Constant-concentration boundary condition: Lessons from the HYDROCOIN variable-density groundwater

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1 WATER RESOURCES RESEARCH, VOL. 33, NO. 10, PAGES , OCTOBER 1997 Constant-concentration boundary condition: Lessons from the HYDROCOIN variable-density groundwater benchmark problem L. F. Konikow, W. E. Sanford, and P. J. Campbell U.S. Geological Survey, Reston, Virginia Abstract. In a solute-transport model, if a constant-concentration boundary condition is applied at a node in an active flow field, a solute flux can occur by both advective and dispersive processes. The potential for advective release is demonstrated by reexamining the Hydrologic Code Intercomparison (HYDROCOIN) project case 5 problem, which represents a salt dome overlain by a shallow groundwater system. The resulting flow field includes significant salinity and fluid density variations. Several independenteams simulated this problem using finite difference or finite element numerical models. We applied a method-of-characteristics model (MOCDENSE). The previous numerical implementations by HYDROCOIN teams of a constant-concentration boundary to represent salt release by lateral dispersion only (as stipulated in the original problem definition) was flawed because this boundary condition allows the release of salt into the flow field by both dispersion and advection. When the constant-concentration boundary is modified to allow salt release by dispersion only, significantly less salt is released into the flow field. The calculated brine distribution for case 5 depends very little on which numerical model is used, as long as the selected model is solving the proper equations. Instead, the accuracy of the solution depends strongly on the proper conceptualization of the problem, including the detailed design of the constant-concentration boundary condition. The importance and sensitivity to the manner of specification of this boundary does not appear to have been recognized previously in the analysis of this problem. 1. Introduction within the groundwater system. However, for groundwater transport problems it is rare to encounter a field condition that Boundary conditions must always be specified to enable the could cause the concentration of a solute to remain fixed regoverning partial differential equations for groundwater flow or solute transport to be solved mathematically (and initial gardless of stresses or changes in the accompanying flow field or in the concentration gradient. Nevertheless, specified-value conditions must also be specified for transient problems). boundary conditions are frequently implemented in both flow Mathematically, boundary conditions must include the values of the dependent variable or its derivative normal to the and transport models of groundwater systems. When a constant-value condition is imposed on a particular boundary segboundary. For groundwater-model applications the boundary ment in a model and that value differs from values either conditions are generally of three types: (1) specified value (for specified or calculated at adjacent calculation points, then a example, head, concentration, or temperature), (2) specified flux will be maintained into or out of the boundary (depending flux (corresponding to a specified gradient of head, concentra- on the gradient direction). Specifically, the maintenance of a tion, or temperature), or (3) value-dependent flux (or mixed boundary condition, in which the flux across a boundary is related to both the normal derivative and the value) [e.g., Mercer and Faust, 1981]. Franke et al. [1987, p. 2] state that the fixed concentration in the presence of a concentration gradient yields a solute flux, which must be balanced by the addition or removal of an equivalent mass of solute at the location of the fixed concentration boundary condition. selection of boundary conditions "is probably the most critical We examine the nature of constant-concentration boundary step in conceptualizing and developing a model of a ground- conditions using numerical experiments applied to a wellwater system." known and relatively complex benchmarking problem. (Al- The types of boundaries appropriate to a particular field though we focus on a solute-transport problem, our analysis problem require careful consideration. In groundwater flow and results are also relevant to heat-transport problems.) The problems the first type, a specified-head or specified-pressure Hydrologic Code Intercomparison (HYDROCOIN) project was started in 1984 as an international cooperative effort for condition, is often considered to represent a reasonable analog studying groundwater modeling in the context of radioactive to parts of groundwater systems that are bounded by surface waste disposal. The stated objective of HYDROCOIN level 1 water bodies that remain relatively unaffected by stresses studies was to "verify the accuracy of groundwater flow codes" Now at Radian Corporation, Herndon, Virginia. for a variety of hydrogeological problems [Organisation for Economic Co-operation and Development (OECD), 1987]. We This paper is not subject to U.S. copyright. Published in 1997 by the focus on their case 5 problem, which was designed to evaluate American Geophysical Union. a system in which groundwater flow is influenced by significant Paper number 97WR density variations caused by variability in the concentration of 2253

