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SAN D93-7007 Distribution
Unlimited Release Category UC-721Printed October 1993
The SECO Suite of Codes forSite Performance Assessment*
Patrick J. Roache
Ecodynamics Research AssociatesP.O. Box 9229
Albuquerque, NM 87119
ABSTRACT
Modeling for Performance Assessment of the Waste Isolation Pilot Plant (WIPP) has led todevelopment of the SECO suite of codes for groundwater flow, particle tracking, and transport.Algorithm and code developments include the following areas: facilitation of grid convergence tests inmultiple domains; correct treatment of transmissivity factors for unconfined aquifers; efficientmultigrid algorithms; a formulation of brine Darcy flow equations that uses freshwater head as thedependent variable; boundary-fitted coordinates; temporal high order particle tracking; an efficient andaccurate implicit Finite Volume TVD algorithm for radionuclide transport in (possibly) fracturedporous media; accurate calculation of advection via a flux-based modified method of characteristics;and Quality Assurance procedures.
* Work performed under Contract Number 87-6727 for Performant;e Assessment Division (6342),Sandia National Laboratories.
MASTER_._.
I31gilRi_Uf,(j,t_ (..;;i-:l Piii:. b,...'...,_.;;.,',L?,_i i_ UNLIMITED
PREFACE
This SAND report is a reproduction of the identically titled paper published in:
America,. Nuclear Society and American Society of Civil Engineers. High-Level Waste Management:Proceedings of the Fourth Annual International Conference in Las Vegas, NI,', April 26- 30, 1993.American Nuclear Society, La Grange, IL; American Society of Civil Engineers, New York, NY.
iii
THE SECO SUITE OF CODES
FOR. SITE PERFORMANCE ASSESSMENT *
Patrick J. Roache
Ecodynamics Research AssociatesP.O. Box 9229
Albuquerque, NM 87119(505) 843-?445
ABSTRACT regional and local grid simulations; 2-D and 3-D, carte-sign and boundary-fitted coordinate versions; confined
Modeling for Performance Assessment of the Waste (._rtesian) and unconfined (,water table) conditions, in-Isolation pilot Plant (WIPP) has led to development eluding recharge of dry cel_s mad correct treatment ofof the SECO suite of codes for groundwater flow, parti- transmissivity factors; parameterized climatic variations;cle tracking, and transport. Algorithm and code devel- options for node-on-boundary or cell-edge-on-boundaryopments include the following areas: facilitation of grid configurations; spatial grid packing near features; timeconvergence tests in multiple domains; correct treatment step packing near events; automatic time-step error esti-oftransmissivityfactorsforunconfinedaquifers;efficient marionandadaptivetimestepping;semicoarseningmulti-multigridalgorithms;a formulationofbrineDarcy flow gridsolversforefficiency;and highorderaccuracypar-equationsthatusesfreshwaterhead as the dependent ticletracking.The two-phaseflowproblemsincludethevariable;boundary-fittedcoordinates;temporalhighor- followingfeatures:multicomponentchemistry;unusuallyder particletracking;an efficientand accurateimplicit highcapillarypressuresinverytightformations;complexFinite Volume TVD algorithm for radionuclide transport geometries; and super-critical pressures, above the ther-in (possibly) fractured porous media; accurate c_cula- modynamic dome. Modifications to existing FDM andtion of advection via a flux-b_cd modified method of FEM. codes, as well as development of new codes wherecharacteristics; and Quality Assurancc procedures, necessary, have been accomplished. Verification bench-
INTRODUCTION marking exercises and QA (Quality Assurance) aspectsof the software project are significant factors.
As authorized by Congress (Public Law 96-164, 1979),the U.S. Department of Energy (DOE) is developing the SECO-FLOW CODESWaste Isolation Pilot Plant (WIPP) as a facility for the The SECO-FLOW codes [27, 29] perform groundwa-management, storage, and long-term disposal of defense- ter hydrology simulation by solving the time-dependentrelated transuranic (TRU)wastes. Use of the WIPP for partial differential equation for hydraulic head using awaste disposal is contin.gent on performance, assessments, fully implicit formulation. The guiding philosophy for the(PA) to evaluate compliance with applicable regulatmns SECO codes is to produce solutions of verified accuracy ;of the U.S. Environmental Protection Agency. to utilize modern structured programming; to design for
The WIPP is a full-scale mined geologic repository future maintenance; to make the basic problem defini-for TRU radioactive waste located in southeastern New tion efficient; to make the perturbed problem definition
efficient (e.g. for parametric sensitivity studies}; to facil-Mexico, 655 meters below the surface in 255 million year itate grid convergence studies; and to faciiitatelocal areaold salt deposits. Under support from Sandia National simulations within the larger regional area simulation.Laboratories, Ecodynamics has been involved for severalyears in the Computational Fluid Dynamics (CFD) rood- An important aspect of this philosophy is the decou-cling for performance assessments of the disposal system, piing of the problem definition (rock properties, bound-Compliance with environmental regulations requires con- ary conditions, well descriptions, etc.) from the com-sideration of future inadvertent intrusions due to resource putational grid, and of the problcm discretization in theexploration, and significant gas production in waste ma- regional computational grid from the local computationalterial due to corrosion, bacterial action, and radiolysis, grid, both in space and in time.Performance assessment [10] results in a CCDF (Com-plimentary Cumulative Distribution Function) curve for Additional features include: general boundary con-normalized cumulative releases to the accessible enel- ditions; time truncation error estimation and solutionronment. The modeling problems include single phase adaptive time step; parameterized boundary conditionDarcy flow in the overlying aquifer, and two-phase Darcy variations; confined or water table conditions with cor-flow in the collapsed rooms containing the waste. The rect recharge of dry cells; flexible specification of initial
conditions, boundary conditions, rivers/lakes; variableSECO (Sandia - ECOdynamics) single-phase groundwa-ter flow codes developed include the following features: density and viscosity due to variations in salt concentra-decoupling of the problem definition (,_quifer properties, tion; high accuracy particle tracking via adaptive O(At) sboundary conditions, weUs/from the computational grids; algorithms; efficient semi-coarsening multigrid solvcrs.
