Comments on Constant-concentration boundary condition: lessons from the. HYDROCOIN variable-density groundwater benchmark problem by L.F.

Transcription

1 Comments on Constant-concentration boundary condition: lessons from the HYDROCOIN variable-density groundwater benchmark problem by L.F. Konikow, W.E. Sanford, and P.J. Campbell Ekkehard Holzbecher Inst. of Freshwater Ecology and Inland Fisheries (IGB), Berlin, Germany Rudower Chaussee 6A, Building 21.2, Berlin, Germany

2 As one of the members of the pilot group that defined the salt-dome case tackled in the paper I want to add some comments and respond to critical remarks on the HYDROCOIN test case definition. The case definition was set up with reference to a subsurface formation located above a saltdome in Lower Saxony in Germany. In the salt-dome near Gorleben the German high level radioactive waste repository is projected. Safety assessment requires studies on the migration of radionuclides in the aquifers above the salt formation. For that purpose the flow pattern above the salt needs to be known [Bütow et al., 1985]. The concerned HYDROCOIN test-case is a highly simplified representation of an aquifer above a salt-dome cap. In the HYDROCOIN project [HYDROCOIN, 1988] the definition of the salt-dome problem has been worked out by Bütow [1984] assisted by L. Heredia at Technical University Berlin. In HYDROCOIN the salt-dome problem was treated as a test case in level 1 (as case 5) and in level 3 (as case 4). In the level 3 task mainly a sensitivity analysis for the former problem was required [Bütow and Holzbecher, 1986]. The reference case in level 3 is identical with the level 1 case except that diffusivity has a nonzero value: D= m 2 /s. This change was made as reaction to bad experience with zero diffusivity in the level 1 test case. Herbert et al. [1988] provide a good discussion about the difficulties. Most modelers who treated the problem after HYDROCOIN used their own values for diffusivity and dispersivities. The discussion within the HYDROCOIN workshop focused on the question of whether the eddies, which could be recognized on the output from some models, are reasonable or not. Finally it turned out that all of the participating teams got eddies in their numerical solutions. There were quite big differences concerning the extension and circulation of the eddies. With their first attempt in their paper Konikow et. al. [1997] apply the MOCDENSE code and find a solution with the same flow characteristic. The concentration contours fit very well with those obtained by the HYDROCOIN participants.

3 The doubts of Konikow et. al. [1997] concerning their own and the HYDROCOIN solutions start during a sensitivity analysis where they model the unrealistic case of zero diffusivity D and zero dispersivities α L, α T. Unexpectedly salt enters the active aquifer, which is impossible if diffusivity and dispersivities all vanish. In order to resolve the contradiction Konikow et. al. [1997] question all other results instead of questioning their own approach. In fact they are led to statements that can easily be disproved. The crucial statement in the paper is that specifying a constant-concentration boundary condition... allows a significant salt influx due to both advection and dispersion. The statement is made without reference to special codes - obviously the authors mean all numerical approaches as they are used by HYDROCOIN participants. My first comment is that such a far reaching statement cannot be made without looking into details of the codes. For some approaches it will be shown that there is no advective flux of salt on no-flow boundaries with a Dirichlet boundary condition for salt concentration. Fig. 1 depicts the location of nodes and boundaries for block-centered and node-centered rectangular grids. The approach on the left side (Fig. 1a) is used by several transport codes, for example by SWIFT and FAST-B(2D). The numerical algorithm is derived from a mass balance summing the advective and diffusive/dispersive contributions from all 4 block edges. The main point is the following: if a no-flow boundary condition is specified for a certain edge, the advection term is set to zero. With this finite volume idea the block-centered approach naturally delivers no advective contribution across the boundary edge. The described block-centered approach for boundary edges fits perfectly with the finite volume idea of mass conservation for specified blocks or volumes. When pressures or hydraulic heads are calculated at block centers, the simplest finite difference formula delivers normal velocity components at block edges. Exactly these are needed for the block mass balance.