2 2254 KONIKOW ET AL.: CONSTANT-CONCENTRATION BOUNDARY CONDITION 0 p= 105 Pa kg/m3... "... P = 0 Pa 3OO X DISTANCE (m) The HYDROCOIN level 1, case 5 problem is designed to simulate groundwater flow and solute transport in a twodimensional vertical plane in which fluid density depends on the dissolved-solids concentration. The problem was developed to represent a rough approximation of the subsurface geologic conditions existing at the Gorleben salt dome in Germany. The assumptions and framework of the problem are described by OECD [1988, 1992] and in the several reports that present results of HYDROCOIN participants [e.g., Herbert and Jackson, 1986; Herbert et al., 1988; Leijnse and Hassanizadeh, 1989; U.S. Nuclear Regulatory Commission, 1988; Yabusaki et al., 1988]. The OECD [1988] report also presents and compares the results by six project teams that simulated the problem using six different models. More detailed descriptions of the specific models and (or) particular results of several of the various teams are also presented in the separate reports by the HYDROCOIN participating teams. This problem was also recently reanalyzed by Oldenburg and Pruess [1995] using the TOUGH2 code. Case 5 considers the problem of groundwater flow over a salt dome. The basic geometry and boundary conditions are illustrated in Figure 1. The problem is simplified to a 900-m-long, two-dimensional cross section. The aquifer is assumed to be 300 m thick, homogeneous, isotropic, and recharged by freshwater at the surface. The lateral and bottom boundaries are no-flow conditions, and the top boundary is represented by a linearly varying specified-pressure condition, as shown in Figure 1. This induces flow frøm left to right, but does not predetermine the rates of recharge and discharge, nor the separation point between recharge and discharge zones (which depends on the solution to th e problem). It is further assumed that the concentration associated with any recharge is 0.0. Flow is assumed to be isothermal, and fluid viscosity is assumed to be constant. The problem formulation is based on the assumption that Figure 1. Geometry and boundary conditions of HYDRO- COIN case 5 salt dome problem. the top of the salt dome is exposed to circulating groundwater in the overlying aquifer, slowly releasing saturated brine by a diffusive/dispersive process. Thus, in the original definition and evaluation of this problem, salt is assumed to enter the flow dissolved salt. Although the HYDROCOIN project officially ended several years ago, this particular problem continues to be a subject of continuing interest [see, e.g., Oldenburg and Pruess, 1995; Konikow et al., 1996; Johns and Rivera, 1996]. Case 5 represents an idealized shallow groundwater flow system that overlies a subcropping salt dome, which represents the source of salt and brine in the active flow system. The salt is assumed to be essentially impermeable, so that salt is resystem by transverse dispersion only along the central third of the bottom boundary [OECD, 1988]. This represents the only source of solute to the flow system. It is widely recognized that if the fluid velocity is zero within the salt, then no lateral dispersion should be occurring within the salt; the assumption represents a simple and convenient simplification to approximate a diffusive release of salt into the active flow field. However, in several subsequent studies, analysts have redefined the leased into the active flow system only by diffusive or dispersive original problem to include additionally molecular diffusion in processes that are driven by a concentration gradient. Therefore the exposed top of the salt dome has usually been represented by a constant-concentration boundary condition for the solute-transport equation that coincides with a no-flow boundary condition for the flow equation. The purpose of this paper is to examine closely the use of the constant-concentration boundary condition and evaluate solute release mechanisms coincident with this boundary specifithe system so that both the release of salt into the flow system from the constant-concentration boundary nodes and the transport of brine within the flow system would be affected accordingly [e.g., OECD, 1992; Herbert et al., 1988; Oldenburg and Pruess, 1995]. The various teams aimed to develop steady state solutions to the flow and transport equations. The properties assigned to the flow and transport system are summarized in Table 1. cation. Our approach is to accomplish this through a reevaluation and reanalysis of the procedures, assumptions, and results of the well-known HYDROCOIN case 5 benchmarking 3. Governing Equations problem. The HYDROCOIN level 1, case 5 problem is designed to evaluate and compare steady state solutions to the stipulated problem. As described by OECD [1988], flow is assumed to be 2. Description of Salt Dome Flow Problem isothermal and governed by Darcy's law. Permeability is assumed to be homogeneous and isotropic. Fluid density varies as a function of brine concentration. Under these assumptions, the fluid mass balancequation can be written as Table 1. Summary of Physical Parameters for Original Definition of Case 5 Parameter Permeability (k) Porosity (e) Viscosity Gravitational acceleration (g) Reference freshwater density (p f) at C = 0.0 Saturated brine density (Ps) at C = 1.0 Longitudinal dispersivity (al) Transverse dispersivity (a r) Effective molecular diffusivity (Dm) Initial fluid pressure at (0, 0) Modified from OECD [1988]. Value m Pas 9.81 m s kg m kg m m 2m 0m2s -1 l0 s Pa

3 KONIKOW ET AL.: CONSTANT-CONCENTRATION BOUNDARY CONDITION 2255 k V. p (vp- pvz) = o. where p is the fluid pressure [M/Lt2], p is the fluid density [M/L 3], k is the porous medium intrinsic permeability tensor (2nd order) [L2], /. is the fluid viscosity [M/Lt], # is the gravitational constant [L/t2], and z is elevation of the reference point above a standard datum [L]. Likewise, the solute mass balance equation used in the model is V'(apDijVC) - V' (8pVC) = 0, (2) Dij = (Dm 4- arlvl)sij + (am- at) IvI ' (5) 4.1. Constant-Concentration Bounda Condition where 80 = 1 if/ = j, 80 = 0 if/ - j, and D m is the effective molecular diffusivity of the solute [L 2t- 1]. Most documented An important conceptual basis of the original problem forapplications of transport models to groundwater problems mulation and of the subsequent model design was that there is have been based on this conventional formulation. It is also assumed that fluid density varies as a function of the mass fraction of brine, as 1/p = (C/ps) + [(1 - C)/pi ] (6) where Ps is the reference freshwater density at C = 0.0 and Ps is the saturated brine density at C = Numerical Methods Most available numerical models for groundwater simulation are based on finite difference or finite element methods, and all of the models applied in the original HYDROCOIN studies were in one of these two classes. However, Lagrangian methods offer a reasonable alternative numerical approach for