.Thc work dcscribcd in this report was performed for Sandia National Laboratories under Contract No. 87-6727.
Animation display of time-dependent 2-D head con- The difference between any two _st-order solutions is it-tours and particle tracking is provided by the SECO- self a first-order accurate error estimator, provided thatVIEW code built for the Silicon Graphics IRIS 4D-25 we accept the premise, not rigorously proven for finiteworkstation, difference or finite volume methods [17]but well estab-
lished by experience for parabolic problems, that the so-Enhancement modules already built and verified in- lutions from a locally O(At '_ method are globally O(At m
elude: 2-D and 3-D non-orthogonal feature-adapted gridgeneration [15]; 2-D and 3-D tensor conductivity in non- also.) The method is very cheap to implement becauseit does not require smother implicit matrix solution, nororthogonal grids [15]; 3-D animation display of head con- even another explicit stencil evaluation. However, it doestours and particle tracking; and brine transport coupled not include the effects of time-dependent boundary con-to Darcy flow. Versions and enhancements to be com- ditions or wells, lt is also used in an adaptive time steppleted in the near term include non-zero Reynolds num- algorithm.ber (Forscheimer) effects and a 3-D stretched cartesiancode. Enhancements to be completed in the far terminclude: automatic estimation of spatial discretization Transmissivity Factors for Unconfined Aquiferserror via completed Richardson extrapolation; and non-orthogonal feature-adapted grid solutions. The SECO-FLOW codes correct an error found in (at
least) two commonly used groundwater flow codes in theFacilitation of Grid Convergence Tests in Multiple treatment of transmissivity factors (TF) for unconfinedDomains
aquifers (water table conditions). TF for a computationalcell is simply the fraction of the cell vertical thickness
Grid convergence tests, or other reliable truncation occupied by water. The inter-cell volume fluxes drivenerror estimators, are important to amyCFD project (e.g., by head (pressure) gradients as obtained from Darcy'ssee [15, 16]). Unfortunately, in groundwater flow model- law must be reduced by TF appropriately averaged atcell faces. Inter-cell conductance terms are typically ob-ing (as i_ other geophysical studies) such elementary buttime-consuming exercises are exceptional. Unavoidable t_ined by harmonic averaging. The motivation for usingharmonic averaging is very strong in l-D, giving the ex-uncertainty in the governing physical parameters is of- act answer for piece-wise constant properties. The errorten used as the rationale for neglecting grid convergence comes from applying harmonic averaging not only to con-tests. The WIPP PA group tries to improve accuracy ductances but also to the transmissivity factors. This iswhenever possible, not simply a matter of choosing between two different
Accordingly, considerable effort has been expended but consistent discretizations; the harmonic averaging ofin the SECO codes to facilitate the performance of grid transmissivity factors is wrong.
convergence tests by the users. Although these features This is easily shown by considering a de-watered cell,involve few algorithmic innovations, they represent a con-siderable improvement and departure from traditional for which TF = 0. Then application of harmonic averag-ing gives TF = 0 at all the faces of that cell, regardlessgroundwater code design. The SECO codes use two do- of 'IF values in neighboring cells. Now all conductancesm_ns, a local grid embedded in a larger regional grid. into the cell are zero, and the head (pressure) can neverThis allows both higher resolution in the local grid di-rectly over the repository, and closer alignment of grid change. Even though neighboring cells might developarbitrarily large heads, no flow can occur into the cell,boundaries with natural features particular to the site. which is now permanently de-watered. The rational and
The aquifer properties are defined in a separate dis- simple solution is to use harmonic averaging only for thecretization (called the aquifer-defining grid) which is not other terms in the conductsmces, but to use linear aver-used for computation. The properties on the computa- aging for the transmissivity factors.tional grids (local and regional) are then obtained auto-matically byinterpolation using a conservation-preserving Multigrid Algorithmsinterpolation method of Dukowitz [28], enhanced by P.Knupp to handle unaligr.ed boundaries. Solution of the implicit matrix equation by direct LU
Likewise, boundary conditions are not specified in the decomposition is still used in many groundwater codes,computational grids but in the continuum, and the loca- in spite of its intimids.ting operation count, especiallytion and properties of production wells, recharge wells, in 3-D. This is another reason for the coarse grids typi-and surface water features (lakes/rivers) are defined only cally used. The potential for gains in efficiency by imple-once in map coordinates, menting multigrid methods were clear, and have indeed
been realized, but only with considerable developmentTime Error Estimation effort. The primary difficulty is the enormous range of
matrix coefEcients. This is due to both variable gridspacings and to the unusual range of physical parame-
The most reliable method of establishing convergence ters which must be considered in the probabalistic inves-in discretization error is the performance of systematic tigation of uncertainties for WIPP Performance Assess-grid convergence tests in space and time. Tne SECO ment (using Monte Carlo techniques with Latin Hyper-code design philosophy is to facilitate such tests, _ noted cube Sa_mpling). Not only is the range enormous, (e.g.above. The codes also include an inexpensive time errorestimator, using the difference between backward and for- six orders of magnitude in conductivity) but the spatialward time inter:ration, implemented as an extrapolation, variation is not necessarily monotone.
