theta DC over Dt } `+` func { rho sub b dS over dC {partial C} over {partial t} } ~=~ func { grad cdot (theta bold D } func { cdot grad C ) } `-` func { lambda ( theta C `+` rho sub b S ) `+` Q C sub \in `-` QC }  DC.Dt#ddjlbdS.dC,CC.,C.t+z(BD +nC)H  " (  C`  #dd: lbv S )P  Q*CddlinQC$(#(#(#(#`!'#$where /#x#dddggdd94X{`H  "(334b).FUNC { BOLD {V}} ^*~=~ FUNC {BOLD {V}}/ theta  .Vddsc..V ./m./$(#(#<(#(#!'#$Full implementation of the Lagrangian approach implies the solution of Equation 333 using a moving coordinate system. Another method of circumventing the instability problem is to utilize a hybrid EulerianLagrangian approach. Such an approach is implemented in 3DLEWASTE. In the hybrid approach, the advective term of the material derivative is evaluated in a Lagrangian manner by a backwards particle tracking scheme (Figure 3.14). The particle tracking scheme generates a particle starting location and an associated  N& concentration, C*. This concentration, C*, is the starting concentration of each particle which reaches'40*((a'# !4'#pA4'#2a4'#D4'#>"4#'#%4 y2x0*xddWELL3-14.WPG5<<y$(#(#(#(#'#$Figure  3.14.Backward particle tracking to determine the starting point of an advected particle. א? a nodal point at the end of that particular time step. The material term of Equation 333 is then approximated by: s#x6dddddnndd95X{` = !(335).FUNC { DC OVER Dt~=~ {C~~C^*} OVER {DELTA t}} (=DC=.Dtc)>CCdd .C.ts$(#(#((#(#`!'#$The diffusiontype equation is then solved using a fixed coordinate system. Note that for a steadystate simulation, where t, the logic is implemented by multiplying the transient storage terms by zero and evaluating the advection term in a fixed coordinate system. 3.3.2 Boundary Conditions and Transient Source/Sink Terms  Unique solutions to advectivedispersive solute transport problems are generated by solving the governing equation (Equation 319) in conjunction with 1) a set of boundary conditions, defined at the physical edges of the modeled system, and where appropriate, 2) source/sink terms applied within the system (see Figure 3.4). Boundary conditions and source/sink terms available in the 3DLEWASTE model include: @'50*((!'#W5'#q!5@Ԍ Prescribedconcentration (Dirichlet) boundaries  Specifiedflux (Cauchy) boundaries   Specifieddispersiveflux (Neumann) boundaries  Variable boundaries  Point sources  Distributed sources  N Prescribedconcentration or Dirichlet boundaries are defined by prescribing dissolved species concentrations at specified boundary nodes as:  N  #xXddddd dd96X` (336)}9FUNC {C~=~C_d~ LEFT (x_b ,y_b ,z_b ,t RIGHT )~~~~\on~B_d} LCLiLCdd"dTelwLxdd"bL,}Lydd"bL,Lzdd"b%L,LtBLonSLBdd"d} $(#(# (#(#!'#$where Cd is the specified solute concentration, Bd is the portion of the system boundary  N subject to a Dirichlet boundary condition, and (xb,yb,zb) is the spatial coordinate on the boundary. Dirichlet boundaries are typically used to test computer programs by allowing comparisons with analytical solutions. Unlike the analogous constanthead boundaries of flow models, constantconcentration boundaries are generally poor approximations of contaminant source terms. Bodies of fresh water located upgradient from contaminant sources can be approximated using constant concentration nodes. When used to define sources, specified concentrations may be constant or allowed to vary with time, reflecting physical processes such as degradation of the source due to radioactive decay, hydrolysis, biodegradation, or physical removal. Concentration versus time profiles can be defined to account for seasonal  N or other timevariant changes in dissolved species levels.  NP The specifiedflux (Cauchy) boundary represents the portions of the system boundary where infiltration can be quantified. The specifiedflux boundary has many representations including: 1) infiltration due to leachate migration from a landfill or surface impoundment, 2) application of pesticides or fertilizer to fields, and 3) the dilution effects of rainfall or irrigation on previously applied constituents. The specifiedflux boundary condition can be written:  N`  !#x#ddddd dd96X`+(337)FUNC {n CDOT ( FUNC {BOLD {V}} FUNC{ C~~ theta FUNC {BOLD {D}} FUNC{ CDOT GRAD C)~=~q_c~ LEFT (x_b ,y_b ,z_b ,t RIGHT )~~~~\on~B_c}}} LnsLL(;LVLCLLL]LDL%L+LCL)LGLqdd"c2e lULxdd"bL,[Lydd"bL,a Lzdd "b L,g Lt Lon1 LBdd "c $(#(#`(#(#!!'#$where n is an outward unit vector normal to the boundary,  D .+C is the dispersive flux caused  N! by the concentration gradient +C, V C is the advective flux, qc is the specified flux rate  N" (M/T/L2), and Bc is the portion of the system boundary subject to a specifiedflux boundary condition. The specified boundary is simulated by assigning mass flux rates along specified element sides. Flux rate versus time profiles can be defined to account for seasonal or other  N$% timevariant changes in flux rates. In addition to the usersupplied mass flux rates, 3DLEWASTE automatically accounts for the influence of the water entering the system along the boundary. The infiltration of fresh water@'60*((!X'#u6#'#-%!6@ is simulated by applying the specifiedflux boundary condition and setting the mass flux rate to zero. The automatically generated term accounting for water flow normal to the boundary will simulate the dilution due to infiltration.  N` Also available in 3DLEWASTE is a specifieddispersiveflux or Neumann boundary condition of the form:  N A#x ddddd dd97X{`F(338)BiFUNC {n CDOT ( theta FUNC {BOLD {D}} FUNC{ CDOT GRAD C)~=~q_n ~ LEFT (x_b ,y_b ,z_b ,t RIGHT )~\on~B_n}} LnsLL(;LLLDgLL+/LCL)@LLqddS"neq lLxdda"bL,Lyddg"bL,Lzddm"bL, Lt Lon LBdda "nB$(#(#(#(#!A'#$where qn (M/T/L2) is the portion of the boundary flux attributable to the concentration and Bn is the portion of the system boundary subject to a specifieddispersiveflux boundary condition.  N Note that exit boundaries can be declared using this boundary condition and letting qn=0. This physically simulates mass being advected out of the system.  N For solute transport, the variable composite boundary condition represents a combined specifiedflux/dispersiveflux boundary which allows for timevariant infiltration/waterloss rates. The boundary condition during infiltration is:  N a#xddddddd97X{` (339a)QFUNC {n CDOT ( FUNC {BOLD {V}} FUNC{ C~~ theta FUNC {BOLD {D}} FUNC{ CDOT GRAD C)~=~ n CDOT FUNC {BOLD {V}} FUNC{ C_v~ LEFT (x_b ,y_b ,z_b ,t RIGHT )~\on~B_v~ \if~ n CDOT FUNC {BOLD {V}} <= 0}}}} LnsLL(;LVLCLLL]LDL%L+LCL)LGLnLLVsLCdd"v^e lLxdd"b# L, Lydd "b) L, Lzdd "b/ L, Ltp Lon LBdd "vlLif}LnLELVL L0Q$(#(#(#(#!a'#$where n. V is the Darcy velocity or discharge normal to the boundary, Cv is the concentration of  Nj the dissolved species in the water entering at the boundary, and Bv is the portion of the boundary subject to a variable boundary condition. When water is exiting at the boundary, the boundary condition defaults to the dispersive flux condition: `#xzddddd dd97X{`(339b)iFUNC {n CDOT ( theta FUNC {BOLD {D}} FUNC{ CDOT GRAD C)~=~0~~~\on~B_v~\if~ n CDOT FUNC {BOLD {V}}~ >~0}} LnsLL(;LLLDgLL+/LCL)@LL0-Lon>LBdd"v)Lif: Ln L LV L>\ L0`$(#(#(#(#!'#$and mass is advected out of the system. Like the specifiedflux boundary condition, the variable boundary can represent: 1) infiltration due to leachate migration from a landfill or surface impoundment, 2) application of pesticides or fertilizer to fields, and 3) the dilution effects of rainfall or irrigation on previously applied constituents. When the boundary being modeled may be either an exit or an infiltration boundary, such as a precipitation/ evapotranspiration boundary or a seepage face, the variable boundary condition is the proper choice. The variable boundary condition can also be used in a manner similar to the dispersiveflux condition to simulate strictly exit nodes.  N Internal source/sink terms, as represented by the term QCin in Equation 319 are also accounted for in 3DLEWASTE. As with the boundary conditions, the source/sink terms can be constant or allowed to vary with time. Both the fluid flux rate, Q, and the injected fluid species  Nr# concentration, Cin, are allowed to vary with time. Two source/sink options are available in the code. The first is a point source/sink option and the second, a distributed source/sink option. The first option is generally used to represent production or injection wells. The fluid fluxes in  N% wells are represented as volumetric water fluxes, q1 (L3/T), applied at a nodal point or to better represent a screened interval, a column of nodal points (see Figure 3.6). If vertically adjacent nodes are used to represent the screened interval of a well, the volumetric flux must beP'70*((1 '# A7'#a7z'#n7P distributed among the nodes. The most appropriate method of doing this is discussed in Section 3.1.2. Note that the applied fluid fluxes must match those used in the associated flow simulation. The distributed source option is a source intensity that is integrated over the volume of an  N8 element. For a distributed source element, the user defines a fluid source intensity, q2  N (L3/T/L3), or fluid flux rate per unit volume for each distributed source element. This option allows a user modelling a large area to approximate the influence of a well field within an element.  Np Timevariant boundary conditions and source/sink flux or flux intensity rates are defined by a series of paired time and value points. This paired data is used to assemble a lookup table from which appropriate values are obtained using linear interpolation at specified times of analysis. Constant values can be specified by assigning the same value to a set of two time/data point pairs, making sure that the simulation time is fully spanned. 3.3.3 Initial Conditions The solution of the governing equation for solute transport in a porous medium also requires the initialization of concentration values such that:  N #xhddddddd98X{` (340)E"FUNC {C~=~C_i (x,y,z,t=~0)~~\in~R} LCLiLCdd"i L(oLxL,7LyL,LzcL,Lt+LL0<L)2LinCLRE$(#(#(#(#!'#$where Ci is the initial concentration distribution and R is the region of interest. The initial conditions are used to define the starting water quality and soil concentration levels for determining the fate of the dissolved constituents. Besides providing a frame of reference for transient analyses, the initial conditions are used to set the storage parameters for Freundlich and Langmuir isotherms at the beginning of nonlinear simulations. For transient problems, an appropriate set of initial concentration values may either be input directly or derived from a steadystate simulation. For more information on these options see Section 4.2.10. 3.3.4 SteadyState  When looking for a bounding solution to determine the maximum possible concentration levels that may be reached in a solute transport problem, a steadystate option may be employed. In the steadystate case, the time derivatives in Equation 319 are discarded and the equation, including the advective term, is solved in an Eulerian or fixedcoordinate framework. Note that any solute source prescribed as a boundary condition or source term becomes modeled as an infinite source. For many systems this upper bound may be highly conservative. The steadystate option is of no use if the source is solely defined by initial conditions. 3.4 NUMERICAL APPROXIMATION IN 3DLEWASTE  The 3DLEWASTE model was developed to simulate advectivedispersive solute transport in variablysaturated porous media. In the model, the hybrid EulerianLagrangian governing equation (Equation 333) is approximated using the Galerkin finite element technique. The time integral term in Equation 333 is approximated using backwards differencing in time. The0'80*((h'#\80 nonlinearity of the system is treated using Picard iteration and the generated set of linearized equations is solved using a block iterative method. 3.4.1 Galerkin Formulation In 3DLEWASTE, the diffusion equation is approximated using the Galerkin finite element method where the dependent variable, concentration, is approximated by a trial function of the form:  N T#xH ddddd dd99X{`(341)4FUNC {C~=~N_j (x_i ,t)C_j (t)~~~j~=~1,2, DOTSLOW ,n} LCLiLNdd"j L(oLxdd"iL,uLtL)=LCdd"jL(CLtL)LjLAL1L, L2m L, L5 L, LnT$(#(#(#(#!'#$where Nj(xi,t) are the threedimensional shape functions and Cj(t) are nodal values of concentration at time t for the n nodes of which the finite element grid is comprised (see Figure 3.9). Substituting the trial functions into Equation 333 and applying the Galerkin criterion, a set of weighted residual minimization equations of the form:  NX #xddddddd99x{` (342)3func { int from R sub s W sub i } left [ func { ( theta `+` rho sub b K sub d ) DC over Dt } `` func { grad cdot ( theta bold D } func { cdot grad C) `` lambda ( theta `+` rho sub b K sub d ) C } `+` func { Q C sub \in `` QC } right ] func { dR ~=~ 0 }  dd4R<<LspWddi]u]|5(#ddb%Kddd)DY0DCY|DtH+"(ND+zC)T  . (  ~ #dd b Kdd d )$ C QtCddindQCdR03$(#(#X(#(#`!'#$are generated for the linear isotherm case, where Wi are the weighting functions. For the Galerkin method, the weighting functions are the same as the shape functions and therefore, Equation 342 can be written in the form: #xddddddd99x{` (343)#func { int from R sub s N sub i } left [ ( func { theta `+` rho sub b K sub d ) DC over Dt `` }func { grad cdot ( theta D cdot grad C ) `` lambda ( theta `+` } func { rho sub b K sub d ) C `+` Q C sub \in QC } right ] func { dR ~=~ 0 }  dd4R<<LspNddi]u~]|5(#ddb%Kddd)DY0DCY|DtH+"(ND+zC)T  . (  ~ #dd b Kdd d )$ C QtCddinRQCdRk0#߰$(#(#(#(#`!'#$ Integration by parts can be applied to the dispersive term to eliminate all second order terms in Equation 343, leaving an equation of the form:  N { !#x( ddddd`dd99x{`&(344) Jfunc { int from R sub s N sub i ( theta `+` rho sub b K sub d ) N sub j DC sub j over Dt dR } `+` func { int from R sub s grad N sub i cdot theta bold D } cdot func { grad N sub j dR C sub j `` int from B sub s bold n } cdot func { theta bold D } cdot func { grad N sub j dB C sub j } # `+` func { int from R sub s lambda ( theta `+` rho sub b K sub d ) N sub i N sub j dR C sub j } `` func { int from R sub s { partial theta} over { partial t} N sub i N sub j dR C sub j } `` func { int from R sub s Q C sub \in N sub i dR `+` int from R sub s Q N sub i N sub j dR C sub j ~=~ 0 }  ZZddYR<<syNdd Oi\y(y6yy#ddObLyKddOdy)RyNddOj "DCddjADt=ydRyddR<<sy+R yNdd Oi yX y yD y y+ yNddN Oj ydRR yCdd Ojy}}dd|B<<synAyy yDmyy+5yNddOjydByCddOj!dd3R<<s\(6#ddbLKddd)RNddiNddZjdR^Cddjdd3R<<s/,{/{,{{tNddZ i Ndd j8 dR Cddf j + + dd* 3R<<g s Q CddU in Ndd3iodRIdd3R<<s QNddi&NddjdRCddj{(0 { $(#(#(#(#!!'#$where Bs is the entire region boundary. The integrals given in Equation 344, which are taken over the entire region being modeled, can be replaced by the summation of integrals taken over the volumes and surfaces of individual elements of the finite element grid. This finite element approximation generates a set of n nodal equations of the form: `H$90*((AH '#<9'#9'#9 '#%!9`ԌA#xdddddtdd9:X{`(345a)FUNC { STACK {A_{ij}~{dC_j} OVER {dt}~+~ LEFT (B_{ij}~+~ E _{ij} RIGHT )C_j~=~R_{ij} ~~~~~~~i~=~1,2,...n # ALIGNR j~=~1,2,...n}} )AddijYKndCdd8ijdt)ePl)Bddij)t)Eddij)Cddjk))Rdd|ij )i )P )1 ), )2| ), ).D).). )n -j -D -1 -, -2p -, -.8-.-.-nߔ$(#(#(#(#`!A'#$where a#xddddddd9:X{`c (345b)6jFUNC { A_{ij}~=~ SUM FROM {k=1} TO m ~ INT from R sub e LEFT ( theta ~+~ rho _b K_d RIGHT )N_i^e N_j^e dR} Addsij4IddmddVkddVddZV1dd 4R<<Jene-l>#ddObKddd]Ndd.eddiNdd.edd~jdR6$(#(#8(#(#`!a'#$ #x ddddd]dd9:X{`F (345c)sFUNC {B_{ij}~=~ SUM FROM {k=1} TO m~ INT from R sub e~ GRAD N_i^e CDOT theta BOLD {D} } FUNC { CDOT GRAD N_i^e dR} Bddsij4IddmddVkddVddZV1dd 4R<<Je+Ndd.eddi:Df+NddI.edd0idR߭$(#(#p(#(#`!'#$ Y#xXddddd dd9:x{`l( (345d)func { E sub ij ~=~ sum from { k=1} to m int from R sub e N sub i sup e } ~ left [ func { lambda ( theta `+` rho sub b K sub d ) `+` Q } right ] ~ func { N sub j sup e dR }  Edduij6Idd mddVkdd Vdd\V1dd4R<<e'Ndd.eddi+szK(#ddcbKdddA)-Q Ndd .eddl j dRY$(#(# (#(#`!'#$and  N #xddddd) dd9:X{`$(345e)QFUNC {R_{ij}~=~ SUM FROM {k=1} TO m~ } LEFT [~ FUNC { INT from R sub e ~QC_{\in}~ N_i^e dR~+~ INT from B sub e ~ n CDOT theta BOLD {D} } FUNC { C_j GRAD N_j^e dB } RIGHT ] Rddsij4IddmddVkddVddZV1n vL xn wn }L n ~{{ddz4R<<e$QCddinNdd,.eddihdRy''dd&4B<<cen4   D` Cdd j +f Ndd .edd j! dBQ$(#(#(#(#`!'#$where m is the number of elements into which the system is discretized and Ne are the elemental shape functions. Note that for a steadystate simulation, the full Eulerian approach is used. The Lagrangian term DC/Dt is replaced by ,C/,t and the Eulerian term:  N =#xxddddddd9:x{`  (346)ifunc { sum from { k=1} to M int from R sub e N sub i sup e bold V } func { cdot grad N sub j sup e dR }  IddLMddVkddLVddV1dd4R<</eSNdd.eddiVr+:Ndd.eddjdR=$(#(#(#(#`!'