solving variable-density groundwater flow and transport problems. We apply the MOCDENSE model [Sanford and Konikow, 1985], based on the method of characteristics, to this EXPLANATION where e is the effective porosity [dimensionless], C is the mass fraction of solute in the fluid phase [dimension!ess ], D ij is the Impermeable Salt o Active Node of Numerical Grid coefficient of hydrodynamic dispersion (2nd-order tensor) No-Flow Boundary e Gonstant-Goncentration Node [L2t-1], and V is the average linear groundwater velocity Figure 2. Schematic representation of lower part of numervector [Lt- 1]. ical grid showing alternative grid designs for implementing a Groundwater velocities can be obtained from Darcy's law by the equation constant-concentration boundary condition in the HYDRO- COIN salt dome problem. (a) Regular quadrilateral finite element or node-centered finite difference grid. (b) Blockv= k centered finite difference grid. (c) Modified grid design for (Vp- pavz). (3) block-centered finite difference method to preclude advective release of brine from the salt dome. The coefficient of hydrodynamic dispersion is typically defined as the sum of mechanical dispersion and molecular diffusion. The dispersivity tensor for an isotropic porous medium same case 5 variable-density problem in order to compare and can be defined in terms of the longitudinal and transverse contrast results. dispersivities, a L and a r. This yields two dispersion coeffi- In general, the method of characteristics yields less numercients oriented with the direction of flow; the longitudinal ical dispersion in advection-dominated problems than do standispersion coefficient, D L; and the transverse dispersion coefdard finite difference and finite element methods for equivaficient, D r: lent discretizations. The solute-transport equation can be DL = IvI DT- TIVI. (4) solved accurately and with maximum efficiency by the methodof-characteristics model if dispersivity is set to zero, whereas Additionally, an effective diffusion coefficient, D m, can be most finite difference and finite element codes cannot solve the included to account for molecular diffusion. The diffusion cotransport equation accurately, if at all, for this advection-only efficient is isotropic and includes the effects of tortuosity. This yields the components of the hydrodynamic dispersion tensor, Du, which may be expressed as case. This enables our numerical analysis to distinguish more clearly between the contributions and effects of advective flux and dispersive flux and to isolate the interaction between these processes and the nature of the constant-concentration boundary condition. no flux of fluid (or brine) out of the salt dome into the overlying aquifer; the only release of salt is by lateral dispersion. Therefore all of the models that were applied to this problem imposed a constant-concentration boundary condition be- tween x = 300 m and x = 600 m on the bottom no-flow boundary (as shown in Figure 1). The dispersive or diffusive salt flux into the active flow system therefore depends on the difference in concentration between the fixed boundary value and the concentration in the active flow system adjacent to that boundary, which changes over time. The specification and locations of the constant-concentration nodes depend partly on the type of numerical method being used in a particular code and partly on the conceptualization of how to implement the problem specification into the grid design. Figure 2 illustrate some of the possibilities. Figure 2a represents either a regular quadrilateral finite element grid or a node-centered (or point-centered) finite difference grid. For these types of numerical codes, nodes are placed directly on the bottom no-flow boundary, and constant-concentration

4 2256 KONIKOW ET AL.: CONSTANT-CONCENTRATION BOUNDARY CONDITION Table 2. Summary of Codes Applied to Case 5 Team* Code Numerical Method Original Study? 1 METROPOL finite element 2 SWIFT finite difference 3 NAMMU finite element 4 CFEST finite element 5 SWIFT2 finite difference 6 SUTRA finite element Recent Studies$ ß.- TOUGH2 integral finite difference ß -. MOR3D finite difference very small, the resulting advective solute flux would also be negligible. Regardless of the magnitude of dvective salt release, it is clear that the case 5 problem definition requires that it be zero. Unfortunately, the presumption Of zero advective salt release probably cannot be rigorously tested or evaluated using the models of the original HYDROCOIN studies because the standard finite difference and finite element models cannot accurately distinguish between advective and dispersive solute fluxes because an accurate solution cannot be easily obtained for cases in which no dispersion is allowed in the system. Thus we use the MOCDENSE model to make this evaluation. This Study Another grid design, as represented in Figure 2c, improves MOCDENSE method of characteristics consistency with the definition of the problem. This design, shown here for a block-centered finite difference grid, extends *Team numbers are provided only for "original study" to facilitate the middle third of the active grid for a distance of one row of cross reference between codes and results shown in Figure 3.,OECD [1988]. cells or elements below the specified depth of the active $Oldenburg and Pruess [1995] and Oldenburg et al. [1996]. groundwater flow system (effectively into the top of the salt dome). The constant-concentration conditions would then be specified only at nodes representing locations within the salt, conditions can be specified at those nodes that define the middle third of that boundary. This represents the basic grid design that appears to have been implemented in all of the models benchmarked in the original HYDROCOIN studies. If where the permeability of the media could simultaneously be defined at a value that is many orders of magnitude lower than in the active groundwater flow system above the salt. This should effectively preclude any flow or advective transport a block-centered finite difference method is used, the constant- through the constant-concentrationodes yet still allow the concentration conditions could be specified as