S. Schaffer [18[ has developed semicoarsening multi- VE- (O,O,_E/dz) (5)grid algorithms in "2-D and 3-D for the single phase Darcyflow problem (with a Symmetric Positive Definite or SPD Brine concentration also enters into the evaluation of Kmatrix) which are robust and achieve true multigrid per- through viscosity/z. The brine continuity equation !s justformance, even with 12 orders of magnitude variation in
coel_cients, and proved tobe less sensitive to computer Sail! V (6)round-off errs'." accumulation than line SOR, J. Ruge u_ =
$
and S. McCormick also developed variants of AMG (al-gebraic multigrid) methods [19] which were successfully but the specific storativity So depends on true (environ-installed into the STAFF2D [20] finite element code, but mental) head for a.n unconfined aquifer. We assume S, isthe AMG solvers have not proven to be as robust as de- not affected by brine concentration.
The combined equations (4) and (6) are convenientsired on problems with large property variations, in the interior, but some of the appeal may be lost when
The two-phase flow problem at the repository itself boundary conditions are considered. There still is an in-(caused by brine flow in the Formation and gas genera- herent appeal to formulating the equations in such a waytion from the waste) has additional difficulties because a as to isolate the effect of a physically small perturbation.system of two nonlinear equations is involved, Newton- However, we stress that this is not a "small perturbation"Raphson llnearization is used, the ranges of grid spacings equation, in spite of the fact that it looks reminiscent ofand cell aspect ratios are extreme, and again physical the Boussinesq approximation. There is no approxima-l:,arameters vary over orders of magnitude. After much tion involved. Also, to allay the concerns of readers whowork, J. Ruge and J. Jones achieved multigrid perfor- know of misinterpretations caused by [21], we emphasizemance using semicoarsening, that there is nothing mathematically significant, about
using freshwater head as the dependent variable, asideBrine Darcy Flow in Terms of Freshwater Head from convenience of interpretation. Freshwater head in
this formulation is just a normalization of pressu_ e usingthe density of fresh water. The density of mercury could
The use of hydrauhc head, rather than pressure, as just as well have been used, and the equations would stillthe dependent variable in constant density Darcy flow be valid.codes has advantages, not the least of which is that headcan be measured directly in the field. Also, head being Boundary-Fitted Coordinatesthe sum of pressure and elevation, it is the gradient of
head, not pressure, which causes flow. Modules have been developed [15]. for non-orthogonalHowever, for a variable density code, e.g. includ- boundary-fitted (or feature-fitted) discretizations of the
ing variable (unsaturated) brine concentration, pressure head equations. Although the use of non-orthogonal co-is used virtually universally. The well-known paper by ordinates in finite difference and finite volume discretiza-Lusczynski [21] indicates that, even for variable density tions is now commonplace, we paid attention to severalflows, the horizontal flow is driven only by gradients in special aspects of the present problems, notably tensorfreshwater head. Although true, we find this mislead- properties with step discontinuities, strict conservation,ing (like some other theorems). It is true for 3-D flows demonstrated convergence [15], and symmetry. A sur-but only for strictly horizontal coordinates in quasi-2-D prising result by S. Steinberg [24] is a proof that matrixcodes, whereas real aquifers have some dip. symmetry cannot be maintained for a nearest-neighbor
discretization in agrid that is non-orthogonal at a bound-Nevertheless, we find some advantage in formulating ary where Robin (mixed) boundary conditions are crnl-the variable density flow equations using freshwater head
as the dependent variable. This results l'n the addition of unfed to second order accuracy.a bouyancy term involving brine to the usual Darcy equa- Near term enhancements to the SECO-FLOW codestion. With z vertical (colinear with g), define freshwater now also include full tensor conductivity and non-zerohead as Reynolds number (Forscheimer) effects, and plan for
special 2-D enhancements such as aquitard leakanceShavebeen dropped in favor of the full 3-D code, presently un-
H I = P/(p, [g 1) + E (1) der verification.
where P is pressure, Pt is freshwater density, and eleva- SECO-TRACKER
tion E is measured along z. Define the usual hydraulic The SECO-TRACKER code is built to provide tem-conductivity poral high-order particle tracking.