#$is added to Bij. 3.4.2 Solution Techniques To solve the series of linearized ordinary differential equations represented by Equation 345a, the time differential is replaced by a finite difference formulation, resulting in working equations for 3DLEWASTE of the form:  :0*((a'#A:'# a: '#3:X'#k:'#:x'#: #xEddddd }dd9;X{`(347)FFUNC { {A_{ij}} OVER { DELTA t}~ LEFT (C_j^{k+1}~~C_{j}^{* ` k} RIGHT )~+~LEFT (B_{ij}^{k+1}~+~E_{ij}^{k+1} RIGHT )~C_j^{k+1}~=~R_i^{k+1}} (=AddijG..twYeYlCddkddKdd1dd~j Cdd.ddukdd~j*YeB YlBddokdddd1dd^~ijlEddkdddd 1dd}~ij Cdd0 kddl dd 1dd! ~j-  RddO kdd dd 1dd@ ~iF$(#(#(#(#`!'#$where k+1 represents the current time step, k the previous time step, and t the length of the current time step. Note that since the transient solution scheme allows only for a backwards difference approximation the associated flow runs should also be solved using backwardsintime approximation. For each time step, the solution method involves an inner iterative scheme (see Figure 3.10) which controls the block iterative method of solving the linear equations. For simulations where the nonlinear Freundlich or Langmuir isotherms are used, the solution method also involves an outer iterative scheme where the iterations control convergence of the nonlinear terms in the linearized set of equations. For each nonlinear iteration, the linearized set of equations is solved using storage terms updated using concentration values from the previous nonlinear (outer) iteration. Storage terms for the first iteration in a time step are based on concentration values from the previous time step, or for the first time step, from the initial conditions. If the outer iterative scheme becomes unstable it may be helpful to damp the iterative changes in the concentration. One method of damping the iterative changes is through the use of an underrelaxation factor. Implementation of the underrelaxation factor for the outer iterations in 3DLEWASTE is as follows:  Nh R!#xddddd dd9;X{`p (348)>FUNC {C_i^{r+1}~=~ LEFT (1~~u_o RIGHT )C_i^r~+~u_o C_i^{r+1}} LCddrdddd1ddu4iL.eMlQL1LLudd"o}LCddrdd4iL.Ludd"oLCddErdddd1dd64iR$(#(#h(#(#!!'#$where uo is the outer underrelaxation factor and r is the iteration number. If damping is needed, values between 0.5 and 0.9 should suffice. Acceleration or overrelaxation  N (1.0sddzdd1dd/4i$(#(# (#(#!A'#$where s denotes the inner iteration number and o is the relaxation factor. In general, the use of acceleration by overrelaxation (1.0 3-hh4Read boundary and pointer arrays. X 4.IBUG = Integer indicating if diagnostic output is desired to help determine problems encountered while executing the code (column 79);(# 0 = no, 1 = yes. X 5.ICHNG = Integer control number indicating if the cyclic change of rainfall-seepage nodes is to be printed (column 80); (# 0 = no, 1 = yes. 4.1.2 Data Set 2: Basic Integer Parameters  XOne record with FREEFORMAT per problem. It contains the following variables:(# X 1.NNP = Number of nodal points. (# X 2.NEL = Number of elements. (# X 3.NMAT = Number of material types. (# X 4.NCM = Number of elements with material property correction. (# X 5.NTI = Number of time steps or time increments (see notes at the end of Data Set 2).(# X 6.KSS = Steadystate control; (# 0 = steadystate solution, 1 = transientstate solution (see note at the end of Data Set 2). X 7.NMPPM = Number of material properties per material; this parameter should be set equal to 6 in the present version of the code (see Data Set 5).(# X 8.KGRAV = Gravity term control; (# 0 = no gravity term, '=0*((Ԍ1 = gravity term included. X 9.ILUMP = Mass lumping control; (# 0 = no, 1 = yes. X10.IMID = Mid-difference control; (# 0 = no, 1 = yes. X11.NITER = Number of iterations allowed for solving the non-linear equation. (# X12.NCYL = Number of cycles permitted for iterating rainfall-seepage boundary conditions per time step. (# X13.NDTCHG = Number of times the timestep size will be reset to the initial timestep size; NDTCHG should be > 1 (see Section 5.1.2.10).(# X14.NPITER = Number of iterations for a pointwise solution. (# X**** NOTE: %NTI can be computed by NTI = I1 + 1 + I2 + 1, where I1 is the largest integer not exceeding Log(DELMAX/DELT)/Log(1+CHNG), I2 is the largest integer not exceeding (RTIME-DELT*((1+CHNG) **(I1+1)-1)/CHNG)/DELMAX, RTIME is the real simulation time, and DELMAX, DELT,and CHNG are defined in data set 3. (# **** NOTE: %A steadystate option may be used to provide either the final state of a system under study or the initial condition for a transientstate calculation. In the former case, KSS = 0 and NTI = 0 in this data set. In the latter case, KSS = 0 and NTI > 0. If KSS > 0, there will be no steadystate calculation. (# 4.1.3 Data Set 3: Basic Real Parameters XOne record with FREEFORMAT per problem. It contains the following variables:(# X 1.DELT = Initial time step size, (T). (# X 2.CHNG = Fractional change in the timestep size in each subsequent time increment, (dimensionless decimalpoint value). (# X 3.DELMAX = Maximum value of DELT, (T). (# X 4.TMAX = Maximum simulation time, (T). (# X 5.TOLA = Steady-state convergence criterion, (L). (# X 6.TOLB = Transient-state convergence criterion, (L). (#'>0*((Ԍ N ԙX 7.RHO = Density of water, (M/L3). (#  N X 8.GRAV = Acceleration of gravity, (L/T2); (e.g., 32.17 ft/s2 or 9.81 m/s2). (# X 9.VISC = Dynamic viscosity of water, (M/L/T). (# X10.W = Time derivative weighting factor; (# 0.5 = Crank-Nicolson central and/or mid-difference, 1.0 = backward difference. X11.OME = Iteration parameter for solving the nonlinear matrix equation; (# 0.0 < OME < 1.0 = under-relaxation, 1.0 = exact relaxation, 1.0 < OME < 2.0 = over-relaxation. X12.OMI = Relaxation parameter for solving the linearized matrix equation pointwise; (# 0.0 < OMI < 1.0 = under relaxation, 1.0 = exact relaxation, 1.0 < OMI < 2.0 = over relaxation. 4.1.4 Data Set 4: Printer and Disk Storage Control and Times for Step Size Resetting XThree records are needed per problem. The first two records are formatted input with FORMAT(2I1). The third record is a FREEFORMAT input. The number of lines for the first two records depends on the value of NTI, the number of time increments. The number of lines for the third record depends on the value of NDTCHG, the number of times to reset the timestep size. (# XRecord 1 FORMAT(2I1): This record contain the following variables:(# X 1.KPR0 = Printer control for steadystate and initial conditions;(# 0 = print nothing, 1 = print the values for the variables FLOW, FRATE, and TFLOW, 2 = print values above plus pressure head H, 3 = print values above plus total head, 4 = print values above plus moisture content, 5 = print values above plus Darcy velocity. X 2.KPR(I) = Printer control for the I-th (I = 1, 2, ..., NTI) time step;(# 0 = print nothing, 1 = print the values for the variables FLOW, FRATE, and TFLOW, 2 = print values above plus pressure head H, 3 = print values above plus total head, 4 = print values above plus moisture content, 5 = print values above plus Darcy velocity. XRecord 2 FORMAT(2I1): This record can be used to store 3DFEMWATER output in'?0*(( a binary file for use in plotting or as input to 3DLEWASTE. It contains the following variables:(# X 1.KDSK0 = Auxiliary storage control for the steadystate or initial condition; (# 0 = no storage, 1 = store on logical unit LUSTO. X 2.KDSK(I) = Auxiliary storage control for the I-th (I = 1, 2, ..., NTI) time step; (# 0 = no storage, 1 = store on logical unit LUSTO. XRecord 3 FREEFORMAT: This record contains the following variables:(# X 1.TDTCH(I) = Time when the I-th (I = 1, 2, ..., NDTCHG) timestep-size resetting is needed. (# 4.1.5 Data Set 5: Material Properties XEither hydraulic conductivity or permeability can be input in this data set. The flag KCP in data set 6A is used to indicate which of the two is being used. A total of NMAT records are needed per problem, one for each material.(# X(# XRecord I (I = 1, 2, ..., NMAT) FREEFORMAT: Each record contains following variables:(# X 1.PROP(1,I) = Saturated xxhydraulic conductivity or permeability of the medium I,  N (L/T or L2). (# X 2.PROP(2,I) = Saturated yyhydraulic conductivity or permeability of the medium I,  Nx (L/T or L2). (# X 3.PROP(3,I) = Saturated zzhydraulic conductivity or permeability of the medium I,  N (L/T or L2). (# X 4.PROP(4,I) = Saturated xyhydraulic conductivity or permeability of the medium I,  N (L/T or L2). (# X 5.PROP(5,I) = Saturated xzhydraulic conductivity or permeability of the medium I,  N (L/T or L2). (# X 6.PROP(6,I) = Saturated yzhydraulic conductivity or permeability of the medium I,  N" (L/T or L2). (# 4.1.6 Data Set 6: Soil Property Parameters 6A.LSoil Property Control Integers (# XOne record per problem. This record is FREEFORMATTED and contains the'@0*(( following variables:(# X 1.X` hp x (#%'0*,.8135@8: 0 and data set 11 is used to input the pre-initial condition, which is required as the starting condition for the steadystate iteration. In order to obtain a steadystate solution, both KSS and NTI are set equal to zero and data set 11 supplies the starting condition for the steadystate solution.(# 4.1.12 Data Set 12: Integer Parameters for Source and Boundary Conditions XOne record per problem is needed. This record is FREEFORMATTED and contains the following variables:(# X 1.NSEL = Number of distributed source/sink elements. (# X 2.NSPR = Number of distributed source/sink profiles (i.e., time histories).(# X 3.NSDP = Number of data points in each of the NSPR source/sink profiles.(# X 4.KSAI = Option for the distributed source/sink profiles to be input analytically. This variable should be set equal to zero in the current version of the code.(# X 5.NWNP = Number of well or point source/sink nodes. (# X 6.NWPR = Number of well or point source/sink profiles (i.e., time histories). (# X 7.NWDP = Number of data points in each of the NWPR profiles. (# X 8.KWAI = Option for the well source/sink profiles to be input analytically. This variable should be set equal to zero in the current version of the code.(# X 9.NDNP = Number of fixedhead (Dirichlet) nodes (NDNP should be > 1). (# X10.NDPR = Number of fixedhead profiles (i.e., time histories) (NDPR should be > 1). (# X11.NDDP = Number of data points in each fixedhead profile (NDDP should be > 2). (# X12.KDAI = Option for the fixedhead boundary value profiles to be input analytically. This variable should be set equal to zero in the current version of the code.(# X13.NVES = Number of variable composite (rainfall/evaporationseepage) boundary element sides. (#'G0*((ԌX14.NVNP = Number of variable composite boundary nodal points. (# X15.NRPR = Number of variable composite profiles (i.e., time histories).(# X16.NRDP = Number of data points in each of the NRPR profiles. (# X17.KRAI = Option for the variable composite profiles to be input analytically. This variable should be set equal to zero in the current version of the code.(# X18.NCES = Number of specifiedflux (Cauchy) boundary element sides. (# X19.NCNP = Number of specifiedflux nodal points. (# X20.NCPR = Number of specifiedflux profiles (i.e., time histories). (# X21.NCDP = Number of data points in each of the NCPR profiles. (# X22.KCAI = Option for the specifiedflux profiles to be input analytically. This variable should be set equal to zero in the current version of the code.(# X23.NNES = Number of specifiedpressurehead gradient (Neumann) boundary element sides. (# X24.NNNP = Number of specifiedpressurehead gradient nodal points. (# X25.NNPR = Number of specifiedpressurehead gradient flux profiles (i.e., time histories). (# X26.NNDP = Number of data points in each of the NNPR profiles.(# X27.KNAI = Option for the specifiedpressurehead gradient profiles to be input analytically. This variable should be set equal to zero in the current version of the code.(# 4.1.13 Data Set 13: Distributed and Point Sources/Sinks XThis data set is used to supply data for both distributed sources/sinks and well (point) sources/sinks.(# 13A.Distributed Sources/Sinks XThe following three subdata sets are needed if and only if NSEL in data set 12 is greater than zero. The first subdata set is used to specify the distributed source/sink profiles. The second subdata set is used to read the global element numbers of the distributed source/sink elements. The third subdata set is used to assign a source/sink profile to each distributed source/sink element.(# X(a)X` hp x (#%'0*,.8135@8: 3-hh4Read boundary and pointer arrays. X 4.IBUG = Integer indicating if diagnostic output is desired (column 80); (# 0 = no, 1 = yes. 4.2.2 Data Set 2: Basic Integer Parameters XOne record with FREEFORMAT per problem. It contains the following variables:(# X 1.NNP = Number of nodal points. (# X 2.NEL = Number of elements. (# X 3.NMAT = Number of material types. (# X 4.NCM = Number of elements with material property correction. (# X 5.NTI = Number of time steps or time increments (see notes at the end of Data Set 2).(# X 6.KSS = Steadystate control; (# 0 = steadystate solution, 1 = transientstate solution (see note at the end of Data Set 2). X 7.NMPPM = Number of material properties per material; this parameter should be set equal to 8 in the present version of the code (see Data Set 5).(# X 8.KVI = Velocity input control; (# -1 = velocity and moisture content read from data set 17,  1 = !steadystate velocity and moisture content input read from FEMWATER binary file, (#  2 = !transient velocity and moisture content input read from FEMWATER binary file. (# X 9.ILUMP = Mass lumping control; (# 0 = no, 1 = yes. X10.IWET = Weighting function control; (# 0 = Galerkin weighting, 1 = upstream weighting. X11.IOPTIM = Optimization control; (# 1 = upstream weighting optimization factor is to be computed, 0 = factor is to be set equal to 1.0.'V0*((ԌX12.NITER = Number of iterations allowed for solving the non-linear equation. (# X13.NDTCHG = Number of times the timestep size will be reset to the initial timestep size; NDTCHG should be > 1 (see Section 5.2.2.9). (# X14.NPITER = Number of iterations for a block or pointwise solution. (# X15.KSORP = Sorption model control; (# 1 = linear isotherm, 2 = Freundlich isotherm, 3 = Langmuir isotherm. X**** NOTE: #NTI can be computed by NTI = I1 + 1 + I2 + 1, where I1 is the largest integer not exceeding Log(DELMAX/DELT)/Log(1+CHNG), I2 is the largest integer not exceeding (RTIME-DELT*((1+CHNG)** (I1+1)-1)/CHNG)/DELMAX, RTIME is the real simulation time, and DELMAX, DELT, and CHNG are defined in data set 3. (# **** NOTE:@ #A steadystate option may be used to provide either the final state of a system under study or the initial condition for a transientstate calculation. In the former case, KSS = 0 and NTI = 0 in this data set. In the latter case, KSS = 0 and NTI > 0. If KSS > 0, there will be no steadystate calculation. (# 4.2.3 Data Set 3: Basic Real Parameters XOne record with FREEFORMAT per problem. It contains the following variables:(# X 1.DELT = Initial time step size, (T). (# X 2.CHNG = Fractional change in the timestep size in each subsequent time increment, (dimensionless decimalpoint value). (# X 3.DELMAX = Maximum value of DELT, (T). (# X 4.TMAX = Maximum simulation time, (T). (# X 5.OME = Iteration parameter for solving the nonlinear matrix equation; (# 0.0 < OME < 1.0 = under-relaxation, 1.0 = exact relaxation, 1.0 < OME < 2.0 = over-relaxation. X 6.OMI = Relaxation parameter for solving the linearized matrix equation pointwise; (# 0.0 < OMI < 1.0 = under relaxation, 1.0 = exact relaxation, 1.0 < OMI < 2.0 = over relaxation. X 7.TOLB = Transient-state convergence criterion, (L). (#'W0*((ԌX 8.TOLA = Steady-state convergence criterion, (L). (# 4.2.4 Data Set 4: Printer and Disk Storage Control and Times for Step Size Resetting  XThree records are needed per problem. The first two records are formatted input with FORMAT(2I1). The third record is a FREEFORMAT input. The number of lines for the first two records depends on the value of NTI, the number of time increments. The number of lines for the third record depends on the value of NDTCHG, the number of times to reset the timestep size. (# XRecord 1 FORMAT(2I1): This record contain the following variables:(# X 1.KPR0 = Printer control for steadystate and initial conditions; (# X0 = print nothing, (# X1 = print values for the variables FLOW, FRATE, and TFLOW, (# X2 = print values above plus concentration, (# X3 = print values above plus material fluxes. (# X 2.KPR(I) = Printer control for the Ith (I = 1, 2, ..., NTI) time step; 0 = print nothing, (# X1 = print values for the variables FLOW, FRATE, and TFLOW, (# X2 = print values above plus concentration, (# X3 = print values above plus material fluxes. (# XRecord 2 FORMAT(2I1): This record can be used to store 3DLEWASTE output in a binary file for use in plotting results. It contains the following variables:(# X 1.KDSK0 = Auxiliary storage control for the steadystate or initial condition; (# 0 = no storage, 1 = store on logical unit LUSTO. X 2.KDSK(I) = Auxiliary storage control for the Ith (I = 1, 2, ..., NTI) time step; (# 0 = no storage, 1 = store on logical unit LUSTO. XRecord 3 FREEFORMAT: This record contains the following variables:(# X 1.TDTCH(I) = Time when the Ith (I = 1, 2, ..., NDTCHG) timestep-size resetting is needed. (# 4.2.5 Data Set 5: Material Properties XA total of NMAT records are required for this data set, one for each material. (# XRecord I (I = 1, 2, ..., NMAT) FREEFORMAT: Each record contains the following variables:(#  N' X 1.PROP(1,I) = Distribution coefficient (L3/M) or Freundlich K or Langmuir K for'X0*(( medium I, depending on the value of KSORP in data set 2.