indicated in dispersive or diffusive transfer of salt into the higher velocity Figure 2b. The no-flow boundary is aligned with the lower edge zone of the aquifer. There are other possible combinations of (or face) of the cell rather than being associated with the node at the center of the cell. On the other hand, in Figure 2b the parameter specifications for this basic grid geometry that could yield essentially the same desired effect, both for blockconstant-concentration condition would be imposed at the centered and node-centered (or finite element) approaches. In node rather than on a cell face. With respect to the effective the discretization represented in Figure 2c, on the cell face position of the constant-concentration boundary, this formulation (relative to that in Figure 2a) would also have an effect essentially equivalent to raising the top of the salt dome a small above the constant-concentration node, V z 0 and V x > 0; thus dispersive transfer across that cell face into the active flow field will arise in the numerical algorithm from the only nondistance that is proportional to the vertical grid spacing of that row of finite difference cells. However, there should be no zero term in (5), which is Dzz = ((Vx)2/IvI) --- Vx. significant difference between the final steady state solutions 4.2. Summary of Previous Numerical Results obtained using these two formulations, assuming that grid discretization is appropriately fine. The MOCDENSE model is The solution to case 5 is highly nonlinear and represents a balance among buoyancy effects, advection, and hydrodynamic based on a block-centered finite difference discretization. dispersion. A list of the models used in the original comparison Because the bottom no-flow boundary defines a streamline, are listed in Table 2. An overview and evaluation of their fluid flow can occur parallel to the boundary, along the bottom row of cells and nodes in both Figures 2a and 2b but not across the boundary. Howev. er, the coefficient of hydrodynamic disresults is presented in a summary report of the HYDROCOIN project [OECD, 1988]. On the basis of the nature of the problem, it was expected that the calculated pressure heads would persion (Dii) is also related to the average linear groundwater decrease from left to right across the domain and that the salt velocity (V). For the original case definition in which D m , when V = 0 along the constant-concentration boundary, then D ii = 0 and there will be no dispersive flux into the system. On the other hand, if V > 0 at the constantconcentration nodes in the numerical model, as occurs even at steady state in the case 5 problem, then there must also be an would form a plume flowing to the right and upwards from the part of the domain bottom representing the top of the salt dome [OECD, 1988]. The final results of the various teams showed general agreement in the overall nature of the calculated flow field, in that the primary flow was driven by the imposed pressure gradient advective 'flux of solute present through the constant- along the top surface. However, there were significant differconcentration nodes. Thus the representation of a constantconcentration boundary as depicted in Figures 2a and 2b alences in calculated pressure distributions, particularly in the lower part of the domain, and large differences in the final salt lows a solute flux by both advection and dispersive processes distributions between the results of some individual teams and and an addition of salt m ass into the active flow field that must balance the flux generated by both transport processes. The magnitude of the rate of salt release due to advection is related to the scale of discretization. Some might assume that the average results of all the teams. The results were compared in several different ways, including visual comparisons of contour maps of pressure and concentration distributions, comparisons of pressures and concenbecause.the nodes of Fig ure 2a are points and have no thick- trations along various horizontal lines representing different ness, the advective solute flux would be infinitesimally small and could be ign0,red. Similarly, one might argue that if the vertical grid spacing for the design shown in Figure 2b were depths in the aquifer, comparisons of selected pathlines, and comparisons of calculated velocity fields. For example, OECD [1988] presented a comparison of concentrations calculated by

5 KONIKOW ET AL.: CONSTANT-CONCENTRATION BOUNDARY CONDITION Z O 0.4 Z U.I 0.3 Z O ' I ' I ' I ' I ' I ' I ' I ' I ' I METROPOL... '... 7 MOCDENSE (fig. 4)... m--- 2 SWIFT 8 MOCDENSE (fig. 7) ; 3 NAMMU 4 CFEST _...,.... e... 5 SWIFT2.,,,,-= ,-- 6 SUTRA 1....,, "... ß '" ""L* '... 6''."4' -'-.' X-DIRECTION (m) Figure 3. Example of one type of comparison of results of six HYDROCOIN teams, showing calculated brine concentration at a depth of 200 m (modified from OECD [1988]). Results from two of the MOCDENSE simulations described in this report are added for comparison. five of the teams at a depth of 200 m, as shown in Figure 3. We have added the results from SUTRA and MOCDENSE sim- ulations to allow a more complete comparison. As characterized by OECD [1988], the results fell into three groups. The results of teams 2, 3, 4, and 6 were similar and included all of the physical features expected. The results of team 1 were too diffuse, which was attributed to the use of a coarser grid than the other teams. The results of team 5 had essentially no salt in the domain, which OECD [1988, p. 117] attributed to a difficulty related to representing boundary conditions in their version of the code. Team 3 [Herbert and Jackson, 1986, p. vii concluded, "Our results were the best obtained by the HYDROCOIN participants," whereas a conclusion of team 1 [Leijnse and Hassanizadeh, 1989, p. 32] was "this kind of exercise and intercomparison of models... does not tell us much about the validity of models involved. Even if all models predict the same results, one can only say that those models are equally good or equally bad... " The case 5 problem was also reanalyzed recently by Oldenburg and Pruess [1995] using the TOUGH2 code. They applied their analysis to both the original problem (without molecular diffusion) and to the modified problem (that includes molecular diffusion). They found that the solutions could be characterized as being in one of two flow regimes, depending on the relative strength of the diffusion coefficient. These were called (1) recirculating and (2) swept forward. The former occurred when diffusion was relatively high, and the final solution showed that the brine spread laterally across the entire lower part of the system. This was the characteristic of the numerical , X-DISTANCE (m) Figure 4. Steady state salt concentrations (as mass fraction) for case 5 problem calculated using MOCDENSE with a 75 x 45 node grid. solutions obtained in the original HYDROCOIN comparison. The second type (swept forward) occurred when diffusion was relatively small and was characterized by having brine move only in the same general direction as the regional flow. In this case the final solution showed essentially no brine upstream of the constant-concentration boundary. Similar results were also obtained by Oldenburg et al. [1996] using the MOR3D model for cases with molecular diffusion and zero dispersivities Application of MOCDENSE to Salt Dome Problem The MOCDENSE model uses an implicit finite difference method to solve the flow equation in a block-centered grid and the method of characteristics (MOC) to solve the transport equation (see Sanford and Konikow [1985] for more details). MOC is implemented using particle tracking to represent advective transport and an explicit finite difference method for dispersive transport. For the application to the case 5 variabledensity groundwater flow problem, the generic code was modified to incorporate the density function described in (6). The problem was discretized at several different scales to evaluate grid convergence. It was found that discretizing the domain into a grid consisting of 45 nodes (or cells) in the x direction at a spacing of fioc = 20 m and 75 nodes in the z direction at a spacing of Az - 4 m yielded stable and accurate results. MOCDENSE is designed to solve the transient solutetransport equation V. (spdvc) - V. (spvc) = (spc). (7) A steady state solution, equivalento solving (2) directly, was achieved by marching through time in a transient mode until sufficiently small changes occurred in the system, which required about 400 years. The salt concentrations calculated with MOCDENSE, after having formulated the constant-concentration boundary condition in the manner depicted in Figure 2b, are shown in Figure 4. The results of solving the case 5 problem with MOCDENSE are similar to those of most of the HYDRO- COIN teams (see Figure 3). Most of the flow, as well as the highest velocities, occurs in the upper (freshwater) part of the system. The brine is distributed across the entire bottom of the 0

6 2258 KONIKOW ET AL.: CONSTANT-CONCENTRATION BOUNDARY CONDITION "' 200 I 3001 INTERFACE, X-DISTANCE (m) 5. Analysis of Constant-Concentration Boundary Condition An important conceptual basis of both the original and modified problem formulation and model design was that there is no flux of fluid (or brine) out of the salt dome into the overlying aquifer; the only release of salt is by lateral dispersion or molecular diffusion, processes which are driven by the concentration gradient. Therefore all of the HYDROCOIN model teams imposed a constant-concentration boundary between x = 300 m and x m on the bottom no-flow boundary. All viable solutions for this case 5 problem geometry, regardless of whether mechanical dispersion or molecular diffusion are included separately or jointly, will indicate some nonzero fluid velocity over the part of the bottom boundary representing the top of the salt dome. Because V > 0 at the constantconcentration nodes in the numerical model, then there must also be an advective flux of solute emanating out of these nodes, which was not a condition specified in the problem formulation. The boundary condition representing the top of the salt dome in case 5 can be stated mathematically as and (op -pg ) =0onF1 (8) C = t.0 on F1 (9) where F is the boundary segment at z m and 300 < x < 600 m. Equation (8) is equivalento stating Vz - 0 on F. Because this boundary condition yields a flux of salt into the active system that varies with time, it may be more appropriate to express the boundary condition as an equivalent third-type boundary condition in terms of the salt flux normal to the boundary O C Vxa r qc = -Vxctr + VzC - (Cs- C) on F 1 t > 0 Figure 5. Steady state salt concentrations (as mass fraction) calculated using MOCDENSE for modification of case 5 aswhere q½ is the solute flux, C is the specified concentration value, and n is distance normal to the boundary. In the case 5 suming advective release only from constant-concentration problem the dispersive flux will vary over time because the boundary nodes and no dispersion during transport. All con- concentration on the aquifer side (C) of the boundary intours between 0.1 and 0.9 fall so close to the 0.5 contour that creases with time, resulting in a maximum concentration grathey cannot be distinguished in this figure; they essentially dient at t = 0 and a minimum concentration gradient when represent an interface between brine and fresh groundwater. equilibrium is achieved. The advective flux is supposed to be zero. However, by placing the boundary condition of (9) or (10) at a location coincident with an active streamline, where domain during the transient evolution of the system. This V ) 0, the operative boundary condition effectively becomes MOCDENSE solution closely matches that of Herbert et al. Vxa r [1988, Figure t0] for a case having identical dispersivity values qc = (Cs- C) + VxC on Fi Y/ plus D, = 5 x 10-8 m2/s. The solution in Figure 4 is also t > 0 (tt) similar in form and magnitude to those obtained by Oldenburg and Pruess [1995, Figure tt] and by Oldenburg et al. [1996] for That is, a solute flux arises from both dispersion and advection. The latter clearly is not consistent with the case 5 problem a case having a relatively high value of diffusion (Dm = 5 X definition m2/s), although their solutions reflect a somewhat greater amount of spreading and larger total salt mass