K =-klglpj/# (2) For particle tracking, the SECO-TRACKER codesRun e Kutta Feh_Iberuse a g_- - g ODE integrator [221 com-
and the density perturbation bined withlinear velocity interpolation in space and timeto produce particle tracking with verified accuracy [15]
2 5e = (p- Pf)/Pl. (3) of O((Az) ,(At) ). This may appear to be inconsistent
with the flow field accuracy, but is in fact justifiable.Then the Darcy equation with bouyancy (brine) usingfreshwater head is In an intercomparison study of radionuchde transport
codes [23], surprising variability occured in particle track-V = -K[VH_ + eVE] (4] ing results, much more so than the variability in pressure
and velocity fields. Although this poo: perf,',imance was must be treated carefully.noted several times in the analyses of results, no men-tion was made in any of the calculations about the par- Governing Equationsticle tracking algorithms. One might assume that theparticle tracking was performed with algorithms anal-ogous to those used for the Darcy flow solutions, i.e. The radionuclide transport problem consists of NO((Az)2,(At)) or perha.ps O((Az)2, (At) 2) at most. species equations, k = 1,... ,N:
Although plausible at first glance, further thoughtsuggests that this may be inadequate. The trap is that V. [DVCj, - VC_,] = CRk OCk-Ot + CRk)_Cj,the flow codes are Eulerian, whereas the particle track-ing is intrinsically Lagrangian. For a steady Eulerian - ¢Rk-t,X_,-lCk-i - QCk - F_ (7)velocity field with linear vaxiation in space, there is zerotime truncation error in the velocity field. Yet the (La- where the dependent variables are Ck, the concentration
of the k-th radionuclide. Physical parameters includegrangian) solution for paxticle position involves exponen- D(x, t), a 2 x 2 hydrodynamic dispersion tensor (velocity-tim functions in time. The higher time vaxiability of par-ticle position, compared to Eulerian velocity fields, justi- dependent), V(x, t), the Darcy Velocity, ¢(x), the frac-ties the use of higher accuracy tracking algorithms. See ture system porosity, Rk, the retardation coefficient, ,_,also [30]. the species decay constant, and Ck, the concentration of
the k-th injected radionuclide. The well injection rateSECO..TRANSPORT is Q. Detailed physical descriptions of these terms can
An implicit TVD schen'e h_ been developed to solve be found in [1, 2]. A double-porosity (dual-continuum)the two-dimensional radionuclide transport equation irl model [3, 4, 5] requires the additional source term Fk toa (possibly) fractured porous medium [14]. The scheme represent the flux due to the exchange of contaminantuses a finite volume staggered mesh. The 2-D governing between the fracture and matrix domain. The N equ_-equation is solved using an efficient Approximate Factor- tions are linear and, in the earlier code, wer,_ sequentiallyization procedure with an implicit treatment of boundary coupled. A general Robin boundary condition is used,conditions. This method has been applied to test prob-
lems and was verified to be be second order accurate in aCj, + flOCk,space and time for both high and low mesh Peclet hum- On =: ")' (8)hers.
on a planar rectangular domain f2. For various choiceNumerous codes solve the solute transport equations. ' " " " uof a(x), ft(x), and 7rx), one may obtain Dmchlet, Ne -
Many were written over ten years ago and use outdated mann, or Cauchy Boundary conditions on different por-numerical algorithms and coding practices. Typically, tions of the boundary. For example, the commomy usedthese older codes use upstream differencing, which is iri- flux boundary condition isaccurate, and direct (banded) solvers, which are ineffi-cient. In contrast, the SECO-TRANSPORT code uses VCk- D_Ck = Vf(t.) (9)an implicit TVD scheme with three-level time differenc-
ing and directional splitting to produce an efficient code where / is a known function.that is also second-order accurate both in time and space.Problems with moderately-high Peclet number greatly The flow-field V is assumed to be independent ofbenefit from this scheme by avoiding spurious oscilla- the solute concentration. It is essential that the flowtions commonly associated with the central differencin_ field be consistent with the advection differencing used inschemes or the excessive numerical diffusion associated SECO-TRANSPORT; otherwise, false source terms can
with upstream differencing. The long time-scales of the result. With the wide range of parameters sampled inproblems to which the code is to be applied dictate the the WIPP PA project, single precision VAX calculationsuse of fully-implicit _lgorithms. were marginal. In WIPP practice, the flow-field is ob-
" tained from the SECO-FLOW code in double precision.This AF/TVD algorithm originally developed for aero-
dynamics problems required significant adaptions to the The Transport Algorithmfinite volume (or block-centeredfinite difference) grid, es-pecially in regard to ghost cells, cross-derivative terms,and velocity locations staggered with respect to concen- The transport algorithm developed by Salari [14] usestrations (which proved to be advantageous for TVD). The a finite volume mesh (with ghost cells) where fluxes arefinite volume (or block-centered finite difference) grid evaluated at cell faces and concentrations at cell centers.makes SECO-TRANSPORT compatible with the SECOflow codes, and with MODFLOW [31] and SWIFT II Equation (7) is transformed into a stretched Cartesiancoordinate32], whereas the STAFF2D finite element formulationnvolved an inconsistency with these. I_ --'z 7",
The relevant partial differential equation contains ad- z = z((), (10)vection, dispersion, adsorbtion, source, and decay terms.Mesh Peeler numbers of order one in the nominal case y --- y(r/)
and of order ten in the extreme case are expected in the where metric transformations are _ = dy,_, flu = dz_,present application, indicating that the advection term and d = _,%. The transformed equation, with further
algebraic manipulations, was put into a strong conserva- Table 1: Partial list of schemes available
tion form [6] to ensure mass conservation which is essen- "'0 _o Schemes Truncation errorSial here. The transformed equation is given by 1 0 Baler, implicit O( At )
_t 0 0 - { 0 Trapezoidal, implicit O( At 2 )l 3-point-bac}ward, implicit O( At _ )eR_ (Ck)+bTC_)+ = 1 i
0 . 0 . O 8
At
+CRk:_ + CR___a___O,__+ 0 +fi Oa) +_l-Z_ - F_+ Ct;o_)_+ CE_)_^n.where + (_:i). + CF_,),
k-_+ +fi"]
= -y, i+___[+ eRkAO_-' l (13)
/_ = rh,vC'_, The cross derivative terms are time-lagged to facilitatethe factorization of the right-hand side operator. The er-
j_ _ _Dl_ 0C'j, rot introduced by lagging these terms is corrected throughJ 0_ ' _ intra-time step iteration.