4 In the MOCDENSE code [Sanford and Konikow, 1985] which is used by Konikow et al. [1997] the block-centered finite difference approach is also applied. But for the boundary conditions a procedure different from the one described above is implemented. Dirichlet boundary conditions are set in block centers and not at block edges. Fig. 1c illustrates the situation. Several textbooks describe this procedure [Aziz and Settari, 1979; Peaceman, 1977] - so I want to note it here as the traditional approach of block-centered finite differences. In a former report Konikow and Bredehoeft [1978] outline that zero transmissivity blocks are introduced at places with Dirichlet boundary condition for flow. Flux boundary conditions, in contrast, are set at block edges which coincide with boundaries of the model region [Konikow and Bredehoeft, 1978]. No-flow is such a condition with zero flux normal to the boundary prescribed. Following the traditional approach of block-centered finite differences, Dirichlet conditions for salt concentration are set in centers of boundary blocks. At these points there may in fact be a nonzero advective flux, because the no-flow (zero velocity condition) for fluid flow is set at a different location along the boundary edge. Obviously the argumentation of Konikow et al. [1997] is based on the type of boundary condition implementation used in the MOCDENSE code. The main point of comment is that all codes of HYDROCOIN participants treat boundary conditions differently from MOCDENSE. The method of SWIFT as a block-centered FD (finite volume) code is recalled above. FE codes are node-centered and thus have a different implementation, too. The node-centered approach has difficulties with the mass balance in general - not only in elements on the model boundary. Fig. 1c here is a different illustration of the situation shown by Konikow et al. [1997] in Fig. 2a. Quadrilateral finite elements or node-centered finite differences do not necessarily fulfill the mass balance. This well-known problem has led FE programmers to several more or less sophisticated numerical strategies: consistent velocity

5 representation [Voss, 1984]; mixed hybrid formulation [Chavent, 1988]. With these improvements codes deliver better mass balances. I want to illustrate the main point using another finite difference approach. The streamfunction can be used for the description of flow as an alternative to the pressure or head formulations [Holzbecher, 1998]. The block-centered approach provides streamfunction values at the block centers. Simple finite differences lead to parallel velocity components at block edges. The FAST-C(2D) code takes the mean of values at block edges for the parallel velocity component at a block center. Fig. 2 illustrates the procedure for first and second discrete approximations. A formulation with zero advective salt flux is possible. It does not depend on the absolute value of the velocity component but on the velocity component on a block edge. The advective term is set to zero when a block edge is specified as a no-flow model boundary. The following notes an afterthought on the zero-diffusion/zero-dispersion case of Konikow et al. [1997]! Is it really unexpected that salt enters the active domain? The mathematical problem is to find a differentiable function C that takes a high value C 1 on one part of the boundary and a low value C 0 on another part of the boundary. There is no way to fulfill these conditions other than by a transition zone where C takes values between C 0 and C 1. A step function with a jump from C 1 to C 0 is not differentiable. When the requirement of differentiability is skipped, the problem cannot be described by a differential equation. As the differential equation (no. (2) in paper of Konikow et al. [1997]) is derived from a conservation principle, it is not surprising that the salt mass conservation is not fulfilled when the code works on the problem inconsistently formulated. This is the reason why the salt comes into the system, which could not be explained by Konikow et al. [1997]! In the zero-diffusion/zero-dispersion case Konikow et al. [1997] formulate contradicting requirements. The problem formulation is mathematically inconsistent. The numerical results reflect the contradiction of the definition. A zero-diffusion/zero-dispersion case cannot be

6 taken for a verification exercise - neither for MOCDENSE, nor for CFEST, METROPOL, NAMMU or SWIFT - the codes of the HYDROCOIN participants. In a final model variation Konikow et al. [1997] redesign grid and boundary conditions in order to become more consistent with the HYDROCOIN level 1 case definition. A very low permeable boundary layer is introduced at the salt-dome cap. Konikow et al. [1997] do not recognize that with their specification they reduce the velocities and transverse dispersion above the salt cap significantly. The case thus does not represent the conditions stipulated in level 1 case 5. Because transverse dispersion is the only process to dissolve salt into the fluid phase, the outcome of this variation shows less salt in the active system than the results of the HYDROCOIN teams. The test case with vanishing diffusivity has attracted too much attention in the recent discussion among scientists. The discussion should not only focus on parameter values but should be guided also by phenomena which can be observed in field conditions. Codes should be tested and improved with a view of the purpose for which they are set up. A zero diffusivity causes numerical difficulties. The unrealistic setting is only of mathematical interest. It is more important to study solutions with parameter values that are relevant in real problems. An extensive observation program has been made concerning the hydrogeological situation in the subsurface near Gorleben, including geophysical and geochemical characterization [Boehme et al., 1985]. The measurements show clearly that salinity contours qualitatively agree with numerical results for eddy solutions: the contours increase slightly from the downstream water divide to the other boundary. Swept forward type solutions as they are named by Oldenburg and Pruess [1995] are different because salt is transported downstream from the salt-dome cap only. The definition from the HYDROCOIN level 3 case 4 is recommended as reference in future verification studies. Fig. 3 gives a result from a new calculation using the FAST-C(2D) code.