(#  N X 2.PROP(2,I) = Bulk density for medium I, (M/L3).(# X 3.PROP(3,I) = Longitudinal dispersivity for medium I, (L).(# X 4.PROP(4,I) = Transverse dispersivity for medium I, (L).(#  N X 5.PROP(5,I) = Molecular diffusion coefficient for medium I, (L2/T).(# X 6.PROP(6,I) = Tortuosity for medium I, (Dimensionless).(# X 7.PROP(7,I) = Decay constant in medium I, (1/L).(# X 8.PROP(8,I) = Freundlich N or Langmuir SMAX for medium I. (# 4.2.6 Data Set 6: Nodal Point Coordinates XCoordinates for NNP nodes are needed only if KVI < 0, where NNP and KVI are defined in data set 2. Usually a total of NNP records are required. However, if a group of subsequent node numbers follows a regular pattern, an automatic generation input option can be used. (# XEach record contains the following variables and is FREEFORMATTED.(# X 1.NI = Node number of the first node in the sequence. (# X 2.NSEQ = Number of subsequent nodes which will be automatically generated. (# X 3.NAD = Increment of node number for each of the NSEQ subsequent nodes. (# X 4.XNI = X-coordinate of node NI, (L). (# X 5.YNI = Y-coordinate of node NI, (L).(# X 6.ZNI = Z-coordinate of node NI, (L).(# X 7.XAD = Increment of x-coordinate for each of the NSEQ subsequent nodes, (L). (# X 8.YAD = Increment of y-coordinate for each of the NSEQ subsequent nodes, (L). (# X 9.ZAD = Increment of z-coordinate for each of the NSEQ subsequent nodes, (L).(# X**** NOTE: #A record with nine zeroes must be used to signal the end of this data set. (# 4.2.7 Data Set 7: Element Incidences XElement incidences for NEL elements, specified in data set 2, are needed if KVI < 0. Usually, a total of NEL records are needed. However, if a group of element numbers'Y0*(( follows a regular pattern, the automatic generation input option can be used.(# XEach record is FREEFORMATTED and contains the following variables:(# X 1.MI = Global element number of the first element in a sequence. (# X 2.NSEQ = Number of subsequent elements which will be automatically generated. (# X 3.MIAD = Increment of MI for each of the NSEQ subsequent elements. (# X 4.IE(MI,1) = Global node number of the first node of element MI. (# X 5.IE(MI,2) = Global node number of the second node of element MI. (# X 6.IE(MI,3) = Global node number of the third node of element MI. (# X 7.IE(MI,4) = Global node number of the fourth node of element MI. (# X 8.IE(MI,5) = Global node number of the fifth node of element MI. (# X 9.IE(MI,6) = Global node number of the sixth node of element MI. (# 10.LIE(MI,7) = Global node number of the seventh node of element MI. (# 11.LIE(MI,8) = Global node number of the eighth node of element MI. (# X12.IEMAD = Increment of IE(MI,1) through IE(MI,8) for each of the NSEQ elements.(# **** NOTE:@ #IE(MI,1) - IE(MI,8) are numbered according to the convention shown in Figure 4.2. The first four nodes start from the front, lower, left corner and progress around the bottom element surface in a counterclockwise direction. The other four nodes begin from the front, upper, left corner and progress around the top element surface in a counterclockwise direction. (# Z0*(( yA ) 0*xddWELL4-2.WPG[< y # Figure 4.2. Node numbering convention for the elements. * 4.2.8 Data Set 8: Subregional Data XThis data set is needed only if KVI < 0, where KVI is defined in data set 2.(# 8A.LSubregion Control Integer(# XOne FREEFORMATTED record is needed for this subdata set. It contains the following variable: (# L1. !NREGN = Number of subregions. (# 8B.LNumber of Nodes in Each Subregion (# XNormally, NREGN records are required. However, if the sequence of node numbers follows a regular pattern between sequential subregions, the automatic generation input option can be used. (# Each record is FREEFORMATTED and contains the following five variables: X1. !NK = Subregion number of the first subregion in a sequence.(# L2. !NSEQ = Number of subsequent subregions which will be automatically generated. (# 3. !NKAD = Increment of NK in each of the NSEQ subsequent subregions.(# X 4. !NODES = Number of nodes in the subregion NK. (# X 5. !NOAD = Increment of NODES in each of the NSEQ subsequent subregions. (# '[0*((ԌX**** NOTE:-A record with five zeroes must be used to end the input of this subdata set. (# 8C.LMapping between Global Nodes and Subregion Nodes (# XThis subdata set should be repeated NREGN times, once for each subregion. For each subregion, normally, the number of records equals the number of nodal points in the subregion. Automatic generation can be used, however, if the subregional node numbers follow a regular pattern. (# Each record contains five variables and is FREEFORMATTED. X 1. !LI = Local node number of the first node in a sequence.(# X 2. !NSEQ = Number of subsequent local nodes which will be generated automatically.(# X 3. !LIAD = Increment of LI for each of the NSEQ subsequent nodes.(# X 4. !NI = Global node number of local node LI.(# X 5. !NIAD = Increment of NI for each of the NSEQ subsequent nodes. (# X**** NOTE:-A record with five zeroes must be used to signal the end of this subdata set. (# **** NOTE:` -Local node numbers have values between one and the total number of nodes in a subregion (i.e., 1,2,...,NODES). Global node numbers are associated with the entire grid and are entered using data set 6.(# 4.2.9 Data Set 9: Material Type Correction XThis data set is required only if NCM > 0 and KVI < 0, where NCM and KVI are defined in data set 2. Normally, NCM records are required. However, if a group of element numbers follows a regular pattern, automatic generation may be used. (# Each record is FREEFORMATTED and contains the following variables: X 1.MI = Global element number of the first element in the sequence. (# X 2.NSEQ = Number of subsequent elements which will be generated automatically. (# X 3.MAD = Increment of element number for each of the NSEQ subsequent elements. (# X 4.MITYP = Type of material for element MI. (# '\0*((ԌX 5.MTYPAD = Increment of MITYP for each of the NSEQ subsequent elements. (# X**** NOTE: #A record with five zeroes must be used to signal the end of this data set. (# 4.2.10 Data Set 10: Card Input for Initial or PreInitial Conditions XNNP records (i.e., one record for each node) are normally needed. However, if a group of node numbers follow a regular pattern, automatic generation can be used. (# Each record is FREEFORMATTED and contains the following variables: X 1.NI = Global node number of the first node in the sequence. (# X 2.NSEQ = Number of subsequent nodes which will be generated automatically. (# X 3.NAD = Increment of node number for each of the NSEQ nodes. (#  N X 4.CNI = Initial or pre-initial concentration of node NI, (M/L3). (#  N0  5.LCAD = Increment of CNI for each of the NSEQ nodes, (M/L3).(# X 6.CRD = Geometrical increment of CNI for each of the NSEQ subsequent nodes (i.e, CNI**CRD).(# X**** NOTE: #A record with six zeroes must be used to signal the end of this data set. (# **** NOTE:@ #The initial condition for a transient calculation may be obtained in two different ways: 1) it can be read directly from data set 10, or 2) the code can perform a steady-state simulation using time-invariant boundary conditions before beginning the transient computations. For the first case, both KSS and NTI in data set 2 should be greater than zero. In the latter case, KSS = 0 and NTI > 0 and data set 10 is used to input the pre-initial condition, which is required as the starting condition for the steadystate iteration. In order to obtain a steadystate solution, both KSS and NTI are set equal to zero and data set 11 supplies the starting condition for the steadystate solution.(# 4.2.11 Data Set 11: Integer Parameters for Sources and Boundary Conditions XOne record per problem is needed. This record is FREEFORMATTED and contains the following variables:(# X 1.NSEL = Number of distributed source/sink elements. (# p#]0*(( X 2.NSPR = Number of distributed source/sink profiles (i.e., time histories) (NSPR should be > 1). (# X 3.NSDP = Number of data points in each of the NSPR source/sink profiles (NSDP should be > 2). (# X 4.KSAI = Option for the distributed source/sink profiles to be input analytically. This variable should be set equal to zero in the current version of the code.(# X 5.NWNP = Number of well or point source/sink nodes. (# X 6.NWPR = Number of well or point source/sink profiles (i.e., time histories). (# X 7.NWDP = Number of data points in each of the NWPR profiles. (# X 8.KWAI = Option for the well source/sink profiles to be input analytically. This variable should be set equal to zero in the current version of the code.(# X 9.NDNP = Number of prescribedconcentration (Dirichlet) nodes (NDNP should be > 1). (# X10.NDPR = Number of prescribedconcentration profiles (i.e., time histories) (NDPR should be > 1). (# X11.NDDP = Number of data points in each prescribedconcentration profile (NDDP should be > 2). (# X12.KDAI = Option for the prescribedconcentration boundary profiles to be input analytically. This variable should be set equal to zero in the current version of the code.(# X13.NVES = Number of variable composite boundary element sides.(# X14.NVNP = Number of variable composite boundary nodal points.(# X15.NRPR = Number of variable composite profiles (i.e., time histories). (# X16.NRDP = Number of data points in each of the NRPR profiles. (# X17.KRAI = Option for the variable composite profiles to be input analytically. This variable should be set equal to zero in the current version of the code.(# X18.NCES = Number of specifiedflux (Cauchy) boundary element sides.(# X19.NCNP = Number of specifiedflux boundary nodal points.(# X20.NCPR = Number of specifiedflux profiles (i.e., time histories). (# '^0*((ԌX21.NCDP = Number of data points in each of the NCPR profiles. (# X22.KCAI = Option for the specifiedflux profiles to be input analytically. This variable should be set equal to zero in the current version of the code.(# X23.NNES = Number of specifieddispersiveflux (Neumann) boundary element sides.(# X24.NNNP = Number of specifieddispersiveflux boundary nodal points.(# X25.NNPR = Number of specifieddispersiveflux profiles (i.e., time histories). (# X26.NNDP = Number of data points in each of the NNPR profiles. (# X27.KNAI = Option for the specifieddispersiveflux profiles to be input analytically. This variable should be set equal to zero in the current version of the code.(# 4.2.12 Data Set 12: Distributed and Point Sources/Sinks XThis data set is used to supply data for both distributed sources/sinks, and point (well) sources/sinks. (# 12A.LDistributed Sources/Sinks (# XThe following three subdata sets are needed if and only if NSEL in data set 11 is greater than zero. The first subdata set is used to specify the distributed source/sink profiles. The second subdata set is used to read the global element numbers of the distributed source/sink elements. The third subdata set is used to assign a source/sink profile to each distributed source/sink element.(# X(a)Sources/Sink Profiles(# X(# XNSPR records (see data set 11) are needed. Each record contains NSDP data points, defined in data set 11. Three numbers, representing the time, source flow rate, and source concentration, respectively, are associated with each data point.(# XRecord I (I = 1, 2, ..., NSPR) FREE FORMAT: Each record contains the following variables:(# X1. !TSOSF(J,I) = Time of J-th data point in I-th profile, (T). (# X2. !SOSF(J,I,1) = Source/sink flow rate of the J-th data point in the I-th profile,  N" (L3/T/L3); positive for source and negative for sink. (# X3. !SOSF(J,I,2) = Source/sink concentration of the J-th data point in the I-th  N % profile, (M/L3).(# ` `  ! # %-. ` `  ! # %-. ` `  !Up to NSDP data points.'_0*((Ԍ(b)LGlobal Element Number of All Distributed Source/Sink Elements(# XOne record is needed for this subdata set. The number of lines in the record depends on NSEL, defined in data set 11. The record is FREEFORMATTED and contains the following variables:(# X1. !LES(1) = Global element number of the first distributed source/sink element. (# X2. !LES(2) = Global element number of the second distributed source/sink element. (# Xd` ` , !d # %X-.(# Xd` ` , !d # %X-.(# Xd` ` , !d # %X-.(# ` `  !Up to NSEL numbers. X(c)Source/Sink Profile Type Assigned to Each Element(# XUsually NSEL records are needed. However, automatic generation can be used. Each record is FREEFORMATTED and contains the following variables:(# X1. !MI = Compressed element number of the first element in the sequence. (# X2. !NSEQ = Number of subsequent elements which will be automatically generated. (# X3. !MAD = Increment of element number for each of the NSEQ elements. (# X4. !MITYP = Source/sink profile associated with element MI. (# X5. !MTYPAD = Increment of MITYP for each of the NSEQ subsequent elements. (# X **** NOTE:-A record with five zeroes must be used to signal the end of this subdata set. (#  **** NOTE:` -Compressed element numbers have values between one and the total number of distributed source/sink elements. Compressed element one corresponds to the first element listed in 12A(b), compressed element two corresponds to the second global element, etc.(# 12B.Point (Well) Source/Sink X(# XThe following three subdata sets are needed if and only if NWNP in data set 11 is greater than zero. The first subdata set is used to specify the point source/sink profiles. The second subdata set reads the source/sink global node numbers, and the third assigns a source/sink profile type to each node.(#'`0*((ԌX(a) Source/Sink Profiles(# XNWPR records (see data set 11) are needed. Each record contains NWDP data points, defined in data set 11. Three numbers, representing the time, source flow rate, and source concentration, respectively, are associated with each data point.(# XRecord I (I = 1, 2, ..., NWPR) FREE FORMAT: Each record contains the following variables:(# X1. !TWSSF(J,I) = Time of J-th data point in I-th profile, (T). (# X2. !WSSF(J,I,1) = Source/sink flow rate of the J-th data point in the I-th profile,  N (L3/T); positive for source and negative for sink. (# X3. !WSSF(J,I,2) = Source/sink concentration of the J-th data point in the I-th  N profile, (M/L3).(# ` `  ! # %-. ` `  ! # %-. ` `  ! # ` `  !Up to NWDP numbers. X(b)Global Node Number of All Point (Well) Source/Sink Nodes(# XOne record is needed for this subdata set. The number of lines in this record depends on NWNP, defined in data set 11. The record is FREEFORMATTED and contains the following variables:(# X1. !NPW(1) = Global node number of the first point source/sink node. (# X2. !NPW(2) = Global node number of the second point source/sink node.(# Xd` ` , !d # %X-.(# Xd` ` , !d # %X-.(# ` `  ! # Xd` ` , !Up to NWNP numbers.(# X(c)Source/Sink Profile Type for Each Node(# XUsually one record per node (i.e., NWNP records) are needed. However, automatic generation can be used. Each record is FREEFORMATTED and contains the following variables:(# X1. !NI = Compressed point source/sink node number of the first node in a sequence. (# X2. !NSEQ = Number of subsequent nodes which will be automatically generated. (# X3. !NIAD = Increment of NI for each of the NSEQ nodes. (# 'a0*((ԌX4. !NITYP = Source/sink profile associated with node NI. (# X5. !NTYPAD = Increment of NITYP for each of the NSEQ subsequent nodes. (# X**** NOTE:-A record with five zeroes must be used to signal the end of this subdata set.