Er.2 _%D1_ 0_Tk The convective t.-rms are modeled using the following= d 0-"_' TVD flux developed for staggered meshes by Sahxri [14_.The flux is a combination of upwind and centered scheme.
F_l = 1
s 0_ E___,_= _(1- ej_,__)[CC_,_+ c__,,_),,L_,__Dm 0(7_, u"L,- r/_ , -(eh-c"
J Or/ _-a,_,) [ i-[,_ []
= Q_____ +_ ___,_(q,_+ c___,_)(_.b__,_,__,_(_4)J ,
g, = -.r where -. 2((.)_,_(_b..,,_2 (_.)_-_,_= (_.)j,,+ (_)__,,_
Equation (11) is solved using an implicit Approximate The function q' is called a limiter function. There are aFactorization procedure [71. The convective terms are number of limiter functions available ranging from verymodeled by TVD [81and the remaining terms by central compressive (Roe superbee) to very dissipative (minmod)differencing. A general two-level implicit finite volume [8].scheme, in delta {orm [7], can be written as After the explicit portion (RHS) of equation (13) has
been evalr _.ed, the solution at the new time level is ob-OAt ., At . ,,
eR_A(7_' = 1+_(¢R_,AC_,)t + _+-_w(¢RkCk)_ tained through the following sequence(I + a=L_)A6'i,_ = RHS, (15)
+I--_(¢R_AO_'-')¢(12) (Z+_L.)A6s,_ = AO,,_, (_6)_"+' "" "" (17)where "-'i.k = Ci., + ACi._,
AC_' = 6"_'+_- 0_' where I is an identity matrix and L,_, L_ are the x andThe AC_' can be thought of as a correction to advance the y operators, respectively. The first sweep in either thesolution to a new time.level (n+l). The time difference x or y direction produces intermediate results, denotedequation (12), with appropriate choice of the parameters by CTi._. The second sweep uses the intermediate results0 and _o, produces many two- and three-level implicit to complete the cycle. The order of the sweep can beschemes as shown in Table 1. Applying equation (12) to symmetrized by alternating the direction. After bothequation CIi) we ha ,. sweeps are complete, the solution is updated.
Th," boundary conditions (Dirichlet, Neumann, at:dRobin)are ali implicitly imple,nented in the 1-D operator
OAt [-(AE")¢ - (A._'"), in both directions. This ensures the second-order accu-eR_A6'_ = 1 +---_ racy of the scheme. Tbe implicit construction of bound-
+ (A/_),, + (AF_). -¢R_,_AC_'] ary conditions requires an intermediate boundary condi-tion for the initial sweep. The intermediate boundary
OAt _E_ )_+ (iF:,).1
Table 2: Convergence results, uniform grid PA transport calculations, even allowing high-resolutionS'ize Az At RMS .....RMS ratio calculations of the effects of clay linings, a problem which
---20x20 .05 .25 7.697E-3 could not be calculated i previous PA's with STAFF2D.
40x40 .025 .125 1.954E-3 3.94 The algorithmic breakthrough consists of a directly80x80 .0125 .0625 4.921E,-4 3.97 coupled (simultaneous, vs. iterative or sequential) solu-160x160 .00625 .03125 1.234E-4 3.99 tion of both the fracture system and matrix equations
within the constrictions of the AF (approximate factor-ization) algorithm. It is analogous to the coupled solutionin STAFF2D, which is itself a fairly complex algorithm.
condition is subtle, and is evaluated by applying either The complicating factors in,the SECO-TRANSPORT al-the x or y operator, depending on the boundary, to the gorithm for incorporation of direct coupling are the stag-equation of the ghost cell. The stencils of these operators gered finite volume mesh (compared to the co-locatedwill be different r.ear the boundaries, variables used in STAFF2D), the use of ghost cells, and
the delta-formulation of the AF algorithm.Transport Accuracy, Single Porosity
With the source term of Equation (7) evaluated asF = -),Dr=, the basic algorithmic steps following Huyakorn
The SECO-TRANSPOP_ code was verified for tem- et. al [4] for the double-porosity solution are as follows.poral and spatial accuracy using an exact solution for anunsteady problem. Details may be found in [14]. Ta- • Algebraically perform forward elimination on dis-ble 2 presents the computed solution at time=25 for four cretized equations to get matrix solution in termsdifferent grid sizes and time steps. By examining the ra- of unknown interface concentration ui.rio of Root Mean Square (RMS) of errors, it is evidentthat the overall solution is secondlorder accurate in time • Express source term as P = ctut +/3, where a andand space. Also in [14] are comparisons with the two- /3 depend on matrix solution.
dimensional convection-dispersion model problem of [9] • Substitute source term in fracture system equation;for both low and moderate mech Peeler numbers (Pe = put ux on operator side of equation.2, 10).