7 With input parameters from the HYDROCOIN level 3 case 4 reference case the model obtains an eddy solution. The streamfunction and salinity contours coincide very well with those published by Herbert et al. [1988] for the same input parameters. The result shown was obtained using a transient simulation for density-driven flow, because the model did not converge with the steady state option. Using SWIFT the same problem appeared during the HYDROCOIN project. The influence of numerical parameters on FAST-C(2D) results will be discussed in a forthcoming paper. Aside from the critical remarks, I appreciate the paper of Konikow et al. [1997] because it is the first team demonstrating that the MOC - a powerful method, underrepresented in scientific literature - can be used successfully to treat the salt-dome problem.

8 References Aziz, K., and A. Settari, Petroleum reservoir simulation, Applied Science Publishers, London, Boehme, J., and G. Delisle, K. Fielitz, W. Giesel, S. Keller, R. Ludwig, K. Schelkes, F. Schildknecht, G. Schmidt, and H. Vierhuff, Grundwasserbewegung im Deckgebirge über dem Salzstock Gorleben, Abschlußbericht Projekt Sicherheitsstudien Entsorgung, Fachband 17, Berlin, Bütow, E., Salt water distribution in a saturated porous medium, Proposal for a test problem HYDROCOIN level 1 case 5, HYDROCOIN project, Bütow, E., and G. Brühl, M. Gülker, L. Heredia, S. Lütkemeier-Hosseinipour, R. Naff, and S. Struck, Modellrechnungen zur Ausbreitung von Radionukliden im Deckgebirge, Abschlußbericht Projekt Sicherheitsstudien Entsorgung, Fachband 18, Berlin, Bütow, E., and E. Holzbecher, Proposal for definition of HYDROCOIN level 3 case 4: sensitivity analysis for the flow over a salt dome, HYDROCOIN project, Chavent, G., Advanced finite element methods, IGWMC expert meeting on: New Developments in Groundwater Modelling, Delft, 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), , Holzbecher, E., Modellierung dynamischer Prozesse in der Hydrologie: Grundwasser und ungesättigte Zone, Springer Publ., Heidelberg, Holzbecher, E., Modeling density driven flow in porous media, Springer Publ., New York, HYDROCOIN, The International HYDROCOIN Project, Level 1: code verification, OECD, Paris, 1988.

9 Konikow, L.F., and J.D. Bredehoeft, Computer model of two-dimensional solute transport and dispersion in ground water, U.S. Geol. Survey, Techniques of Water-Ressources Invest., Book 7, Chapter C2, Konikow, L.K., W.E. Sanford, and P.J. Campbell, Constant-concentration boundary condition: lessons from the HYDROCOIN variable-density groundwater benchmark problem, Water Resour. Res., 33(10), , Oldenburg, C.M., and K. Pruess, Dispersive transport dynamics in a strongly coupled groundwater-brine flow system, Water Resour. Res., 31(2), , Peaceman, D.W., Fundamentals of numerical reservoir simulation, Elsevier, Amsterdam, Sanford, W.E., and L.F. Konikow, A two-constituent solute-transport model for ground water having variable density, U.S.-Geological Survey, Water-Res. Investigations Report , Voss, C.I., SUTRA: A FE simulation model for saturated-unsaturated, fluid-densitydependant groundwater flow with energy transport or chemically-reactive singlespecies solute transport, U.S. Geol. Survey, Water Resources Invest. Rep , Codes: MOCDENSE: see Sanford and Konikow [1985]; SUTRA: see Voss [1984]; CFEST, METROPOL, NAMMU, SWIFT: see references in HYDROCOIN [1988]; FAST-B(2D): see Holzbecher [1986]; FAST-C(2D): see Holzbecher [1998]

10 Figure Captions Fig. 1: Illustration of location of nodes and boundary conditions for various finite difference approximations for rectangular 2-dimensional grids; a: block-centered / finite volume; b: node-centered; c: block-centered / traditional approach (compare: Holzbecher [1996]) Fig. 2: Scheme for calculating the vertical velocity component at the center of block i Fig. 3: Results for the reference case HYDROCOIN level 3 case 4 using FAST-C(2D) code; top fig.: streamlines - not equally spaced; bottom fig.: normalized salt concentrations 0.05(0.05)0.85

11 a b c nodes boundary conditions

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