(# 4.2.13 Data Set 13: Variable Composite Boundary Condition XThe following four subdata sets are required if and only if NVES in data set 11 is greater than zero. The first subdata set is used to specify the concentration profiles. The second subdata set is used to assign a concentration profile type to each of the variable composite boundary element sides. The third subdata set is used to specify the variable composite boundary element sides. The fourth subdata set is used to read the global nodal number of all the variable composite boundary nodes. (# 13A.LConcentration Profiles(# XThere will be NRPR records (see data set 11) in this subdata set. The number of lines in each record depends on NRDP, defined in data set 11.(# XRecord I (I = 1, 2, ..., NRPR) FREEFORMAT: Each record contains the following variables: (# X1. !TCRSF(1,I) = Time of the first data point in the I-th profile, (T).(#  N L2. !CRSF(1,I) = Concentration of the first data point in the I-th profile, (M/L3). (# X3. !TCRSF(2,I) = Time of the second data point in the I-th profile, (T).(# L4. !CRSF(2,I) = Concentration of the second data point in the I-th profile,  N( (M/L3). (# ` `  ! # %-. ` `  ! # %-. ` `  ! # ` `  !Up to NRDP data points. 13B.LConcentration Profile Type Assigned to Each Boundary Element Side (# XUsually one record per variable composite boundary element side is needed. However, automatic generation can be used. (# XRecord I (I = 1, 2, ..., ) FREEFORMAT: Each record contains the following variables: (# X1. !MI = Compressed variable boundary element side of the first side in a sequence. (# 'b0*((ԌL2. !NSEQ = Number of subsequent sides which will be generated automatically. (# L3. !MIAD = Increment of MI for each of the NSEQ subsequent sides. (# L4. !MITYP = Type of concentration profile assigned to side MI.(# L5. !MTYPAD = Increment of MITYP for each of the NSEQ subsequent sides. (# X**** NOTE:-A record with five zeroes must be used to signal the end of this subdata set. (# 13C.LSpecification of Variable Composite Boundary Element Sides(# XNormally, NVES records are required, one each for a variable boundary element side. However, if a group of variable composite boundary element sides appears in a regular pattern, automatic generation may be used. (# XRecord I (I = 1, 2, ..., ) FREEFORMAT: Each record contains the following variables:(# X1. !MI = Compressed variable composite boundary element side number of the first side in a sequence. (# L2. !NSEQ = Number of subsequent element sides which will be generated automatically. (# X3. !MIAD = Increment of MI for each of the NSEQ element sides. (# L4. !I1 = Global node number of the first node of element side MI. (# X5. !I2 = Global node number of the second node of element side MI. (# L6. !I3 = Global node number of the third node of element side MI. (# X7. !I4 = Global node number of the fourth node of element side MI. (# L8. !I1AD = Increment of I1 for each of the NSEQ subsequent element sides. (# L9. !I2AD = Increment of I2 for each of the NSEQ subsequent element sides. (#  10. !I3AD = Increment of I3 for each of the NSEQ subsequent element sides. (#  11. !I4AD = Increment of I4 for each of the NSEQ subsequent element sides.(# X**** NOTE:-A record with 11 zeroes must be used to signal the end of this subdata set. (# 'c0*((Ԍ13D.LGlobal Nodal Number of All Variable Composite Boundary Nodes(# XAt most, NVNP records (see data set 11) are needed for this subdata set. (# XRecord I (I = 1, 2, ..., ) FREEFORMAT: Each record contains the following variables:(# X1. !NI = Compressed variable boundary node number of the first node in the sequence. (# L2. !NSEQ = Number of subsequent nodes which will be generated automatically. (# L3. !NIAD = Increment for NI for each of the NSEQ nodes. (# X4. !NODE = Global nodal number of the node NI. (#  X5. !NODEAD = Increment of NODE for each of the NSEQ subsequent nodes.(# X**** NOTE:-A record with five zeroes must be used to signal end of this subdata set. (# 4.2.14 Data Set 14: PrescribedConcentration (Dirichlet) Boundary Condition XThis data set is required if and only if NDNP in data set 11 is greater than zero. It consists of three subdata sets. The first subdata set is used to specify the prescribedconcentration profiles, the second is used to read the prescribedconcentration boundary nodes, and the third is used to assign a concentration profile type to each of the Dirichlet nodes.(# 14A.LPrescribed-Concentration Profiles(# XThere will be NDPR records (see data set 11) in this subdata set. The number of lines in each record depends on NDDP, defined in data set 11.(# XRecord I (I = 1, 2, ..., NDPR) FREEFORMAT: Each record contains the following variables: (# X1. !TCDBF(1,I) = Time of first data point in I-th profile, (T). (#  N X2. !CDBF(1,I) = Concentration of first data point in I-th profile, (M/L3). (# X3. !TCDBF(2,I) = Time of second data point in I-th profile, (T). (#  NH$ X4. !CDBF(2,I) = Concentration of second data point in I-th profile, (M/L3). (# ` `  ! # %-. ` `  ! # %-. ` `  ! # ` `  !Up to NDDP data points.'d0*((Ԍ14B.LGlobal Node Number of All the PrescribedConcentration Nodes(# XOne FREEFORMATTED record is needed for this subdata set. The number of lines in this record depends on NDNP, defined in data set 11.(# X1. !NPDB(1) = Global node number of the first compressed prescribedconcentration node. (# L2. !NPDB(2) = Global node number of the second compressed prescribedconcentration node. (# Xd` `  ! # % X-.(# Xd` ` , ! #, % X-.(# Xd` ` , !d #(# XX` ` , !Up to NDNP numbers.(# 14C.LType of Concentration Profile Assigned to Each Dirichlet Node(# XNormally one record per Dirichlet node (i.e., a total of NDNP records) is needed. However, if the Dirichlet node numbers follow a regular pattern, automatic generation may be used. (# XRecord I (I = 1, 2, ..., ) FREEFORMAT: Each record contains the following variables:(# X1. !NI = Compressed Dirichlet node number of the first node in the sequence. (# L2. !NSEQ = Number of subsequent Dirichlet nodes which will be automatically generated. (# L3. !NIAD = Increment of NI for each of the NSEQ nodes. (# L4. !NITYP = Dirichlet concentration profile type assigned to node NI and NSEQ subsequent nodes. (# X5. !NTYPAD = Increment of NITYP for each of the NSEQ subsequent nodes. (# X**** NOTE:-A record with five zeroes must be used to signal the end of this subdata set. (# 4.2.15 Data Set 15: SpecifiedFlux (Cauchy) Boundary Condition XFour subdata sets are required if and only if NCES in data set 11 is greater than zero. The first subdata set is used to read the specifiedflux profiles. The second subdata set is used to assign the type of specifiedflux profile to each of the specifiedflux boundary element sides. The third subdata set is used to read the specifiedflux boundary element sides. The fourth subdata set is used to read the global nodal numbers associated with the specifiedflux boundaries. (# 'e0*((Ԍ15A.LSpecifiedFlux Profiles(# XThere will be NCPR records (see data set 11) in this subdata set. The number of lines in each record depends on NCDP, defined in data set 11.(# XRecord I (I = 1, 2, ..., NCPR) FREEFORMAT: Each record contains the following variables:(# L1. !TQCBF(1,I) = Time of the first data point in the I-th profile, (T). (# X2. !QCBF(1,I) = Normal specifiedflux of the first data point in the I-th profile,  NH (M/T/L2); positive out of the region, negative into the region.(# L3. !TQCBF(2,I) = Time of the second data point in the I-th profile, (T). (# L4. !QCBF(2,I) = Normal specifiedflux of the second data point in the I-th  N profile, (M/T/L2); positive out of the region, negative into the region. (# Xd` ` , !d # %X-.(# Xd` ` , !d # %X-.(# Xd` ` , !d #(# XX` ` , !Up to NCDP data points.(# 15B.LType of SpecifiedFlux Profile Assigned to Each Boundary Element Side(# XAt most, NCES records (see data set 11) are needed. However, automatic generation can be used.(# XRecord I (I = 1, 2, ..., ) FREEFORMAT: Each record contains the following variables:(# X1. !MI = Compressed specifiedflux boundary element side number of the first side in the sequence. (# L2. !NSEQ = Number of subsequent sides which will be generated automatically. (# X3. !MIAD = Increment of MI for each of NSEQ subsequent sides. (# L4. !MITYP = Type of specifiedflux profile assigned to side MI. (# X5. !MTYPAD = Increment of MITYP for each of the NSEQ subsequent sides. (# X**** NOTE:-A record with five zeroes must be used to signal the end of this subdata set. (# 15C.LSpecifiedFlux Boundary Element Sides(# XNormally, NCES records are required, one for each specifiedflux boundary element'f0*(( side. However, if a group of specifiedflux boundary element side numbers follows a regular pattern, automatic generation may be used. (# XRecord I (I = 1, 2, ..., ) FREEFORMAT: Each record contains the following variables:(# L1. !MI = Compressed specifiedflux boundary element side number of the first element side in a sequence. (# L2. !NSEQ = Number of subsequent element sides which will be generated automatically. (# X3. !MIAD = Increment of MI for each of the NSEQ subsequent sides. (# L4. !I1 = Global node number of the first node of element side MI. (# X5. !I2 = Global node number of the second node of element side MI. (# L6. !I3 = Global node number of the third node of element side MI. (# X7. !I4 = Global node number of the fourth node of element side MI.(# X8. !I1AD = Increment of I1 for each of the NSEQ subsequent element sides.(# L9. !I2AD = Increment of I2 for each of the NSEQ subsequent element sides.(#  10.` ` I3AD = Increment of I3 for each of the NSEQ subsequent element sides.(#`  11.` ` I4AD = Increment of I4 for each of the NSEQ subsequent element sides.(#` X**** NOTE:-A record with 11 zeroes must be used to signal the end of this subdata set. (# 15D.LGlobal Node Number of All Compressed SpecifiedFlux Boundary Nodes(# XUsually NCNP records (see data set 11) are needed for this subdata set. However, automatic generation can be used. (# XRecord I (I = 1, 2, ..., ) FREEFORMAT: Each record contains the following variables:(# L1. !NI = Compressed specifiedflux boundary node number of the first node in a sequence. (# L2. !NSEQ = Number of subsequent nodes which will be generated automatically. (# X3. !NIAD = Increment for NI for each of the NSEQ nodes. (# 'g0*((ԌX4. !NODE = Global nodal number of the node NI. (# L5. !NODEAD = Increment of the global nodal number for each of the NSEQ subsequent nodes. (# X**** NOTE:-A record with five zeroes must be used to signal end of this subdata set. (# 4.2.16 Data Set 16: SpecifiedDispersiveFlux (Neumann) Boundary Condition XThe following four subdata sets are required if and only if NNES in data set 11 is greater than zero. The first subdata set is used to read the specifieddispersiveflux profiles. The second subdata set is used to assign a specifieddispersiveflux profile type to each boundary element sides. The third subdata set is used to read the specifieddispersiveflux boundary element side. The fourth subdata set is used to read the global nodal numbers associated with the specifieddispersiveflux boundaries.(# 16A.LPrescribed SpecifiedDispersiveFlux Profiles(# XThere will be NNPR records (see data set 11) in this subdata set. The number of lines in each record depends on NNDP, defined in data set 11.(# XRecord I (I = 1, 2, ..., NNPR) FREEFORMAT: Each record contains the following variables:(# X1. !TQNBF(1,I) = Time of the first data point in the I-th profile, (T). (# L2. !QNBF(1,I) = Normal specifieddispersive flux of the first data point in the I-th  Nx profile, (M/T/L2); positive out of the region, negative into the region. (# X3. !TQNBF(2,I) = Time of the second data point in the I-th profile, (T). (# L4. !QNBF(2,I) = Normal specifieddispersive flux of the second data point in the  N I-th profile, (M/T/L2); positive out of the region, negative into the region. (# Xd` ` , !d # %X-.(# Xd` ` , !d # %X-.(# Xd` ` , !d #(# Ld` ` , !Up to NNDP data points.(# 16B.LType of SpecifiedDispersiveFlux Profile Assigned to Each Boundary Element Side(# XAt most, NNES records (see data set 11) are needed. However, automatic generation can be used.(# XRecord I (I = 1, 2, ..., ) FREEFORMAT: Each record contains the following variables:(#'h0*((ԌX1. !MI = Compressed specifieddispersiveflux boundary element side of the first side in a sequence. (# L2. !NSEQ = Number of subsequent sides which will be generated automatically. (# L3. !MIAD = Increment of MI for each of NSEQ subsequent sides. (# L4. !MITYP = Type of specifieddispersiveflux profile assigned to side MI. (# X5. !MTYPAD = Increment of MITYP for each of the NSEQ subsequent sides. (# X**** NOTE:-A record with five zeroes must be used to signal the end of this subdata set. (# 16C.LSpecifiedDispersiveFlux Boundary Element Sides(# XNormally, NNES records are required, one for each specifieddispersiveflux boundary element side. However, if a group of specifieddispersiveflux element side numbers follows a regular pattern, automatic generation may be used. (# XRecord I (I = 1, 2, ..., ) FREEFORMAT: Each record contains the following variables:(# X1. !MI = Compressed specifieddispersiveflux boundary element side number of the first element side in a sequence. (# L2. !NSEQ = Number of subsequent sides which will be generated automatically. (# X3. !MIAD = Increment of MI for each of the NSEQ subsequent sides. (# L4. !I1 = Global node number of the first node of element side MI. (# X5. !I2 = Global node number of the second node of element side MI. (# L6. !I3 = Global node number of the third node of element side MI. (# X7. !I4 = Global node number of the fourth node of element side MI.(# L8. !I1AD = Increment of I1 for each of the NSEQ subsequent element sides.(#  X9. !I2AD = Increment of I2 for each of the NSEQ subsequent element sides.(#  10.` ` I3AD = Increment of I3 for each of the NSEQ subsequent element sides.(#`  11.` ` I4AD = Increment of I4 for each of the NSEQ subsequent element sides.(#` 'i0*((ԌX**** NOTE:-A record with 11 zeroes must be used to signal the end of this subdata set. (# 16D.LGlobal Node Number of All Compressed SpecifiedDispersiveFlux Boundary Nodes(# XUsually NNNP records (see data set 11) are needed for this subdata set. However, automatic generation can be used. (# XRecord I (I = 1, 2, ..., ) FREEFORMAT: Each record contains the following variables:(# X1. !NI = Compressed specifieddispersiveflux boundary node number of the first node in a sequence.(# L2. !NSEQ = Number of subsequent nodes which will be generated automatically. (# X3. !NIAD = Increment of NI for each of the NSEQ nodes. (# L4. !NODE = Global nodal number of the node NI. (# X5. !NODEAD = Increment of the global nodal number for each of the NSEQ subsequent nodes. (# X**** NOTE:-A record with five zeroes must be used to signal end of this subdata set. (# 4.2.17 Data Set 17: Hydrological Variables  XThis data set is needed if and only if KVI in data set 2 is less than or equal to zero. When KVI < 0, two subdata sets are needed: one for the velocity field and the other for the moisture content.(# 17A.LVelocity Field(# XUsually NNP records (see data set 2) are needed. However, if the velocity values follow a regular pattern, automatic generation can be used.(# XRecord I (I = 1, 2, ..., ) FREEFORMAT: Each record contains the following variables:(# L1. !NI = Node number of the first node in a sequence.(# X2. !NSEQ = Number of subsequent nodes which will be automatically generated.(# L3. !NIAD = Increment of NI for each of the NSEQ subsequent nodes.(# 'j0*((ԌL4. !VXNI = X-velocity component at node NI, (L/T).(# X5. !VYNI = Y-velocity component at node NI, (L/T).(# L6. !VZNI = Z-velocity component at node NI, (L/T).(# X7. !VXAD = Increment of VXNI for each of the NSEQ subsequent nodes, (L/T).(# L8. !VYAD = Increment of VYNI for each of the NSEQ subsequent nodes, (L/T).(# L9. !VZAD = Increment of VZNI for each of the NSEQ subsequent nodes, (L/T).(# X**** NOTE:-A record with nine zeroes must be used to signal the end of this subdata set. (# 17B.LMoisture Content Field(# XUsually, NEL records (see data set 2) are needed. However, if the moisture content values follow a regular pattern, automatic generation can be used.(# XRecord I (I = 1, 2, ..., ) FREEFORMAT: Each record contains the following variables:(# L1. !MI = Element number of the first element in a sequence.(# X2. !NSEQ = Number of subsequent elements which will be automatically generated.(# L3. !MIAD = Increment of MI for each of NSEQ subsequent elements.(# X4. !THNI = Moisture content of element MI, (decimal point).