These results are compared to results obtained using • Solve fracture system equations for _.._ each frac-ture system node.imphcit upstream differencing, representative of solutionscomputed by the majority of existing codes. The maxi-mum error usng implicit upstream differencing is about • Perform backward elimination on discretized equa-three times and the RMS about 8 times larger than TVD tions to complete matrix solution.
solution, even for mesh Pe = 2. For moderately high The previously used algorithm [14], like ali sequentialmesh Peclet number, Pe = 10, the difference between the algorithms, was effective only when the coupling betweentwo solutions is dramatic. As expected, the TVD scheme the fracture system and matrix equations was not dotal- -retained a sharp front as opposed to a very diffused front nant. Although each equation was solved efficiently andgenerated by the implicit upstream differencing, stably in the implicit AF algorithm, allowing large time
We have developed an even more accurate transport steps for physically well resolved flows, the s_4uentialalgorithm using a Flux-Based Modified Method of Char- couphng introduced an ezphcit behavior which limitedacteristics (MMOC) methodology thr is fully conserva- the time step. When the coupling dominated, either duetive and allows implementations of various flux limiters to large retardation (R >> 1) or to high resolution ofto avoid spurious oscillations [27, 13]. As yet, it has been the clay linings (Az) << 1 then this exphcit time stepapplied only in 1-D limitation dominated the calculations, reducing the al-
" lowable time step even though the physics became lessTransport E_ciency, Single Porosity consequential, i.e. for R >> 1 the transport shuts down.
The new coupled algorithm solves a somewhat largerCompared to STAFF2D for a single-porosity problem, implicit system than the previous sequential approach,
SECO-TRANSPORT has been benchmarked at 9.6 times but does not require any intra-time step iterations. Evenfaster on a 41x41 grid and 34.5 times faster on an 81x81 for one iteration, the old sequential approach is slower,grid [14]. and for difficult problems it required more than one it-eration. Also, for R >> 1, the new calculation becomes
Direct Solution for the Double-Porosity Equations more stable computationally, even as it does physically.The result is that the previously most difficult cases havenow become the easiest cases.
A major algorithmic development recently built intothe SECO-TRANSPORT code (since [14]) is the direct Performance :or Double Porositysolution for the double-porosity (or dual-continuum) equa.tions. This algorithmic development, with the code run-ning on a modern workstation (IBM Prise/System 0000) With the new algorithm, the complete array of pa-ehminated a former computing bottleneck for the WIPP rameter variations, including highly resolved clay lin-
ings, R >> 1, and dual porosities, are computable, withroughly a factor of 50 increase in speed. (C¢,,npared
to STAFF2D using the same time steps, the new ver- QUALITY ASSURANCE
sion of SECO-TRANSPORT is still roughly a factor of Formal software QA procedures for the WIPP project30 faster.) Some double-porosity cases, with R = 1 are described in [25], and encompass (amongst other tasks)or R > 1, can now be run stably in O(10) time stepswith SECO-TRANSPORT, the same as previously used code Verification studies, establishment oi review com-with STAFF2D. However, as is always the case, accu- mittees, version control, traceability, retrievability, doc-racy must be verified by systematic grid and time-step umentation, etc.convergence studies, which axe now feasible with SECO- SUMMARYTRANSPORT. Note that the operation count for theSECO-TRANSPORT algorithm, which is indicative of This paper has described the SECO suite of codesits execution time, varies optimally with grid size (i.e., for groundwater flow, particle tracldng, and radionuclidelinear with the number of unknowns) whereas the the op- transport, presently used and in ongoing development byeration count for the commonly used direct solvers v,.-ies the WIPP PA. The algorithm and code developments inquadratically, making grid convergence te_:.s expensi, e. the areas of facilitation of grid convergence tests in mul-tiple domains, correct treatment of transmissivity factors
An especially good feature of the ne,_ algorithm is the for unconfined ,_.quifers, efficient multigrid algorithms, ahigh resolution now possible for the clay lining calcula- formulation of brine Darcy flow equations that uses fresh-tions. High resolution r.ear the interface is necessary for wa;er head as the dependent variable, boundary-fittedaccuracy, e.g. 15 nodes in a stretched grid with (Az) = coordinates, temporal high-order particle tracking, an eh1.F_,-06. Like R >> 1, this has the effect _f strongly cou- ficient and accurate implicit Finite Volume TVD algo-piing the fracture system and matrix equations. The new rithm for radionuclide transport in (possibly) fracturedalgorithm allows this calculation to proceed with insignif- porous media, and accurate calculation of advection viaicant penalty in computer time. (A pr,_viously developed a flux-based modified method of characteristics, have ledrepresentation of clay linings by way of a skin resistance to documented significant improvements in both accu-is still a maintained option.) racy and efficiency, making systematic grid convergence
tests feasible, and increasing the confidence in modelingThe 2-D SECO-TRANSPORT, along with the SECO-
FLOW code, has been incorporated into CAMCON [26] computations.(the WIPP PA executive controller _-ystemfor computer Acknowledgement
codes) via a pre- and post-processor af _:,r_;::.chby R. Blaine Thi._ is Sandia National Laboratories report SAND93-lt is now operational and was used !_ the 1992 Prelimi- 7007C. The work was performed for Sandia National Lab-nary Performance Assessment [10], _;.lacing STAFF2D oratories and the United States Department of Energyused in previous years [11, 12]. under Contract DE-AC04-76DP00789.