in the To evaluate further the impact on the final solution of imposing a constant-concentration condition in the conventional problem domain at steady state. This occurs because their manner on the central part of the no-flow boundary, we modassumed value of the diffusion coefficient yields a larger coefficient of hydrodynamic dispersion, by relations indicated in ified the problem specification by setting ct L - ct and D, This would assure that salt can only be released (5), throughout most of the problem domain than do the original dispersivity values alone. and transported by advection. The brine distribution calculated under these conditions is (10) shown in Figure 5. Note that because there is no dispersion in this conceptualization, there is no mixing between freshwater and brine and the 0.5 contour essentially represents the position of an interface between the two fluids. At steady state conditions the thin and relatively dense brine "pool" has become trapped hydrodynamically and the fluid velocities within the brine are effectively zero (and there is no release of salt into the active regional flow field and no discharge of salt from the problem domain). As can be seen in Figure 5, there is much less brine in the system at steady state than is evident in Figure 4. The smaller volume of dense brine allows fresh groundwater to circulate deeper into the system than occurs when more brine is present (Figure 4). The brine content for the no-dispersion case (Figure 5) is indeed significantly greater than zero, an outcome that would not be expected on the basis of the problem statement that salt is released only by lateral dispersion or diffusion. This confirms the important point that specifying a constant-concentration boundary condition at one or more nodes allows a significant salt influx due to both advection and dispersion. This outcome often is not recognized by model users in general, and we could find no indication in the published literature that advective release was recognized by any of those who performed the previous simulations of the case 5 problem. The lack of rec- ognition of the processes releasing salt in the various models may then lead to erroneous interpretations about the relative importance of dispersion or diffusion or of the numerical method itself in controlling the final salt distribution, density variations, and flow field. The goal of this particular simulation (Figure 5) was simply to demonstrate clearly that an advective release is associated

7 KONIKOW ET AL.: CONSTANT-CONCENTRATION BOUNDARY CONDITION 2259 with the formulation of the constant-concentration boundary in this problem as implemented by the HYDROCOIN teams, and it is important not to confuse release mechanisms with transport processes in the analysis. Regardless of which release mechanisms are operative, it is not appropriate in the case 5 problem formulation to simulate transport without dispersion and (or) diffusion occurring simultaneously. Therefore we conducted an additional set of simulations in which we used the grid design shown in Figure 2c to ensure that the release process would be only by advective flux of solute from the constant-concentration boundary cells, but allowed transport throughouthe active flow field (except at those cells adjacent Figure 7. Steady state salt concentrations (as mass fraction) calculated using MOCDENSE for case 5 in which constantconcentration boundary was modified by incorporating it into an additional row of cells between x = 300 m and x = 600 m to the constant-concentrationodes) to be subject to both advection and dispersion for the parameters listed in Table 1. at a depth of 300 m to assure that brine release from salt dome was only by dispersion process and to preclude any advective These results (Figure 6) yield a final mass of salt in the release of brine. (Because Az is only 4 m, these extra cells system that is much greater than shown in Figure 5 but less cannot be shown clearly at the scale of this illustration.) All than that shown in Figure 4. The total salt mass is much greater other parameters and conditions are identical to the simulation than for the case in which both the release and the transport depicted in Figure 4. are controlled only by advection because when the salt released by advection is also subjecto dispersion during transport, the salt spreads through a larger volume of the media, some salt discharges in the outflow area, and the concentrations downgradient from the constant-concentration boundary cells are then lower than they would otherwise be. This, in turn, results in a different density and pressure distribution than for the Figure 5 case, so that even when steady state conditions are attained, there is still a positive velocity and advective solute flux at the constant-concentration boundary cells. The syneractive flow field by lateral dispersion (or by diffusion in the modified case 5 problem) in the z direction as a consequence of the positive x velocity in the deepest part of the active flow system immediately above these near-impermeable boundary cells representing the salt dome. The resulting steady state numerical solution is shown in Figure 7. The total salt mass in the system at steady state when the only release mechanism is lateral dispersion (Figure 7) is somewhat less than that calcugistic feedback effect of dispersive transport on the advective- lated for the advection-only release case shown in Figure 6, only source mechanism enables an ultimate release of a signif- and it is significantly less than that computed in the simulations icantly larger mass of salt from the constant-concentration based on the original constant-concentration boundary condiboundary source than for the no-dispersion case having the identical release mechanism. In our final analysis we redesigned the grid and boundary conditions for the MOCDENSE simulation to represent more accurately the conditions that were stipulated for case 5, that is, that the only source of salt to the flow field should be from releases by lateral dispersion (in the original formulation or by diffusion in the modified formulation) across the part of the bottom boundary representing the top of the salt