(# L5. !THNIAD = Increment of THNI for the NSEQ subsequent elements, (decimal point). (# X**** NOTE:-A record with five zeroes must be used to signal the end of this subdata set. (# 4.2.18 Data Set 18: End of Job XIf another problem is to be run, then input begins again with input data set 1. If termination of the job is desired, a blank line must be inserted at the end of the data set.(# H$k0*((    ;SECTION 5 6PARAMETER SELECTION׃ This section provides guidance in selecting values for some of the parameters required as input to the 3DFEMWATER/3DLEWASTE codes. This guidance is not intended in any way to be used as a substitute for data collection. The most accurate model results are obtained from simulations which are based on sitespecific information. In some cases, however, it is not feasible to measure certain parameters, and satisfactory results may be obtained using estimated values taken from the reported ranges presented here. For easy reference, the parameters are grouped according to the data group in which they appear in the input data sets (see Section 4). Concepts, such as initial and boundary conditions, isotherms, distributed and point sources and sinks, and subregional data, were introduced in Section 3 and guidance is not provided in this section for related parameters. 5.1 3DFEMWATER 5.1.1 Data Set 1: Title of the Simulation Run 5.1.1.1 Geometry, Boundary, and Pointer Array Control, IGEOM [] The integer IGEOM has two functions. It is used to specify if geometry, boundary, and pointer arrays should be printed so that the user can examine them. It also controls whether the boundary and pointer arrays are written to or read from binary files. Boundary arrays store data related to the boundary conditions. Pointer arrays store the global matrix in compressed form and are used to construct the subregional block matrices. For large problems, it takes too much time to generate these arrays for each computer execution of a particular scenario. Usually, they should be generated only once and stored in binary files using logical units LUBAR and LUPAR (see Table B1). In order to compute and store the boundary and pointer arrays, the user should choose a value for IGEOM less than or equal to one. In subsequent runs, the boundary and pointer arrays can be read from the binary files by changing the value of IGEOM to a number greater than three. Whenever changes are made to the model which involve the geometry of the problem, the boundary conditions, and the configuration of the subregions, the arrays must be generated and stored again. Note that the option presented in the input to read boundary arrays and compute and write pointer arrays is not used in 3DFEMWATER under normal conditions. For the options explained above, if the number chosen by the user is even, the arrays will be printed as output. If the number is odd, the arrays will not be printed.'l0*((Ԍ5.1.2 Data Set 2: Basic Integer Parameters 5.1.2.1 Number of Material Types, NMAT [] This parameter is the total number of different porous media being modeled. For example, if the region of interest is predominantly sand with clay lenses, then the value of NMAT should be set equal to two. When material properties are assigned to each material type, using data set 5 (see Section 4.1.5), the first material type should be the predominant porous medium (e.g., for the example here, the sand). 5.1.2.2 Number of Elements with Material Property Correction, NCM [] In the code, all the grid elements automatically are initialized as having a material type of one. If the region being modelled is homogeneous, the parameter NCM is set equal to zero. To model a heterogeneous porous medium, NCM and the parameters in data set 10 of the input (see Section 4.1.10) are used to specify which elements have different material types associated with them. The parameter NCM is the total number of elements which have a material type different than the first material type. 5.1.2.3 Number of TimeSteps, NTI [] For a constant timestep size, this number is obtained by dividing the simulation time by the timestep size, DELT. If the timestep size is variable, this number is computed using the formula given in the note at the end of data set 2 in Section 4.1.2. If a steadystate solution is desired, NTI should be set equal to zero. 5.1.2.4 SteadyState Control, KSS [] As noted in Section 4.1.2, a steadystate option may be used to provide either the final state of a system under study or the initial condition for a transientstate calculation. In the former case, both KSS and the number of time steps, NTI, should be set to zero. In the latter case (i.e., when KSS = 0 and NTI > 0), the code performs a steadystate calculation before beginning the transient computations. If KSS = 1, no steadystate calculation is performed. Rather, the code begins transient calculations using initial conditions supplied in data set 11 of the input. 5.1.2.5 Gravity Term Control, KGRAV [] This parameter indicates if the gravity term should be included. For most cases, KGRAV should be equal to 1. For cases when flow due to the pressure gradient is much greater than that due to gravity, KGRAV is set to 0. 5.1.2.6 Mass Lumping Flag, ILUMP [] This parameter indicates if the mass matrix is to be lumped or not. Normally, one should set this parameter to 0. Without lumping, the solution is more accurate. However, for occasions when negative concentrations or oscillating solutions occur, this parameter should be set to 1. It has been suggested that for saturatedunsaturated flow computations, the parameter ILUMP'm0*(( should always be set equal to 1. 5.1.2.7 Middifferencing Flag, IMID [] This parameter indicates if the more accurate middifference method should be used in flow computations. For practical purposes, IMID = 0 should be sufficient. IMID = 1 is used only for research purposes. 5.1.2.8 Number of Iterations for the Nonlinear Equation, NITER [] This parameter is the number of iterations allowed for solving the nonlinear equation. Normally, NITER = 50 should be sufficient. If this number is exceeded and the solution does not converge, the program will issue a warning message. When this occurs, the user should first recheck the input values. If the input is correct, the program can be reexecuted using a larger value for NITER. 5.1.2.9 Number of Cycles, NCYL [] This parameter indicates how many cycles are used for iterating the boundary conditions. A value of 20 should be adequate for most problems. 5.1.2.10 Number of Times to Reset the Time Step, NDTCHG [] This parameter indicates how many times the timestep size should be reset to the initially small timestep size. When we start a simulation, we normally use a small timestep size. However, for every consecutive time step, we may gradually increase the timestep size by some amount specified by the variable CHNG in Data Set 3 in Section 4.1.3. When a steep change in boundary conditions or source/sink conditions occurs, however, the timestep size should be reset to the initially small value. (See the example problem in Section 6.1.1.) NDTCHG tells us how many times we want to reset the timestep size during a simulation. The value of NDTCHG must be at least one. If the user does not want to reset the time step, a value of one should be entered here and a very large number, larger than the total simulation time, should be entered for TDTCH(1) in data set 4 (see Section 4.1.4). 5.1.2.11 Number of Iterations for Pointwise Solution, NPITER [] This parameter is used to input the number of iterations allowed for solving the matrix equations with the block iteration method. A value of 300 should be sufficient for most problems. If this number is exceeded and the solution does not converge, the program will issue a warning message. When this occurs, the users should reexecute the program using a larger value for NPITER. 5.1.3 Data Set 3: Basic Real Parameters 5.1.3.1 Initial TimeStep Size, DELT [T] This is the timestep size used for the first timestep computation if the variable CHNG is not'n0*(( equal to 0.0. It is the timestep size used for every time step if the variable CHNG is set equal to 0.0. It is advisable to choose the value of DELT such that: (F*DELX*DELX)/(DELT*K) < 1 where DELX = the element size (L) K = hydraulic conductivity (L/T) F = specific storage (1/L) For example, if F = 0.001 1/m, K = 0.00001 m/sec, and an element size of 10 m is used, then DELT should be less than 10,000 seconds. 5.1.3.2 Fractional Change in TimeStep Size, CHNG [] This parameter specifies how much of an increase one would like to make to the timestep size for each subsequent time step. Normally, a value from 0.0 to 0.5 can be used. 5.1.3.3 Maximum Allowable Time Step, DELMAX [T] The maximum timestep size allowed depends on how fast the system responds to change. Use of a value one to ten times the size of the initial time step is advised. 5.1.3.4 Maximum Simulation Time, TMAX [T] This is the actual length of time to be simulated. If this time is exceeded before you have made NTI step computations, the simulation will be terminated. 5.1.3.5 SteadyState Convergence Criterion, TOLA [L] This is the absolute error allowed for assessing if a steadystate solution for hydraulic head has converged. The value used for TOLA depends on how much the system is disturbed. Normally, setting TOLA equal to onetenthousandth (0.0001) of the maximum disturbance should be sufficient. For example, if one is conducting a simulation of drawdown due to pumping and one expects the maximum drawdown at steadystate will be 1 m, then a value of TOLA equal to 0.0001 m should be sufficient. 5.1.3.6 Transient Convergence Criterion, TOLB [L] This is the absolute error allowed for assessing if the solution for hydraulic heads has converged for each transient time step. A value equal to onehundredthousandth (0.00001) of the maximum disturbance should be sufficient for most problems.  NH$ 5.1.3.7 Density of Water, RHO [M/L3]  N% The density of water, #w, is the ratio of its mass to its volume and has SI units of kg/m3. Density varies with temperature (Table 51) and can be computed using regression equations presented in CRC (1981). Density also varies with the concentration of dissolved chemical 'o0*((ԌC#TABLE 51. WATER DENSITY AS A FUNCTION OF TEMPERATURE  Temperature Density  N   (oC) (kg/m3)  0 999.87 10 999.73 20 998.23 30 995.67 40 992.24 50 988.07 60 983.24 70 977.81 80 971.83 90 965.34 100 958.38  Source: Mercer et al., 1982; Original Reference: CRC, 1965  species. Water density appears in the definition of specific storage and in the relationship between hydraulic conductivity and intrinsic permeability (Section 5.1.4.2). 5.1.3.8 Dynamic Viscosity of Water, VISC [M/L/T] The viscosity of a fluid is a measure of the forces that work against flow when a shearing stress is applied (Lyman et al., 1982). The more viscous a fluid is, the greater the shear stress needed to maintain a given velocity gradient. Dynamic viscosity is often expressed in terms of poise (gram per centimeter per second) or centipoise (0.01 poise). Water has a  N( viscosity of approximately 1 centipoise at 20oC. Viscosity varies with temperature, as indicated in Table 52, and with concentration of dissolved chemicals. The effect of pressure on fluid viscosity is generally unimportant (Mercer et al., 1982). Note that dynamic viscosity is a term in the relationship between hydraulic conductivity and intrinsic permeability (Section 5.1.4.2). 5.1.3.9 Time Integration Weighting Factor, W [] A value of W equal to 1.0 should be used for most practical problems (see Equation 316). Setting W equal to 0.5 is normally done for research purposes to assess the accuracy of the CrankNicolson scheme. 5.1.3.10 Relaxation Parameter for Solving the Nonlinear Equation, OME [] Normally this parameter should be set to 1.0 (see Equation 317). If the convergence history shows sign of oscillation, then a value of 0.5 should be used. If the convergence history 'p0*((ԌTABLE !52. DYNAMIC VISCOSITY OF WATER AS A FUNCTION OF TEMPERATURE!  Temperature Dynamic Viscosity  N  (oC) (centipoise)  0 1.7921 10 1.3077 20 1.0050 30 0.8007 40 0.6560 50 0.5494 100 0.2838  Source: CRC, 1965  shows monotonic decreases but at a very slow rate, then OME should be set to somewhere between 1.7 to 1.9. 5.1.3.11 Iteration Parameter to Solve the Linearized Matrix Equation, OMI [] Normally this parameter should be set to 1.0 (see Equation 318). If the convergence history shows signs of oscillation, then set OMI to 0.5. If the solution converges monotonically but at a very slow rate, then set OMI to between 1.7 and 1.9. 5.1.4 Data Set 5: Material Properties In the material properties data set, the user must input values for either hydraulic conductivity or permeability for each aquifer/soil material type. The flag which tells the code which of these two properties is being input is the permeability input control, KCP, located in Data Group 6. 5.1.4.1 The Saturated Hydraulic Conductivity Tensor [L/T] Hydraulic conductivity is the coefficient of proportionality which appears in Darcy's Law. It expresses the ease with which a fluid can be transported through a porous medium and is a function of properties of both the porous medium and the fluid (Mills et al., 1985b). It is defined as the volume of water that will move in unit time under a unit hydraulic gradient through a unit area measured at right angles to the direction of flow. For threedimensional flow in an anisotropic medium, hydraulic conductivity varies with direction at any point in space and is expressed as a symmetric secondrank tensor: a#x (ddddddd9qx%` (51)>func { K sub ij ~=~ } left [func { matrix { K sub xx & K sub xy & K sub xz # K sub yx & K sub yy & K sub yz # K sub zx & K sub zy & K sub zz} } right ]d6X@K@d6X@K@d6X@K@4Koow ij@Kooxxb@Kooxy@KooxzGKooyxbGKooyyGKooyzNKoo$zxbNKoo$zyNKoo$zzF4vxxUxxxw}U~>$(#(#p#(#(#8!a'#$0'q0*(( ('#F,aq0Ԍ N where Kij is the hydraulic conductivity tensor and x, y, and z are the coordinate axes of the  N model grid. Because of symmetry, only 6 of the 9 terms are needed (Kxx, Kyy, Kzz, and Kxy =  N Kyx, Kxz = Kzx, and Kyz = Kzy). If the coordinate axes coincide with the principal directions of anisotropy, then the nine  N8 components of the tensor reduce to Kxx, Kyy, and Kzz, with the other components equal to zero. For isotropic media, hydraulic conductivity is independent of the direction of measurement  N (i.e., Kxx = Kyy = Kzz ). Hydraulic conductivity estimates should be based on sitespecific data collection (e.g., pumping tests or piezometer tests). Some typical horizontal hydraulic conductivity values for various materials are shown in Table 53. Note that hydraulic conductivity varies over a very wide range. As a result, values are rarely known with more than an orderofmagnitude accuracy. Hydraulic conductivity values for fractured rock can be found in Mercer et al. (1982). For many materials, the vertical hydraulic conductivity is substantially smaller than the horizontal hydraulic conductivity (assuming horizontal bedding and measurements made along the principal axes) (Mercer et al., 1982). Mills et al. (1985b) state that the ratio of horizontal to vertical conductivity, known as the anisotropy ratio, is from 2 to 10 for alluvium and glacial outwash and from 1.5 to 3 for sandstone. The variability in horizontal and vertical conductivities for a few aquifer materials is shown in Table 54.  Nh 5.1.4.2 The Permeability Tensor [L2] Intrinsic permeability is a property of the porous medium only. It is a measure of the resistance to fluid flow through the medium. The greater the permeability, the less the resistance. Like hydraulic conductivity, permeability is a symmetrical secondrank tensor. Permeability is equal to hydraulic conductivity multiplied by a scalar value, as is seen in the following equation:  N( ` `  ! #kij = Kij %/(#g)hh4`!(#k(52) where  N kij = permeability (L2)  N Kij = hydraulic conductivity (L/T) % = dynamic viscosity (M/L/T)  N8 # = density (M/L3)  N g = acceleration of gravity (L/T2) As was true for hydraulic conductivity, permeability estimates should be based on sitespecific data collection. Ranges of values for permeability are shown in Table 53 and in Table 55. Permeability is sometimes expressed in units of darcies. Conversion from darcies to other units can be done by using the conversion factors provided at the bottom of Table 53.  %r0*(( TABLE 53.RANGE OF HYDRAULIC CONDUCTIVITY VALUES FOR VARIOUS GEOLOGIC MATERIALS (Freeze and Cherry, 1979) ya2x`0*xddFIG5-3.WPGs<<y$(#(#(#(#a'#$ #^Hp@ƒQ ^@# ` ` Conversion Factors for Permeability ` ` and Hydraulic Conductivity Units 20  R&H ` `  !Permeability, k*hh4< Hydraulic conductivity, K ` ` 20 <C20 (#(#p R&8 X` hp x (#%'0*,.8135@8: 0), the code performs a steadystate calculation before beginning the transient computations. If KSS = 1, no steadystate calculation is performed. Rather, the code begins transient calculations using initial conditions supplied in data set 10 of the input. 