The verification exercises indicated the importance ofretaining the flow information of the ghost cells, whichare not naturally retained in the CAMCON data base. ReferencesThe post-processor of the SECO flow codes was modi-fied to retain this information. The verification exercises [1] tIuyakorn, P.S. and Pinder, G.F. Computationalbased on the exact solutions have not only demonstrated Methods in Subsurface Flow, Academic Press, New
that the theoretical O((Az. )._,(A_))2 accuracy is actually York, 1983.
attained, but also provide gmdance to the WIPP PA an- [2] Bear, J. and Bachmat, Y. Introduction to Modelingalysts on selection of grid size, grid clustering about thesource (intrusion), time step size, and sensitivity of the of 7_ranspor_ Phenomena in Porous Media, KluwerAcademic Publishers, Dordrecht, Netherlands, 1990.release (integrated mass discharge and pulse shape) to
dispersivities, velocity, porosity, species decay rate, and [3] Streltsova-Adams, T.D. 'Well Hydraulics in Hetero-retardation, geneous Aquifer Formations' Advances in Hydro-
science, Vol. ll,(Ed., Chow, V.T.), pp. 357-423,Computational Dimensionallty Academic Press, New York, 1978.
lt is noteworthy that a 2-D flow problem produces a [4] Huyakorn, P.S., Lester, B.H., and Mercer, J.W.'An Efficient Finite Element Technique for Model-computati_nally 3-D transport problem when the direc- ing Transport in Fractured Porous Media: _;ingletion into the clay lining and matrix blocks is properly Species Transport' Water Res. Res., Vol. 19, No. 3,resolved (i.e., when the analyst does not simply assign pp. 841 -854, 1983.an arbitrary small number of node points in this direc-tion, but actually performs systematic grid convergence [5] Huyakorn, P.S., Lester, B.H., and Mercer, J.W.tests). Likewise, the 3-D S2]CC-FLOW code will give rise 'An Efficient Finite Element Technique for Model-to a 4-D double-porosity transport problem. In addition, ing Transport in Fractured Porous Media: Nuclidethe number of radionuclides present, while not systemat- Decay Chain Transport' Water Res. Res., Vol. 19,ically increased as axe the number of grid points, can be No. 5, pp. 1286-1296, 1983.substantial; presently [10] we are considering 9 radionu-clides. A computation'ally time-dependent 4-D problem [6] Pulliam, T.H. 'Efficient Solutior, Methods for thewith 9 radionuclides run for Monte Carlo analysis, even Navier-Stokes Equations', Lecture Notes for the Vonwith efficient LHS techniques, makes for a computation- Karman Institute for Fluid Dynamics Lecture .%-ally demandin_ study, especially if accuracy is demanded, ties, Brusses, Belgium, 1986.
[7] Fletcher, C.A.J. Computational Techniques for [18] Schaffer, S. 1991. "An Efficient 'Black Box' Semi-Fluid Dynamics, Volumes I and II, Springer-Verlag, coarsening Multigrid Algorithm for Two and Three1988. Dimensional Symmetric Elliptic PDE's with Highly
Varying Coefficients," Proc. Fifth Copper Mountain[8] Yee, H.C. 'Construction of Explicit and Implicit Conference on Multigrid Methods, March 31-April 5,
Symmetric TVD Schemes and Their Applications' 199I. Also, submitted to SIAM Journal of NumericalJ. Comp. Phys., Vol. 68, pp. 151-179, 1987. Analysis.
[9] Javaaadel, I., Doughty, C., az_dTs_mg, C.F. Ground- [19] Ruge, J. and Steuben, K., "Algebraic Multigridwater Transport: Handbook ,f Mathematical Models, (AMG)", IZconiiers in Applied Mathematics, Vol.American Geophysical Union, Washington, D.C., 3: Multigrid Methods, S. McCormick, cd., SIAM,1984. philadelphia, 1987.
[10] WIPP PA (Performance Assessment)Division, Pre- [20] Huyakorn, P. S., White, Jr., H. O. and Panday,liminary Comparison with 40 CFR Part I9I, Sub- S. 1989. "STAFF2D Solute Transport and Frac-part B for the Waste Isolation Pilot Plant, December ture Flow in Two Dimensions", Hydrogeologic, Inc.,I995, Volume I: Methodology and Results. SAND92- Herndon, VA.0700/1, Sandia National Laboratories, Albuquerque,NM. [21] Lusczynski, N. J., "Head and Flow of Ground Wa-
ter of Variable Density", Journal of Geophysical Re-[11] WIPP PA(Performance Assessment) Division, Pre- search, Vol. 66, No. 12, December 1961, pp. 4247-
liminary Comparison with 30 CFR Part 191, Sub- 4256.part B for the Waste Isolation Pilot Plant, DecemberI991, Volume I: Methodology and Results. SAND91- [22] Shampine, L."F., Watts, W. A., and Davenport, S.,0893/1, Sandia National Laboratories, Albuquerque, "Solving Non-stiff Ordinary Differential Equations -NM. The State of the Art", Sandia Laboratories Report
SAND 75-0182, 1975, Sandia National Laboratories,[12] Bertram-Howery, S. G., M. G. Marietta, R.P. Albuquerque, New Mexico. Also, SIAM Review, Vol.