dome. To implement this, we added another row of cells between x = 300 m andx = 600 m below the previous bottom, as depicted in Figure 2c. In these cells we assigned C = 1 and a hydraulic conductivity several orders of magnitude lower than in the active flow field. This assured that advection through the constant-concentration cells would be essentially zero but still allow an upward release of salt from the salt dome into the tion (Figure 4). This is reflected also in Figure 3, which shows that the concentrations at a depth of 200 m for this simulation (curve 8) are less than all other results (except curve 5, which was considered erroneous). The resulting pattern is of the swept forward type in the classification scheme of Oldenburg and Pruess [1995]. Our solution for concentration and flow velocity using MOCDENSE, shown in Figure 7, agrees very closely with that presented by Oldenburg and Pruess [1995, Figure 7] and Oldenburg et al. [1996] for the same set of conditions. Their implementations of the constant-concentration boundary condition in TOUGH2 and MOR3D, respectively, were very similar to ours in that they also attached an extra row of grid blocks below the active flow system (analogous to the design in Figure 2c) to preclude advective solute flux through or out of these nodes and to assure that the only release mechanism for brine would be by dispersion and diffusion (K. Pruess, Lawrence Berkeley Laboratory, written communication, 1996). However, in contrast to our approach of assigning arbitrarily small hydraulic conductivity values to the constant-concentration cells, their im- 100 plementation was accomplished by arbitrarily assigning extremely large volumes and extremely small vertical grid distances to the constant-concentration cells to achieve the X-DISTANCE (m) Figure 6. Steady state salt concentrations (as mass fraction) calculated using MOCDENSE for modification of case 5 assuming advective release only from constant-concentration boundary nodes, but that dispersion is an active process during transport. E same net effect. X-DISTANCE (m) We conclude that because the brine plume shown in Figure 7 is based on a model grid that allows no advective release of salt, the solute distribution calculated using this design for the constant-concentration boundary condition represents a more proper and more accurate solution to the case 5 problem as it was originally described and formulated [OECD, 1988] than any of the solutions generated by the HYDROCOIN project participants and documented in the HYDROCOIN reports.

8 2260 KONIKOW ET AL.: CONSTANT-CONCENTRATION BOUNDARY CONDITION Much of the analysis, discussion, and debate in the literature HYDROCOIN case 5 salt dome problem, the final solution is about the HYDROCOIN case 5 problem focused on evaluat- much more sensitive to variations in the specific design of the ing which model (or numerical method) gave the most accurate results, whether or not steady state conditions were truly achieved and whether diffusion should be included as a separate process. Although having an accurate numerical method is important and must be assured, the differences in accuracy for this problem among the solvers are minimal (or even trivial) relative to the magnitude of the difference in the final solution that is generated by different treatments of the constantconcentration boundary condition. Similarly, given a reasonbottom constant-concentration boundary condition than had been recognized previously. This recognition will make it easier to resolve ongoing debate about the impact on the solution of using alternative numerical methods, time-stepping schemes, grid dimensions, convergence criteria, or dispersion versus diffusion. This analysis of the effect of boundary conditions also leads to a cautionary warning about the worth of comparing models. Intercode comparisons for complex problems, such as those ably accurate model and the demonstrated significance of the implemented in HYDROCOIN, are indeed worthwhile exerdesign details of the constant-concentration boundary condition, the distinction between a solution that is approximately at steady state and one that is exactly at steady state is not significant. The magnitude of the dispersivities or diffusion coefficients can have a very significant effect on the final solution, but this is largely a separate issue from the treatment of the boundary condition (except to the extent that the magnitude of cises but are limited in value for assessing the accuracy and reliability of a generic groundwater model. Using the "average" of all solutions as a basis of comparison or as a standard of reference for model evaluation may cause the important distinction between accuracy and precision to be blurred or forgotten by analysts. If the "true" solution is significantly different from the mean of all the numerical solutions, then a these coefficients will affect the mass of brine that enters the bias is present and this reflects a loss of accuracy. The deviation active flow field from the salt dome, in addition to affecting the of individual results from the mean is a measure of model rate of spreading during transport within the active flow field). precision; although precision (small deviation) is desirable, it is insufficient as a measure of model "verification." For problems 6. Conclusions such as the case 5 salt dome system, model evaluations based on intercode comparisonshould more properly be termed as In general, if a constant-concentration boundary condition is applied in a numerical solute-transport model at a node in an active flow field, a solute flux can occur by both advective and dispersive processes. This does not seem to be widely recognized, as evidenced by previous analyses of the HYDROCOIN case 5 salt dome problem, in which a significant advective flux benchmarking, rather than as verification, because there is no known analytical solution (or "ground truth") for such complex problems. Comparison of results is primarily an assessment of model consistency rather than an evaluation of model accuracy. In general, we recommend that those applying soluteof solute was associated with the constant-concentration transport models to field problems