5.2.2.5 Mass Lumping Flag, ILUMP [] This parameter indicates if the mass matrix is to be lumped or not. Normally, one should set this parameter to 0. Without lumping, the solution is more accurate. However, for occasions when negative concentrations or oscillating solutions occur, this parameter should be set to 1. 5.2.2.6 Weighting Function Control, IWET [] This parameter indicates if the upstream weighting function is to be used. For the present version of code, this parameter does not affect the solution when a transient solution is sought. If a steadystate solution is desired, one should set this parameter to 1. Thus, it is advisable to always set this parameter to 1 for the present version of the computer code. 5.2.2.7 Optimization Flag, IOPTIM [] This parameter specifies whether the upstream weighting factor is to be optimized. This parameter does not affect the solution if a transient solution is sought. For a steadystate solution, it is advisable to set IOPTIM to 1. When IOPTIM is set to 0, an upstream weighting factor of 1.0 is assumed.'z0*((Ԍ5.2.2.8 Number of Iterations for the Nonlinear Equation, NITER [] This parameter is the number of iterations allowed for solving the nonlinear equation. Normally, a value of NITER equal to 40 should be sufficient. If this number is exceeded and the solution does not converge, the program will issue a warning message. When this occurs, the users should reexecute the program using a larger value of NITER. 5.2.2.9 Number of Times to Reset the Time Step, NDTCHG [] This parameter indicates how many times one should reset the time step size to the initially small timestep size. When we start a computation, we normally use a small timestep size. However, for every consecutive time step, we may gradually increase the timestep size by some amount specified by CHNG in Data Set 3 in Section 4.2.3. When we have a steep change in boundary conditions or in source/sink conditions, we will need to reset the timestep size to the initially small value. NDTCHG tells us how many times we want to reset the timestep size. The value of NDTCHG must be at least one. If the user does not want to reset the time step, a value of one should be entered here and a very large number, larger than the total simulation time, should be entered for TDTCH(1) in data set 4 (see Section 4.2.4). 5.2.2.10 Number of Iterations for Pointwise Solution, NPITER [] This parameter is used to input the number of iterations allowed for solving the matrix equations with the block iteration method. NPITER = 300 should be sufficient for most problems. If this number is exceeded and the solution does not converge, the program will issue a warning message. When this occurs, the user should first recheck the input values. If the input is correct, the program can be reexecuted using a larger value for NPITER. 5.2.2.11 Sorption Model Control, KSORP [] Although the Freundlich isotherm option can be used to simulate a linear isotherm by setting the value of the exponent, n, equal to one, it is recommended that linear isotherms be simulated using only the linear isotherm option. This is because the linear isotherm option makes use of retarded seepage velocities, which result in a more accurate solution for the particle tracking scheme used in 3DLEWASTE than the pore velocities used in conjunction with the nonlinear adsorption models. 5.2.3 Data Set 3: Basic Real Parameters 5.2.3.1 Initial TimeStep Size, DELT [T] This is the timestep size used for the first timestep computation if the variable CHNG is not equal 0.0. It is the timestep size used for every time step if the variable CHNG is set equal to 0.0. For a steadystate computation, DELT should be chosen such that no particle travels more than one element in one time step. For example, if an element has a size of 10 m and the averaged velocity over this element is 0.00001 m/sec, then DELT should be less then 1,000,000 seconds. For transient computations, one should choose a timestep size as large as possible with DELT less than DELX*DELX/D, where DELX is the size of the element and D is the dispersion coefficient. For example, if the element size is 10 m and the dispersion'{0*((  N coefficient is 0.00001 m2/sec, then DELT should be less than 10,000,000 seconds. 5.2.3.2 Fractional Change in TimeStep Size, CHNG [] This parameter specifies how much of an increase one would like to make to the timestep size for each subsequent time step. Normally, a value from 0.0 to 0.5 can be used. 5.2.3.3 Maximum Allowable Time Step, DELMAX [T] The maximum timestep size allowed depends on how fast the system responds to change. Use of a value one to ten times the size of the initial time step is advised. 5.2.3.4 Maximum Simulation Time, TMAX [T] This is the actual length of time to be simulated. If this time is exceeded before you have made NTI step computations, the simulation will be terminated. 5.2.3.5 Relaxation Parameter for Solving the Nonlinear Equation, OME [] Normally this parameter should be set to 1.0 (see Equation 348). If the convergence history shows sign of oscillation, then a value of 0.5 should be used. If the convergence history shows monotonic decreases but at a very slow rate, then OME should be set to somewhere between 1.7 to 1.9. 5.2.3.6 Iteration Parameter to Solve the Linearized Matrix Equation, OMI [] Normally this parameter should be set to 1.0 (see Equation 349). If the convergence history shows signs of oscillation, then set OMI to 0.5. If the solution converges monotonically but at a very slow rate, then set OMI to between 1.7 and 1.9. 5.2.3.7 Transient Convergence Criterion, TOLB [] This is the relative error allowed for assessing if a solution has converged for each time step. Setting TOLB equal to 0.000001 should be sufficient for most problems. 5.2.3.8 SteadyState Convergence Criterion, TOLA [] This is the relative error allowed for assessing if a steadystate solution has converged. TOLA = 0.00001 should be sufficient for most problems. 5.2.4 Data Set 5: Material Properties  Np# 5.2.4.1 Distribution Coefficient [L3/M] Freeze and Cherry (1979) state that adsorption/desorption reactions for contaminants in groundwater are normally viewed as being very rapid relative to the flow velocity and that the amount of contaminant adsorbed is commonly a function of concentration in the solution. At constant temperature and lowtomoderate concentrations, the functional relationship between'|0*((  N the adsorbed concentration, S (M/L3), and the dissolved concentration, C (M/L3), is often approximated by the Freundlich equilibrium isotherm (Helfferich, 1962):  N ` `  %-hh4S = KCn`!(#k(55) where the coefficients K and n depend on several factors, including the solute species and the nature of the porous medium. If the isotherm is linear, n = 1, K is known as the distribution  N coefficient, Kd. The derivation of the distribution coefficient, which is different for each constituent, is discussed briefly in Section 3.3.1.  Np 5.2.4.2 Bulk Density [M/L3] Bulk density can be defined as the mass of a unit volume of dry soil. The soil bulk density directly influences the retardation of solutes and is related to the structure and texture of a soil (Mercer et al., 1982).  N The bulk density of soils typically range between 1.3 and 2.0 g/cm3, but Mercer et al. (1982)  NX state that the bulk density can be as low as 0.3 g/cm3 for soils high in organics or aluminum and iron hydroxides. Representative values for five different types of soils are shown in Table 58. In addition, values of bulk density for a large number of soils can be obtained from the interactive computer program DBAPE, which was discussed in Section 5.1.5.1.1. The bulk density of aquifer materials may differ significantly from that of soils. Therefore, data on the ranges of bulk density for various geologic material are presented in Table 59. If sitespecific data are not available, the bulk density of the saturated zone can be derived using an exact relationship between porosity, particle density and the bulk density (Freeze and Cherry,  N 1979). Assuming the particle density to be 2.65 g/cm3, this relationship can be expressed as:  N ` `  %-hh4#b = 2.65(1 )`!(#k(56) where  NP #b ` ` = bulk density of the soil (g/cm3)  ` ` = saturated moisture content (porosity) () }0*((  N TABLE !58.` ` MEAN BULK DENSITY (g/cm3) FOR FIVE SOIL TEXTURAL  N ` ` CLASSIFICATIONS!a,b  ` hp x (#<Soil Texture7Mean Valueȗ<YRange Reported  Silt Loams6:1.32|<[0.86 1.67Ѓ Clay and Clay Loams6:1.30|<[0.94 1.54 Sandy Loams6:1.49|<[1.25 1.76 Gravelly Silt Loams6:1.22|<[1.02 1.58 Loams6:1.42|<[1.16 1.58 All Soils6:1.35|<[0.86 1.76Ѓ   N <a Baes, C.F., III and R.D. Sharp. 1983. A Proposal for Estimation of Soil Leaching Constants for Use in Assessment Models. J. Environ. Qual. 12(1):1728 (Original reference).  N b From Dean et al. (1989)  ` hp x (#5.2.4.3 Longitudinal and Transverse Dispersivity [L] Hydrodynamic dispersion is a nonsteady, irreversible mixing process by which a contaminant spreads as it is transported through the subsurface. It results from the effects of two components: molecular diffusion and mechanical dispersion. The larger the hydrodynamic dispersion term is, the larger the spreading of an initially localized contaminant. Molecular  N4 diffusion is discussed in Section 5.2.4.4. Mechanical dispersion, D , is caused by variations in pore velocities in a soil or aquifer material. In addition, variations in the rate of advection caused by aquifer inhomogeneity and spatiallyvariable hydraulic conductivities results in plume spreading, which is often confused with dispersion (Keely, 1989). Although mechanical dispersion is a second rank tensor, by assuming that a material is isotropic with respect to dispersion, the dispersion tensor can be expressed in terms of the average groundwater velocity and two constants: the longitudinal and transverse dispersivity  N (see Equation 320). Longitudinal dispersivity, L, is defined as the characteristic mixing  N length in the direction of groundwater flow and lateral dispersivity, T, is the mixing length in the directions perpendicular to flow. Values for dispersivity are difficult to determine. Research has shown that the values are dependent on the scale of the problem being studied (EPRI, 1985). This can be seen in Figure 5.1. Usually, dispersion is determined by adjusting the dispersivity values until modeling results match historical data (Mercer et al., 1982). Transverse dispersivity values are commonly thought to be lower than longitudinal dispersivity values by a factor of 3 to 20. However, recent studies suggest that transverse dispersivity values should be at least an orderofmagnitude smaller than longitudinal dispersivity values (Gelhar et al., 1992) and may even be close to zero (U.S. EPA, 1989).>'~0*((ԌTABLE !59.` ` RANGE AND MEAN VALUES OF DRY BULK DENSITY FOR VARIOUS ` ` GEOLOGIC MATERIALS!   N ` hp x (#X (#%'0*,.8135@8: 0), or a transient computation using usersupplied initial conditions (KSS = 1, NTI > 0). GW3D calls subroutine DATAIN to read and print input data; subroutine PAGEN to generate pointer arrays; subroutine ALLFCT to obtain source/sink and boundary values; subroutine SPROP to obtain the relative hydraulic conductivity, water capacity, and moisture content from the pressure head; subroutine VELT to compute Darcy velocity; subroutine BCPREP to determine if a change of boundary conditions is required; subroutine ASEMBL to assemble the element matrices over all elements; subroutine BC to implement the boundary conditions; subroutine BLKITR to form and solve the subregional block matrix equations; subroutine SFLOW to calculate flux through all types of boundaries and water accumulated in the media; subroutine PRINTT to print out the results; and subroutine STORE to store the flow variables for input to 3DLEWASTE or for plotting. A.1.9 Subroutine PAGEN This subroutine is called by subroutine DATAIN to preprocess pointer arrays that are needed to store the global matrix in compressed form and to construct the subregional block matrices. The pointer arrays automatically generated in this subroutine include the global node connectivity (stencil), GNOJCN(J,N), regional node connectivity, LNOJCN(J,I,K), total node number for each subregion, NTNPLR(K), the bandwidth indicator for each subregion, LMAXDF(K), and a partial fillup for the mapping array between global node number and local subregion node number, GNPLR(I,K), with I = NNPLR(K) + 1 to NTNPLR(K). Here GNOJCN(J,N) is the global node number of the Jth node connected to the global node N; LNOJCN(J,I,K) is the local node number of the Jth node connected to the local node I in the Kth subregion; NTNPLR(K) is the total number of nodes in the Kth subregion, including the interior nodes, the global boundary nodes, and intraboundary nodes; LMAXDF(K) is the maximum difference between any two nodes of any element in the Kth subregion; and GNPLR(I,K) is the global node number of the Ith localregion node in the Kth subregion. These pointer arrays are generated based on the element connectivity, IE(M,J), the number of nodes for each subregion, NNPLR(K), and the mapping between global node and localregion node, GNLR(I,K), with I = 1, NNPLR(K). Here IE(M,J) is the global node number of Jth node of element M; NNPLR(K) is the number of nodes in the Kth subregion, including the interior nodes and the global boundary nodes, but not the intraboundary nodes. A.1.10 Subroutine PRINTT This subroutine, which is called by GW3D, is used to lineprint the flow variables. These include the fluxes through variable boundary surfaces, the pressure head, total head, moisture content, and Darcy velocity components.  Figure 1 A Figure 1  A.1.11 Subroutine Q4S'0*((ԌThis subroutine is called by subroutines BC, BCPREP, and SFLOW to compute the surface node flux of the type: #x`ddddd dd9i` (A1)E@FUNC { RQ(I) ~ = ~ int from B sub e to \ N sub i sup e q dB ~ }  RQ(;I)Ldd  dd4B<<6eZNdd.eddiqydBE$(#(#(#(#`!'#$where q is either the specifiedflux, specifiedpressurehead gradient flux, or gravity flux; B is the  N global boundary of the region of interest; Nie is the basis functions for nodal point i of element e; and RQ(I) is a 3DFEMWATER code parameter. A.1.12 Subroutine Q8 This subroutine is called by the subroutine ASEMBL to compute the element matrix given by: 4#xddddd dd9h` (A2a)RFUNC { QA(I,J) ~ = ~ int from {R sub e} to \ N sub i sup e F N sub j sup e dR ~ }  QA(;I,Jg)dd dd4R<<e"Ndd.eddiFANdd.eddjdR4$(#(#d (#(#`!'#$ B#xddddd dd9h7#dp@~fQ0@#`(A2b)FUNC { QB(I,J) ~ = ~ int from {R sub e} to \ (grad N sub i sup e) cdot bold K sub s }FUNC { k sub r cdot (grad N sub j sup e) dR ~ }  QB(;I,Jg)dd dd4R<<e"(+Nddi.eddPi) mKddskddur(y+Ndd\ .eddC j ) dRB$(#(#0(#(#`!'#$ where F ` ` = soil property function  N Nje ` ` = basis function for nodal point j of element e  N  Ks ` ` = saturated hydraulic conductivity tensor  N kr ` ` = relative hydraulic conductivity R` ` = region of interest + ` ` = del operator indicating gradient +` ` = del operator indicating divergence and where QA(I,J) and QB(I,J) are 3DFEMWATER code parameters. Subroutine Q8 also calculates the element load vector given by: A#xp!ddddd dd9h`(A2c)FUNC { RQ(I) ~ = ~ int from {R sub e} [(grad N sub i sup e) cdot bold K sub s } FUNC{ k sub r cdot (grad z) ~ ~ N sub i sup e q] dR ~ }  RQ(;I)Ldd4R<<6eZ[("+Ndd.eddiA) KddoskddrM(+yz) 7 Ndd .edd i qV ] dRA$(#(#(#(#`!'#$ where q is the source/sink. `#0*((A`'# '#I'#!'