Rechard, P. N. Swift, D. R. Anderson, B. L. Baker, 18, No. 3, pp 376-411, July 1976.J. E. Bean, Jr., W. Beyeler, K. F. Brinster, R. V.Guzowski, J. C. Helton, R. D. McCur_ey, D.K. [23] "The International HYDROCOIN Project, Ground-Rudeen, J. D. Schreiber, and P. Vaughn, Prelimi. water Hydrology Mode_ng Strategies for Perfor-nary Comparison with ,40 CFR Part 19I, Subpart B m_nce Assessment of Nuclear Waste Disposal, Levelfor the Wast, Isolation Pilot Plant, December 1990. 1: Code Verification".SAND90-2347, Sandia National Laboratories, Albu-querque, NM. [24] Steinberg, S. and Roache, P., "Symmetric Operators
in General Coordinates", Ecodynamics Research As-I13] Roache, P. J., "Validation Exercises of a One- sociates, 1990. To be submitted for publication.
Dimensional Flux-Based Modified Method of Char-acteristics", Proc. IX International Conference on [25] Rechetrd, R. P., Roache, P. J., Blaine, R. L.,Computational Methods in Water Resources, Den- Gilkey, A. P., and Rudeen, D. K., "Quality Assur-vet, Colorado, 9-12 June I992, pp. 69-76. ance Procedures for Computer Software Support-
ing Performance Assessments of the Waste Isola-[14] Salari, K., Knupp, P., Roache, P. and Steinberg, S., tion Pilot Plant", SAND90-1240, Sandia National
"TVD Applied to Radionuclide Transport in l,'rac- Laboratories-Albuquerque, April 1991.turcd Porous Media", Proc. IX International Con-ference on Computational Methods in Water Re- [26] Rechard, R. P., "CAMCON: Computer System forsources, Denver, Colorado, 9-12 June 1992, pp. 141- Assessing Regulatory Compliance of the Waste Iso-148. lation Pilot Plant", Proc. Probabalistc Safety and
Management Plan (PSAM) Conf., Beverly Hills,[15] Roache, P. J., Knupp, P. M., Steinberg, S., Blaine, Calif., Feb. 4-7, I991. Elsevier Science publishers,
R. L. 1990. "Experience with Benchmark Test Cases Amsterrdam, pp. 899-904.for Groundwater Flow." A_:ME FED Vol. 93, Bench-mark Test Cases for Computational Fluid D_nam. [27] "Computational Fluid Dynamics Algorithms andics, I. Celik and C. J. Freitas, Eds., Boer No. Codes Developed for WIPP Site Simulations", Com-H00598, June 1990, pp. '19-56. putational Mechanics, Proc. Asian Pacific Conf. on
Computational Mechanics, Hong Kong, 11-13 Dec.[16] Roache, P. J., "Need for Control of Numerical Ac- 1991, Y. K. Cheung, J.H.W. Lee, and A.Y.T.Leung,
curacy", Journal of Spacecraft and Rockets, Vol. 27, eds., A. A. Balkema, Rotterdam, Vol. 2, pp. 1325-No. 2, 1990, pp. 98-102. 1335.
[17] Russell, T. F., and Wheeler, M. F., "Finite Element [28] Dukowitz, J. D., "Conservative Rezoning Algorithmand Finite Difference Methods for Continuous Flows for Generalized Two-Dimensional Meshes", J. Com-in Porous Media", Mathematics of Reservoir Simu- putational Physics, Vol. 59, 1985, pp. 193-199.lation, R. E. Ewing, cd., SIAM Publications, 1983,pp. 35- 106.
[29] Roache, P. J., "Computational Fluid Dynami,:s Al-orithms Developed for WIPP Site Simulatmns",roc. IX International Conference on Computa-
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[30] Baptista, A. M., Solution of Advectio_-Dominated7"_ansport by Eulerian-La#rangian Methods using theBackward Methods of Characteristics, Ph.D. Thesis,Dept. of Civil Engineering, M.I.T., 1987.
[31] McDonald, M. G. and Haxbaugh, A. W., A ModttlarThree-Dimensional Finite-Difference Ground- Wa_erFlow Model, Techniques of Water-Resources Inves-tigations of the United States Geological Survey,Book 6, Modeling Techniques, Chapter Al. Scien-tific Software Group, P.O. Box 23401, Washington,D.C. 20026-3041.
[32] Reeves, M., Ward, D. S., Johns, N. D., ;,hd Cran-well, R. M., Theory and Implementation for SWIFTII The Sandia Waste-Isolation Flow,c.nd TransportModel for Fractured Media, SAND8_-l159, SandiaNational Laboratories, Albuquerque, New Mexico,August 1986.
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Nationale Genossenschaft f_r die InternalLagerung Radioaktiver Abf_lle (2) MS OKg.Attn: S. Vomvoris 0101 0001 A Narath
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