avoid the use of a constantboundary. The lack of recognition of the processes releasing salt in the various models may then lead to erroneous interconcentration boundary condition. Strictly analogous field conditions are rare. Instead, solute boundary conditions should be pretations about the relative importance of dispersion or dif- formulated as a source concentration associated with a fluid fusion or of the numerical method itself in controlling the final salt distribution, density variations, and flow field. Although our conclusions are derived from an analysis of a variabledensity flow and transport problem, the potential for advective release of solute from a constant-concentration boundary is flux or as a third-type boundary condition for solute. An example of an analogous boundary condition used in groundwater flow models is the head-dependent flux boundary commonly used to represent leakage and stream-aquifer interaction. An advantage of the third-type boundary condition also of importance in more common constant-density prob- is that the mass flux and cumulative mass inflow associated lems too. The magnitude of the numerically calculated advective flux depends on the scale of discretization as well as on physical parameters. In the case 5 problem, if the source-release mechanism is only by advection, but subsequentransport through the flow field is subjecto dispersion and (or) diffusion, then a synergistic feedback effect enables an ultimate release into the flow field of a significantly larger mass of salt from the constant-concentration boundary source than for the case having the identical release mechanism and no dispersion or diffusion during transport. The final calculated solute distribution pattern for case 5 depends very little on which numerical model is used, as long as the selected model is solving the proper equations and includes an adequate representation of the conceptual model. Given a class of appropriate numerical codes, the accuracy of the solution depends much more strongly on the proper conceptualization of the problem and on the scale of discretization than it does on which appropriate code is selected. In general, care must be taken to avoid using parameter values or boundary conditions simply because they are optimal or efficient for the particular code or numerical method being used. In the with the boundary are simply and explicitly calculated, making their magnitudes readily apparent to the model user. When a first-type boundary condition is specified, the flux and cumulative inflow are still present but are typically hidden from scrutiny by the model user. Acknowledgments. The authors appreciate the helpful suggestions provided by Cliff Voss, Al Shapiro, and Alden Provost of the U.S. Geological Survey. We also thank Bryan Travis (Los Alamos National Laboratory), Anne Carey (University of Alabama), and Carl Mendoza (University of Alberta) for their helpful review comments for Water Resources Research. C. Oldenburg and K. Pruess (LBL Laboratories), C. Cole (Battelle PNL), and C. Voss (USGS) generously provided valuable clarifications of their numerical methods and results. References Franke, O. L., T. E. Reilly, and G. D. Bennett, Definition of boundary and initial conditions in the analysis of saturated ground-water flow systems--an introduction, book 3, chap. B5, Techniques of Water- Resources Investigations of the United States Geological Survey, U.S. Geol. Surv., Denver, Colo., Herbert, A. W., and C. P. Jackson, A study of salt transport in a porous medium: The application of NAMMU to HYDROCOIN level 1

9 KONIKOW ET AL.: CONSTANT-CONCENTRATION BOUNDARY CONDITION 2261 case 5, Rep. AERE R 12147, U.K. At. Energy Auth., Harwell Lab., Oxford, Oxfordshire, England, Herbert, A. W., C. P. Jackson, and D. A. Lever, Coupled groundwater flow and solute transport with fluid density strongly dependent upon concentration, Water Resour. Res., 24(10), , Johns, R. T., and A. Rivera, Comment on "Dispersive transport dynamics in a strongly coupled groundwater-brine flow system" by Curtis M. Oldenburg and Karsten Pruess, Water Resour. Res., 32(11), , Konikow, L. F., P. J. Campbell, and W. E. Sanford, Modeling brine transport in a porous medium: A re-evaluation of the HYDRO- COIN level 1, case 5 problem, in Calibration and Reliability in Groundwater Modeling, edited by K. Kovar and P. Van Der Heijde, , IAHS Publ., 237, Leijnse, A., and S. M. Hassanizadeh, Verification of the METROPOL code for density dependent flow in porous media: HYDROCOIN Project, level 1, case 5 and level 3, case 4, Rep , Natl. Inst. of Public Health and Environ. Prot. (RIVM), Bilthoven, Netherlands, Mercer, J. W., and C. R. Faust, Ground-Water Modeling, Natl. Water Well Assoc., 60 pp., Worthington, Ohio, Oldenburg, C. M., and K. Pruess, Dispersive transport dynamics in a strongly coupled groundwater-brine flow system, Water Resour. Res., 31(2), , Oldenburg, C. M., K. Pruess, and B. J. Travis, Reply, Water Resour. Res., 32(11), 34!1-3412, Organisation for Economic Co-operation and Development (OECD), The International HYDROCOIN project--background and results, Paris, Organisation for Economic Co-operation and Development (OECD), The International HYDROCOIN project--level 1: Code verification, Rep , Paris, Organisation for Economic Co-operation and Development (OECD), The International HYDROCOIN project--level 3: Uncertainty and sensitivity analysis, Rep , Paris, Sanford, W. E., and L. F. Konikow, A two-constituent solute-transport model for ground water having variable density, U.S. Geol. Surv. Water Resour. Invest. Rep., , U.S. Nuclear Regulatory Commission, NRC model simulations in support of the Hydrologic Code Intercomparison Study (HYDRO- COIN), Rep. NUREG-1249, vol. 1, U.S. Nucl. Regul. Comm., Washington, D.C., Yabusaki, S. B., C. R. Cole, D. J. Holford, A.M. Monti, and S. K. Gupta, HYDROCOIN level 1: Benchmarking and verification test results with CFEST code, Rep. PNL/SRP-6681, Pac. Northwest Lab., Richland, Wash., P. J. Campbell, Radian Corporation, 2455 Horse Pen Road, Herndon, VA L. F. Konikow and W. E. Sanford, U.S. Geological Survey, 431 National Center, Reston, VA (Received January 27, 1997; revised May 27, 1997; accepted June 30, 1997.)