#$` A.1.13 Subroutine Q8DV Subroutine Q8DV is called by subroutine VELT to compute the element matrices given by: " #x8ddddd dd9h` (A3a)PFUNC { QR(I,J) ~ = ~ int from {R sub e} to \ N sub i sup e N sub j sup e dR ~ }  QR(;I,Jg)dd dd4R<<e"Ndd.eddiNdd\.eddCjdR"$(#(#(#(#`! '#$where QR(I,J) is a 3DFEMWATER program variable. Subroutine Q8DV also evaluates the element load vector: ! #xH ddddd dd9h`(A3b)FUNC { QRX(I) ~ = ~ int from {R sub e} to \ N sub i sup e bold i cdot } FUNC { bold K sub s } FUNC { k sub r cdot (grad N sub j sup e) H sub j dR ~ }  QRX;(I)]dd dd4R<<e"Ndd.eddiiAKdd sGkddrM(+Ndd.edd{j)4 Hdd j dR$(#(#(#(#`!! '#$ A #xddddd dd9h`(A3c)FUNC { QRY(I) ~ = ~ int from {R sub e} to \ N sub i sup e bold j cdot } FUNC { bold K sub s } FUNC { k sub r cdot (grad N sub j sup e) H sub j dR ~ }  QRY;(I)]dd dd4R<<e"Ndd.eddijAKdd sGkddrM(+Ndd.edd{j)4 Hdd j dR$(#(#d (#(#`!A '#$ a #xddddd dd9h`(A3d)FUNC { QRZ(I) ~ = ~ int from {R sub e} to \ N sub i sup e bold k cdot }FUNC { bold K sub s }FUNC { k sub r cdot (grad N sub j sup e) H sub j dR ~ }  QRZ;(I)]dd dd4R<<e"Ndd.eddikAKdd sGkddrM(+Ndd.edd{j)4 Hdd j dR$(#(#0(#(#`!a '#$ where  N  Hj ` ` = total head at nodal point j  N   i ` ` = unit vector along the xcoordinate  N  j ` ` = unit vector along the ycoordinate  N`  k ` ` = unit vector along the zcoordinate and where QRX(I), QRY(I), and QRZ(I) are 3DFEMWATER program variables. A.1.14 Subroutine Q8TH This subroutine, which is called by subroutine SFLOW, is used to compute the contribution from an element to the change in water content over time, using the following equation:   #x"ddddd- dd9h`j ^ (A4)bFUNC { QTHP ~ = ~ int from {R sub e} to \ {d theta} over {dh} {partial h} over {partial t} dR ~ }  QTHPdd dd4R<<e$0d0$|dh/0,0h/|,|t dR $(#(#(#(#`! '#$where h` ` = pressure head  ` ` = moisture content QTHP` ` = 3DFEMWATER program variable p$0*((Q8'#m  H '#}! '#IA '#a "'#% p A.1.15 Subroutine READN This subroutine is called by subroutine DATAIN to generate integer numbers for input data sets 8, 9, 12(c), 12(f), 14(b) through 14(d), 15(c), 16(b), 16(c), 17(b), and 17(c), which are described in Section 4.1. A.1.16 Subroutine READR This subroutine is called by subroutine DATAIN to generate real numbers input for data sets 7, 14(e), and 14(f) (see Section 4.1). Automatic generation of regularly patterned data is built into this subroutine. A.1.17 Subroutine SFLOW This subroutine is called by subroutine GW3D. It is used to compute the fluxes through various types of boundaries and the rate at which water content increases in the region of interest. In this subroutine, the function of variable FRATE(7) is to store the flux through the whole boundary enclosing the region of interest. It is given by:  N  #xddddd$ dd9h`nb(A5)$jFUNC { FRATE(7) ~ = ~ int from B to \ (V sub x n sub x ~ + ~ V sub y n sub y ~ + ~ V sub z n sub z)dB ~ }  FRATE(g7)x%q%dd6 dd6"B(VddMxnddxt!Vddyndd)y[ Vdd z nddc z ) dB$߰$(#(#(#(#`! '#$where Vx, Vy, and Vz are Darcy velocity components, and nx, ny, and nz are the directional cosines of the outward unit vector normal to the boundary B. FRATE(1) through FRATE(5) store the flux  N through the Dirichlet boundary, BD, specifiedflux boundary, BC, specifiedpressurehead boundary,  N BN, the seepage/evapotranspiration boundary, BS, and infiltration boundary, BR, respectively, and are given by:  #x(ddddd' dd9h` a(A6a)=sFUNC { FRATE(1) ~ = ~ int from {B sub d} to \ (V sub x n sub x ~ + ~ V sub y n sub y ~ + ~ V sub z n sub z) dB ~ }  FRATE(g1)x&&dd7 dd%4B<<bd(VddPxnddxw$Vddyndd,y^ Vdd z nddf z ) dB=$(#(#x(#(#`! '#$  #x ddddd' dd9h` a(A6b)=sFUNC { FRATE(2) ~ = ~ int from {B sub c} to \ (V sub x n sub x ~ + ~ V sub y n sub y ~ + ~ V sub z n sub z) dB ~ }  FRATE(g2)x&&dd7 dd%4B<<bc(VddPxnddxw$Vddyndd,y^ Vdd z nddf z ) dB=$(#(#D(#(#`! '#$  #x$ddddd' dd9h` a(A6c)=sFUNC { FRATE(3) ~ = ~ int from {B sub n} to \ (V sub x n sub x ~ + ~ V sub y n sub y ~ + ~ V sub z n sub z) dB ~ }  FRATE(g3)x&&dd7 dd%4B<<bn(VddPxnddxw$Vddyndd,y^ Vdd z nddf z ) dB=$(#(# (#(#`! '#$`p#0*((A'# ('#]  '#)$ $'#' ` ! #xddddd' dd9h` a(A6d)=sFUNC { FRATE(4) ~ = ~ int from {B sub s} to \ (V sub x n sub x ~ + ~ V sub y n sub y ~ + ~ V sub z n sub z) dB ~ }  FRATE(g4)x&&dd7 dd%4B<<bs(VddPxnddxw$Vddyndd,y^ Vdd z nddf z ) dB=$(#(#l(#(#`!! '#$ A #xddddd' dd9h` a(A6e)=sFUNC { FRATE(5) ~ = ~ int from {B sub r} to \ (V sub x n sub x ~ + ~ V sub y n sub y ~ + ~ V sub z n sub z) dB ~ }  FRATE(g5)x&&dd7 dd%4B<<br(VddPxnddxw$Vddyndd,y^ Vdd z nddf z ) dB=$(#(#8(#(#`!A '#$ FRATE(6), which is related to the numerical loss, is given by:  a #x| dddddN dd9h`YM(A7)DFUNC { FRATE(6) ~ = ~ FRATE(7) ~ ~ sum from {I=1} to 5 FRATE(I) }  FRATE(g6)x%FRATE(}7);qIddx]5dd<"Iddx"dd"1 FRATE (v I ) $(#(#H (#(#`!a '#$FRATE(8) and FRATE(9) are used to store the source/sink and increased rate of water accumulation within the media, respectively:  #xdddddldd9h% r5 `J > (A8) ă 1FUNC { FRATE(8) ~ = ~ int from R to \ q dR ~ }  FRATE(g8)x%qdd dd"RqKdR ߬$(#(#X(#(#`! '#$and  #x@ddddddd9` (A9)aNFUNC { FRATE(9) ~ = ~ int from {R} to \ F {partial h} over {partial t} dR ~ }  FRATE(g9)x%q%dd6 dd6"RF,yhj,yjtdRa$(#(#(#(#`! '#$If there is no numerical error in the computation, the following equation should be satisfied:  #xxdddoo dd9h`)(A10)c4FUNC { FRATE(9) ~ = ~ [FRATE(7) ~ + ~ FRATE(8)] }  .FRATE.(g.9.)x.%..[.FRATE.(E.7.)V. .FRATE .([ .8 .)# .]c$(#(#(#(#! '#$and FRATE(6) should be equal to zero. A.1.18 Subroutine SOLVE This subroutine is called by the subroutine BLKITR to solve a matrix equation of the type: \ #x#dddoodd9h` ( (A11)FUNC { [C]\{x\} ~ = ~ \{y\} }  .[s.C.];.{.x.}.].{.y%.}\$(#(#`(#(#! '#$where [C] is the coefficient matrix and {x} and {y} are two vectors. {x} is the unknown to be solved, and {y} is the known load vector. The computer returns the solution and stores it in {y}. The computation is a standard banded Gaussian direct elimination procedure. H$0*((q'#Q! '# A  '#a '#+ @'#c x'#d #'#$ A.1.19 Subroutine SPROP This subroutine is called by subroutine GW3D. It calculates the values for moisture content, relative hydraulic conductivity, and water capacity, using the van Genuchten functional relationships (see Equations 33a through 33d). A.1.20 Subroutine STORE This subroutine, which is called by GW3D, is used to store the flow variables in a binary file. The stored data is intended for use in 3DLEWASTE or for plotting. The information stored includes region geometry, subregion data, and hydrological variables such as pressure head, total head, moisture content, and Darcy velocity components. A.1.21 Subroutine SURF Subroutine SURF is called by subroutine DATAIN. It identifies the boundary sides, sequences the boundary nodes, and computes the directional cosine of the surface sides. The mappings from boundary nodes to global nodes are stored in NPBB(I) (where NPBB(I) is the global node number of the Ith boundary node). The boundary node numbers of the four nodes for each boundary side are stored in ISB(I,J) (where ISB(I,J) is the boundary node number of Ith node of Jth side, I = 1 to 4). There are six sides for each element. Which of these six sides is the boundary side is determined automatically in the subroutine SURF and is stored in ISB(5,J). The global element number, to which the Jth boundary side belongs, is also preprocessed in the subroutine SURF and is stored in ISB(6,J). The directional cosines of the Jth boundary side are computed and stored in DCOSB(I,J) (where DCOSB(I,J) is the directional cosine of the Jth surface with Ith coordinate, I = 1 to 3). The information contained in NPBB, ISB, and DOSB, along with the number of boundary nodes and the number of boundary sides is returned to subroutine DATAIN for other uses. A.1.22 Subroutine VELT This subroutine is called by subroutine GW3D. It calls subroutine Q8DV to evaluate the element matrices and the derivatives of the total head. It then sums over all element matrices to form a matrix equation governing the velocity components at all nodal points. To save computational time, the matrix is diagonalized by lumping. The velocity components can thus be solved point by point. The computed velocity field is then returned to GW3D through the argument. This velocity field is also passed to subroutine BCPREP to evaluate the Darcy flux across the seepageinfiltrationevapotranspiration surfaces. "0*(( A.2 3DLEWASTE LEWASTE consists of a main program, LEWAST3D, 30 subroutines, and a function. Figure A.2 shows the structure of the program. The subroutines and function are listed in Tables A2 and A3, respectively, and the purposes of the subroutines are briefly described below. A.2.1 Subroutine ADVBC This subroutine is called by GM3D to implement the boundary conditions. For a Dirichlet boundary, the Lagrangian concentration is specified. For variable boundaries, if the flow is directed out of the region, the fictitious particle associated with the boundary node must come from the interior nodes. Hence the Lagrangian concentration for the boundary node has already been computed from subroutine ADVTRN and the implementation for such a boundary segment is bypassed. For variable boundaries, if the flow is directed into the region, the concentration of incoming fluid is specified. The Lagrangian concentration is then calculate as:  #x0ddddd dd9xh`G (A12)zFUNC { C sub i sup * = int from {B sub v sup {} } to \ N sub i V sub n C sub \in dB / int from {B sub v sup {} } to \ N sub i V sub n dB ~ }  CddDddui**dd; dd$4B<<j`<<a%vNddi0VddnCdd8indBx/dd dd4B<<"`<<%vFNddiVddNndBz$(#(# (#(#`! '#$where  N Ci* ` ` = Lagrangian concentration at the boundary node i  N Vn ` ` = normal vertically integrated Darcy velocity  Nh Cin ` ` = concentration of incoming fluid B` ` = global boundary of the region of interest  N !0LLdd9<<M N sub i sup e LNddeddu4iM ` ` = basis function for nodal point i of element eJHHZ Specifiedflux (Cauchy) boundary conditions are normally applied to the boundary where flow is directed into the region, where the material flux of incoming fluid is specified. The Lagrangian concentration is thus calculate as: ! #xddddd dd9xh` (A13)uFUNC { C sub i sup * = int from {B sub c } to \ N sub i q sub c dB / int from { B sub c} to \ N sub i V sub n dB ~ }  Cdd.ddui%%dd6 dd$4B<<acNddi'qddcdB/dd dd4B<<2cVNddiVdd^ndBߪ$(#(#g(#(#`!! '#$  N3 where Ci* is the Lagrangian concentration at the boundary node i and qc is the Cauchy flux of the incoming fluid. A.2.2 Subroution ADVTRN This subroutine is called by GM3D to compute the Lagrangian concentrations at all nodes. It calls subroutine MPLOC to find which element a fictitious particle is located in. It also calls subroutine XSI3D to compute the local coordinate, given the global coordinate, of the fictitious particle. If the fictitious particle associated with a particular node is located in the @$0*((!0'#{ '#L!! @  r5 #d6X@K0@#` hpH\!"X` hp x (#%'0*,.8135@8:GRAD(C) over an element.(# Q8R` `  %SFLOWhh4<Computes the material integration and element source integration over an element.(# READN` `  %DATAIN hh4X<Automatically generates integer input if required.(#%0*((   TABLE A2. SUBROUTINES INCLUDED IN 3DLEWASTE (concluded)   Subroutine` `  %Called Byhh4<Description READR` `  %DATAIN hh4X<Automatically generates real number input if required.(# SFLOW` `  %GM3D- hh4X<Computes the flux rates through various types of boundaries and the rate at which material increases in the region of interest.(# SHAPE` `  %Q8DV, Q8 hh4X<Computes the base and weighting functions, their derivatives with respect to X, Y, Z, and the Jacobian at a Gaussian point.(# SOLVE` `  %BLKITRhh4<Solves a matrix equation with a band matrix solver.(# STORE` `  %GM3D- hh4X<Stores pertinent quantities on a auxiliary device for future uses (e.g., for plotting). (# SURF` `  %DATAINhh4<Identifies the boundary sides and sequences of the boundary nodes, and computes the directional cosine of the surface sides.(# THNODE` `  %GM3D- hh4X<Computes moisture content at a node.(# XSI3D` `  %ADVTRNhh4<Computes the local coordinate of an element given the global coordinate within that element.(# 0*(( TABLE A3. FUNCTIONS INCLUDED IN 3DLEWASTE  Function` `  %Called Byhh4<Description FCOS` `  %MPLOChh4<Computes the inner product of an outward normal of the surface with a vector connecting a point on the surface and the fictitious particle to determine if the fictitious particle lies inside the surface.(# p0*(( interior of the region, the Lagrangian concentration is obtained by finite element interpolation of the concentration at the previous time step. If the fictitious particle associated with a particular node is outside the region of interest, the Lagrangian concentration is set equal to the previous timestep concentration of the boundary node that is closest to the fictitious particle. A.2.3 Subroutine AFABTA This subroutine, which is called by subroutine GM3D, calculates the values of upstream weighting factors along 12 sides of all elements. A.2.4 Subroutine ALLFCT This subroutine is called by subroutine GM3D to compute values for all the source/sink and boundary nodes and elements. It uses linear interpolation of tabular data to simulate variations in time for these conditions. A.2.5 Subroutine ASEMBL This subroutine is called by subroutine GM3D. After calling subroutine Q8 to evaluate the element matrices, it sums over all element matrices to form a global matrix equation governing the concentration distribution at all nodes. A.2.6 Subroutine BC This subroutine, which is called by subroutine GM3D, incorporates Dirichlet, variable composite, specifiedflux, and specifieddispersive flux boundary conditions. For a Dirichlet boundary condition, an identity algebraic equation is generated for each Dirichlet nodal point. Any other equation having this nodal variable is modified accordingly to simplify the computation. For a variable composite surface, the integration of the normal velocity times the incoming concentration is added to the load vector and the integration of normal velocity is added to the matrix. For the specifiedflux boundaries, the integration of flux is added to the load vector and the integration of normal velocity is added to the matrix. For a specifieddispersiveflux boundary, the integration of gradient flux is added to the load vector. A.2.7 Subroutine BLKITR This subroutine is called by subroutine GM3D to solve the matrix equation with block iteration methods. For each subregion, a block matrix equation is constructed based on the global matrix equation and two pointer arrays, GNPLR and LNOJCN (see subroutine PAGEN). The resulting block matrix equation is solved with the direct band matrix solver by calling subroutine SOLVE. This is done for all subregions for each iteration until a convergent solution is obtained. '0*((ԌA.2.8 Subroutine DATAIN Subroutine DATAIN is called by subroutine GM3D. It reads and prints all data input described in the Section 4.2 except data set 1. It also calls subroutine SURF to identify the boundary segments and boundary nodes and subroutines READR and READN, respectively, to automatically generate real and integer numbers. A.2.9 Subroutine FLUX This subroutine is called by subroutine GM3D. It calls subroutine Q8DV to evaluate the element matrices and the derivatives of concentrations. It then sums over all element matrices to form a matrix equation governing the flux components at all nodal points. To save computational time, the matrix is diagonalized by lumping. The flux components due to dispersion can thus be solved point by point. The flux due to velocity is then added to the computed flux due to dispersion. The computed total flux field is then returned to GM3D through the argument. A.2.10 Subroutine GM3D The subroutine GM3D controls the entire sequence of operations. It performs either a steadystate computation alone (KSS = 0 and NTI = 0), or a transient state computation using the steadystate solution as the initial condition (KSS = 0, NTI > 0), or a transient computation using usersupplied initial conditions (KSS = 1, NTI > 0). GM3D calls subroutine DATAIN to read and print input data; subroutine LELGEN to generate the pointer array element stencil that describes all elements connected to any node; subroutine ALLFCT to obtain sources/sinks and boundary values; subroutine AFABTA to obtain the upstream weighting factor based on velocity and dispersivity (the upstream weighting factor is needed for solving the steadystate option of 3DLEWASTE); subroutine FLUX to compute material flux; subroutine ASEMBL to assemble the element matrices over all elements; subroutine BC to implement the boundary conditions; subroutine BLKITR to solve the resulting matrix equations with block iteration methods; subroutine SFLOW to calculate flux through all types of boundaries and the water accumulated in the media; subroutine PRINTT to print out the results; subroutine STORE to store the results for plotting; subroutine THNODE to compute the value of moisture content plus bulk density times distribution coefficient in the case of a linear isotherm, or the moisture content in the case of a nonlinear isotherm at all nodes; subroutine NDTAU to compute the number of subtime steps nd the subtime step sizes used for integration in the Lagrangian step; ADVTRN to compute the Lagrangian concentrations at all nodes; and subroutine ADVBC to implement boundary conditions in the Lagrangian step. p#0*(( A.2.11 Subroutine LELGEN This subroutine is called by subroutine GM3D to preprocess the pointer array (the global elements stencil), LRL(K,N), where LRL(K,N) is the global element number of the Kth element connected to the global node N. This pointer array is generated based on the element connectivity IE(M,J). Here IE(M,J) is the global node number of the Jth node of element M. This pointer array is needed to facilitate the location of fictitious particles. A.2.12 Subroutine MPLOC This subroutine is called by NDTAU and ADVTRN to locate the fictitious particle associated with a particular node. It uses the function FCOS to compute the product of the outward unit vector with the vector from a node on the surface to the fictitious particle. A.2.13 Subroutine NDTAU This subroutine is called by GM3D to compute the subtimestep size and the number of subtime steps such that no fictitious particle travels over an element within a subtime step. The subtimestep size and the number of subtime steps are used in subroutine ADVTRN.  A.2.14 Subroutine PAGEN This subroutine is called by subroutine DATAIN to preprocess pointer arrays that are needed to store the global matrix in compressed form and to construct the subregional block matrices. The pointer arrays automatically generated in this subroutine include the global node connectivity (stencil), GNOJCN(J,N), regional node connectivity, LNOJCN(J,I,K), total node number for each subregion, NTNPLR(K), the bandwidth indicator for each subregion, LMAXDF(K), and a partial fillup for the mapping array between global node number and local subregion node number, GNPLR(I,K), with I = NNPLR(K) + 1 to NTNPLR(K). Here GNOJCN(J,N) is the global node number of the Jth node connected to the global node N; LNOJN(J,I,K) is the local node number of the Jth node connected to the local node I in Kth subregion; NTNPLR(K) is the total number of nodes in the the Kth subregion, including the interior nodes, the global boundary nodes, and intraboundary nodes; LMAXDF(K) is the maximum difference between any two nodes of any element in the Kth subregion; and GNPLR(I,K) is the global node number of the Ith localregion node in the Kth subregion. These pointer arrays are generated based on the element connectivity, IE(M,J), the number of nodes for each subregion, NNPLR(K), and the mapping between global node and localregion node, GNLR(I,K), with I = 1, NNPLR(K). Here IE(M,J) is the global node number of the Jth node of element M; NNPLR(K) is the number of nodes in the Kth subregion, including the interior nodes and the global boundary nodes, but not the intraboundary nodes. %0*(( A.2.15 Subroutine PRINTT This subroutine, which is called by GM3D, is used to lineprint the simulation results. These include the fluxes through variable boundary surfaces, the concentration, and vertically integrated material flux components. A.2.16 Subroutine Q4ADB This subroutine is called by subroutine ADVBC and implements Dirichlet, specifiedflux, and variable boudary conditions in a Lagrangian step computation. A.2.17 Subroutine Q4BB This subroutine is called by subroutine SFLOW to perform surface integration of the following type: A #xdddddl dd9x3`f (A14)E?FUNC { RRQ(I) = int from {B sub e} to \ N sub i sup e F dB ~ }  RRQ;(I)gdd dd4B<<e,Ndd.eddiFKdBE$(#(#X(#(#`!A '#$where F is the normal flux and RRQ(I) is a 3DLEWASTE program variable. A.2.18 Subroutine Q4CNB This subroutine is called by the subroutine BC to compute the surface node flux of the type:  N a #xxddddd dd9x3`4 (A15a)C>FUNC { RQ(I) = int from {B sub e} to \ N sub i sup e q dB ~ }  RQ(;I)hhddy ddg4B<<eNddG.edd.iqdBC$(#(#(#(#`!a '#$where q is either the specified (or Cauchy) flux, specifieddispersive (or Neumann) flux, or n . V Cv; and RQ(I) is a 3DLEWASTE program variable. It also computes the boundary element matrices:  N > #x ddddd dd9x3`s/ (A15b)[FUNC { BQ(I,J) = int from {B sub e} to \ N sub i sup e bold V }FUNC { N sub j sup e dR ~ }  BQ(;I,Jg)00ddA dd/4B<<leNdd.eddiKVNdd..eddjjdR>$(#(#(#(#`! '#$where A0#LLdd9<<M N sub j sup e LNddeddu4jM is the basis function for nodal point j of element e, R is the region of interest, V is the Darcy velocity, and BQ(I,J) is a 3DLEWASTE program variable. P"0*((1'#=A x'#a  '## P A.2.19 Subroutine Q8 This subroutine is called by the subroutine ASEMBL to compute the element matrix given by: 6 #x8ddddd dd9x3`s/ (A16a)RFUNC { QA(I,J) = int from {R sub e} to \ N sub i sup e theta N sub j sup e dR ~ }  QA(;I,Jg)00ddA dd/4R<<leNdd.eddiKNdd..eddjjdR6$(#(#(#(#`! '#$  #x ddddd6 dd9x3`Y (A16b)fFUNC { QAA(I,J) = int from {R sub e} to \ N sub j sup e rho sub b {dS} over {dC} N sub j sup e dR ~ }  QAA;(I,gJ)/dd dd4R<<eNdds.eddZj#ddbh}0dS}|dCZNdd.eddjdR߆$(#(#T(#(#`! '#$  #xddddd dd9x3`m (A16c)cyFUNC { QB(I,J) = int from {R sub e} to \ (grad N sub i sup e) cdot theta bold D cdot }FUNC { (grad N sub j sup e) dR ~ }  QB(;I,Jg)00ddA dd/4R<<le(+XNdd.eddi)w?D(k+NddN.edd5j)dRc$(#(# (#(#`! '#$  #xddddd dd9x3`g (A16d)gFUNC { QV(I,J) = int from {R sub e} to \ N sub i sup e bold V cdot }FUNC { (grad N sub j sup e) dR ~ }  QV(;I,Jg)00ddA dd/4R<<leNdd.eddiKV(w+NddZ.eddAj)dRߌ$(#(# (#(#`! '#$ 1! #xhdddddV dd9x3` (A16e)~FUNC { QC(I,J) = int from {R sub e} to \ N sub i sup e [lambda ( theta + rho sub b {dS} over {dC} ) + Q] N sub j sup e dR ~ }  QC(;I,Jg)00ddA dd/4R<<leNdd.eddiK[(w?#ddb 0dS |dC)NQ ]z Ndd .edd j5 dR1$(#(#(#(#`!! '#$where  N Cw` ` = dissolved concentration at the previous iteration  N  D ` ` = dispersion coefficient tensor  ` ` = moisture content S` ` = species concentration in the adsorbed phase Q` ` = source rate of water  N* #b` ` = bulk density of the porous medium ` ` = material decay constant + ` ` = del operator indicating gradient +` ` = del operator indicating divergence and where QA(I,J), QAA(I,J), QB(I,J), QV(I,J), and QC(I,J) are 3DLEWASTE program variables.  N: Note that dS/dC should be evaluated at Cw. Subroutines Q8 also calculates the element load vector given by: p~ 0*((Q8'#m  '#9 '# '# h'#! p  N IA #xddddd dd9xA`z(A16f)FUNC { QR(I) = int from {R sub e} to \ N sub i sup e [ lambda rho sub b (S sub w {dS} over {dC} C sub w ) + Q C sub \in ] dR ~ }  QR(;I)hhddy ddg4R<<eNddG.edd.i[K#ddbO(SddwU0dS|dCCdd*wf). Q Cdd inp ] dRI$(#(#(#(#`!A '#$where Cw and Sw are the dissolved and adsorbed concentrations at the previous iteration, respectively, and QR(I) is a program variable.  A.2.20 Subroutine Q8DV Subroutine Q8DV is called by subroutine FLUX to compute the element matrices given by: a #x ddddd' dd9xA`a (A17a)LFUNC { QB(I,J) = int from {R sub e} to \ N sub i sup e N sub j sup e dR ~ }  QB(;I,Jg)00ddA dd/4R<<leNdd.eddiKNdd.eddjdR$(#(#H (#(#`!a '#$ Subroutine Q8DV also evaluates the element load vector:  #xddddd dd9xA` (A17b)fFUNC { QRX(I) = int from {R sub e} to \ N sub i sup e bold i cdot }FUNC { theta bold D cdot } FUNC { (grad N sub j sup e) C sub j dR ~ }  QRX;(I)g00ddA dd/4R<<leNdd.eddiKiwD?(+Ndd.eddmj)&CddjdRf$(#(# (#(#`! '#$  #xhddddd dd9xA` (A17c)eFUNC { QRY(I) = int from {R sub e} to \ N sub i sup e bold j cdot }FUNC { theta bold D cdot }FUNC { (grad N sub j sup e) C sub j dR ~ }  QRY;(I)g00ddA dd/4R<<leNdd.eddiKjwD?(+Ndd.eddmj)&CddjdRe$(#(#(#(#`! '#$  #x4ddddd dd9xA` (A17d)fFUNC { QRZ(I) = int from {R sub e} to \ N sub i sup e bold k cdot } FUNC { theta bold D cdot }FUNC { (grad N sub j sup e) C sub j dR ~ }  QRZ;(I)g00ddA dd/4R<<leNdd.eddiKkwD?(+Ndd.eddmj)&CddjdRf$(#(#(#(#`! '#$where  N Cj ` ` = concentration at nodal point j  N   i ` ` = unit vector along the xdirection  Nn   j ` ` = unit vector along the ycoordinate  NH   k ` ` = unit vector along the zcoordinate and where QRX(I), QRY(I), AND QRZ(I) are program variables. A.2.21 Subroutine Q8R This subroutine, which is called by subroutine SFLOW, is used to compute contributions to FRATE(8), FRATE(9), FRATE(1), and FRATE(14), discussed in Section A.2.24, by performing material integration and element source integration over an element. pN$0*((Q'#A  '#-a '# h'# 4'#i p a #xddddddd9xA` !(A18a),FUNC { QRM = int from R to \ theta C dR ~ }  QRM;qdd dd"RaCdRa$(#(#(#(#`! '#$ J #x|ddddddd9xA` !(A18b)&FUNC { QDM = int from R to \ S dR ~ }  QDM;qdd dd"RSadRJ$(#(#(#(#`! '#$ ! #xH ddddddd9xA`h (A18c)yWFUNC { SOSM = int from R to \ [Q C sub \in (1 + sign(Q)) + QC (1 sign (Q))]/2 dR ~ }  SOSMqdd dd"Ra[Q)Cddin(k13sign('Q))SQC ( 1G  sign; ( Q )g ) ]//2dRy$(#(#(#(#`!! '#$ where QRM, QDM, and SOSM are 3DLEWASTE program variables. A.2.22 Subroutine READN This subroutine is called by subroutine DATAIN to generate integer numbers for the input data sets if required. A.2.23 Subroutine READR This subroutine is called by subroutine DATAIN to automatically generate real numbers for the input data sets if required. Automatic generation of regularly patterned data is built into this subroutine. A.2.24 Subroutine SFLOW This subroutine is called by subroutine GM3D. It is used to compute flux rates through various types of boundaries and the rate at which material increases in the region of interest. In this subroutine, the variable FRATE(7) stores the flux through the whole boundary. It is given by:  N EA #x8"ddddddd9xA` (A19)OFUNC { FRATE(7) = int from B to \ ( F sub x n sub x + F sub y n sub y ) dB ~ }  FRATE(g7)/qdd dd"B(UFddxndd]xFddcynddyA)dBE$(#(#(#(#`!A '#$where B is the global boundary of the region of interest; Fx, and Fy are the vertically integrated  N! flux components and nx and ny are the directional cosines of the outward unit vector normal to the  N" boundary B. FRATE(1) stores the flux rates through a Dirichlet boundary Bd. FRATE(2) and FRATE(3) store the flux rate through specifiedflux (Cauchy) and specifieddispersiveflux (Neumann) boundaries, respectively. ` %0*((A'# |'#  H '#k! 8"'#[%A ` \a #xddddd dd9x`T (A20a)UFUNC { FRATE(1) = int from {B sub d} to \ (F sub x n sub x + F sub y n sub y) dB ~ }  FRATE(g1)/dd dd4B<<d(XFddxndd`xFddfynddyD)dB\$(#(#(#(#`!a '#$ \ #xddddd dd9x`T (A20b)UFUNC { FRATE(2) = int from {B sub c} to \ (F sub x n sub x + F sub y n sub y) dB ~ }  FRATE(g2)/dd dd4B<<c(XFddxndd`xFddfynddyD)dB\$(#(#8(#(#`! '#$ \ #x ddddd dd9x`T (A20c)UFUNC { FRATE(3) = int from {B sub n} to \ (F sub x n sub x + F sub y n sub y) dB ~ }  FRATE(g3)/dd dd4B<<n(XFddxndd`xFddfynddyD)dB\$(#(#p(#(#`! '#$FRATE(4) and FRATE(5) store incoming flux and outgoing flux rates, respectively, through the  N variable boundaries Bvé and Bv+, as given by: w #x0ddddd dd9xhA`? (A20d) ^FUNC { FRATE(4) = int from {B sub {v sup } } to \ (F sub x n sub x + F sub y n sub y) dB ~ }  FRATE(g4)/dd dd4B<<v<<1(Fddx#nddx)Fddyndd1ym)dBw$(#(# (#(#`! '#$  NL v #xddddd dd9xA`? (A20e)^FUNC { FRATE(5) = int from {B sub {v sup +} } to \ (F sub x n sub x + F sub y n sub y) dB ~ }  FRATE(g5)/dd dd4B<<v<<1(Fddx#nddx)Fddyndd1ym)dBv$(#(#L(#(#`! '#$where a0i}LLdd9<<PB sub {V sup } LBdds"V<<BP and 0y i}LLdd9<<PB sub {V sup +} LBdds"V<<BP are that part of variable boundary where the fluxes are directed into the region and out from the region, respectively. The integration of Equations A20a through A20e is carried out by the subroutine Q4BB. FRATE(6), which is related to the numerical loss, is given by: #x|ddddd* dd9xA`(A21)>FUNC { FRATE(6) = FRATE(7) sum from {I=1} to {5} FRATE(I) }  FRATE(g6)/FRATE(7O)qIddT]5dd"IddT"dd"1FRATE (R I )$(#(#(#(#`!'#$FRATE(8) and FRATE(9) store the accumulate rate in the dissolved and adsorbed phases, respectively, as given by: !#x"ddddddd9xA`/ (A22) 1FUNC { FRATE(8) = int from R to \ theta C dR ~ }  FRATE(g8)/qdd dd"RUCdR ߗ$(#(#(#(#`!!'#$ A#xp&ddddddd9xA` i (A23)5FUNC { FRATE(9) = int from R to \ rho sub b S dR ~ }  FRATE(g9)/qdd dd"R#ddUbSdR߫$(#(#!(#(#`!A'#$FRATE(10) stores the rate loss due to decay and FRATE(11) through FRATE(13) are set to zero as given by: $a#xddddddd9x` (A24)JFUNC { FRATE(10) = int from R to \ lambda ( theta C + rho sub b S) dR ~ }  FRATE(g10/)qdd dd"RU(CI#ddbSM)dR$ #xdddoo dd9xA`2(A25)O-FUNC { FRATE(11)=FRATE(12)=FRATE(13) = 0 ~ }  .FRATE.(g.11/.)..FRATE.(O.12.){..FRATE .(7 .13 .)c . .0O FRATE(14) is used to store the source/sink rate as:'0*(( '#a '#  '#U 0'#e '#1 '#s "'#%!p&'#)A'#a'# Ԍ$(#(#(#(#!a'#$Й9#x ddddd dd9xA`n (A26)XFUNC { FRATE(14) = int from R to \ [Q C sub \in (1+ sign(Q)) + QC(1 sign(Q))]/2 dR ~ }  FRATE(g14/)qdd dd"RU[QCddin(_1'sign( Q ) )G  QCs ( 1;  sign/(Q)[)]#/2dR9$(#(#(#(#`!'#$ If there is no numerical error in the computation, the following equation should be satisfied: #x|ddddddd9xA` (A27)-FUNC { sum from {I=7} to {14} FRATE(I) = 0 }  qIdd.]14dd"IddL"dd"7FRATE(JI)v0߂$(#(# (#(#`!'#$ and FRATE(6) should be equal to zero. A.2.25 Subroutine SHAPE This subroutine is called by subroutines Q8DV and Q8 to evaluate the value of the base and weighting functions and their derivatives at a Gaussian point. A.2.26 Subroutine SOLVE This subroutine is called by the subroutine BLKITR to solve a matrix equation of the type: \#xdddoodd9xA` q !(A28)FUNC { [C] \{x \} = \{ y \} }  .[s.C.];.{.x.}g..{/.y.}\$(#(# (#(#!'#$where [C] is the coefficient matrix and {x} and {y} are two vectors. {x} is the unknown to be solved, and {y} is the known load vector. The computer returns the solution {y} and stores it in {y}. The computation is a standard banded Gaussian direct elimination procedure. p0*((Q'#a'#  '# '#'#p A.2.27 Subroutine STORE This subroutine, which is called by subroutine GM3D, stores the simulation results in a binery file for use in plotting. The information stored includes regional geometry, concentrations, and vertically integrated material flux components at all nodes for any desired time step. A.2.28 Subroutine SURF Subroutine SURF is called by subroutine DATAIN. It identifies the boundary sides, sequences the boundary nodes, and computes the directional cosine of the surface sides. The mappings from boundary nodes to global nodes are stored in NPBB(I) (where NPBB(I) is the global node number of the Ith boundary node). The boundary node numbers of the four nodes for each boundary side are stored in ISB(I,J) (where ISB(I,J) is the boundary node number of the Ith node of the Jth side, I = 1 to 4). There are six sides for each element. Which of these six sides is the boundary side is determined automatically in the subroutine SURF and is stored in ISB(5,J). The global element number, to which the Jth boundary side belongs, is also preprocessed in the subroutine SURF and is stored in ISB(6,J). The directional cosines of the Jth boundary side are computed and stored in DCOSB(I,J) (where DCOSB(I,J) is the directional cosine of the Jth surface with Ith coordinate, I = 1 to 3). The information contained in NPBB, ISB, and DOSB, along with the number of boundary nodes and the number of boundary sides is returned to subroutine DATAIN for other uses. A.2.29 Subroutine THNODE  N This subroutine is called by GM3D to compute ( +#bdS/dC) for the linear isotherm model or  for the Freundlich and Langmuir nonlinear isotherm models. A.2.30 Subroutine XSI3D This subroutine is called by ADVTRN to compute the local coordinate of an element given the global coordinate within that element. With the local coordinate, the Lagrangian concentration can then easily be interpolated from those on the nodes of the element. 0*((    :APPENDIX B׃ 3INPUT AND OUTPUT DEVICES H 0*(( TABLE B1. LOGICAL UNITS USED IN 3DFEMWATER  Logical Unit` ` Number-Purpose LUSTO` ` 11 %-Logical unit for storing binary output for ` `  %-use in 3DLEWASTE or for plotting purposes. LUBAR` ` 13 %-Logical unit for storing binary boundary ` `  %-arrays, if they are generated in the present ` `  %-job, for use in subsequent executions of ` `  %-the same scenario. LUPAR` ` 14 %-Logical unit for storing binary pointer ` `  %-arrays, if they are generated in the present ` `  %-job, for use in subsequent executions of ` `  %-the same scenario. LUINP` ` 15 %-Logical unit for reading input data. LUOUT` ` 16 %-Logical unit for writing output data.  TABLE B2. LOGICAL UNITS USED IN 3DLEWASTE  Logical Unit` ` Number-Purpose LUFLW` ` 11 %-Logical unit for reading flow data from the ` `  %-3DFEMWATER simulation. LUSTO` ` 12 %-Logical unit for storing binary output for ` `  %-use in 3DLEWASTE or for plotting purposes. LUBAR` ` 13 %-Logical unit for storing binary boundary ` `  %-arrays, if they are generated in the present ` `  %-job, for use in subsequent executions of ` `  %-the same scenario. LUPAR` ` 14 %-Logical unit for storing binary pointer ` `  %-arrays, if they are generated in the present ` `  %-job, for use in subsequent executions of ` `  %-the same scenario. LUINP` ` 15 %-Logical unit for reading input data. LUOUT` ` 16 %-Logical unit for writing output data.   '0*((Ԍ :APPENDIX C׃ 'DEFAULT VALUES FOR THE MAXIMUM CONTROL PARAMETERS 0*((   TABLE C1. MAXIMUM CONTROL PARAMETERS USED IN 3DFEMWATER  Parameter Definition CppKRDefault Value  iLocation  L8  Maximum ControlIntegers for the Spatial Domain X` hp x (#%'0*,.8135@8: