1-D problem in a heterogeneous medium with capillary effects

interface within the transition element is positive Fig. 5. Since the discrete phase flow is propor- tional to the product of phase mobility l a and w n,iþ1 ¹ w n,i , the Standard Galerkin method produces a flow of the non-wetting phase from node i þ 1 to node i although at node i þ 1 the non-wetting phase is immobile or not present S n ¼ 0. This non-physical process produces a peak of the non-wetting phase saturation distribution at node i and probably a negative saturation at node i þ 1. The saturation level at the interface necessary for penetration of the non-wetting phase into the low permeability zone is achieved, at the same rate, or, sooner than it would be for the physical solution, because of an unphysical flow. This gives numerical results where the non-wetting phase reaches the entry pressure and enters a low permeability lens although physically this does not happen. The scheme implies a continuous distribu- tion of the saturation at the interface which yields in the case of the Brooks–Corey model a discontinu- ous approximation of the capillary pressure. • The test function of the Petrov–Galerkin method for the non-wetting phase equation at node i is simi- lar to the one for the Standard Galerkin method because the gradients of w n at this node are opposite Fig. 7. Thus the problems arising with the Petrov– Galerkin approach are comparable to the ones with the Standard Galerkin method. • Because the Fully-Upwind Galerkin method uses the fully upstream value of the mobility as defined in eqn 23, a flow of the non-wetting phase across the interface will not occur unless the pressure at the interface has reached the entry pressure. No unphysical flow occurs that could produce oscilla- tions in the numerical solution. At the interface, mobilities are approximated discontinuously, the capillary pressure continuously. To solve the system of non-linear equations eqn 17 for the corresponding discretization methods, a residual-based Newton–Raphson iterative concept is used. 6 NUMERICAL EXAMPLES 6.1 1-D problem in a homogeneous medium without capillary effects First we will show the performance of SG, PG, and FUG, with respect to flow in a homogeneous medium. The Buckley–Leverett problem 29 is a simple test problem without capillary pressure effects whose analytical solution is easily found and which is excellent for investigating the resolution of discontinuities for each method. A non-wetting phase displaces a wetting phase from left to right. The initial total velocity of the two-phase system, defined as the sum of the phase Darcy velocities, is 1·0 m s ¹ 1 , the ratio of the dynamic viscosities is one, residual saturations are zero and the Brooks–Corey function l ¼ 2·0 is used for the relative permeabilities. A space–time discretization of Dx ¼ 0·025 m and Dt ¼ 0·005 s is chosen. Fig. 9 shows the saturation profiles at time t ¼ 0·4 s for the three methods and the analytical solution. The SG method does not converge to the analytical solution because only numerical schemes using a form of upwind technique are able to capture correctly convection-dominated processes. 30

6.2 1-D problem in a heterogeneous medium with capillary effects

The case of a vertical column with a zone of fine sand Fig. 9. One-dimensional displacement of a wetting phase in a homogeneous medium t ¼ 0·4 s. Comparison of Galerkin-type discretization techniques 705 between two layers of a coarse high permeable sand is used to verify how well these methods simulate multiphase flow in heterogeneous media Fig. 10. Table 1 shows the material properties and the parameters for each model used in the following simulations. The van Genuchten parameters are obtained by using the conversion formulas of Lenhard et al. 27 eqn 30 and eqn 31. DNAPL infiltrates at the upper boundary with a flow rate of 0·005 kg s ¹ 1 . The densities and dynamic viscosities are ̺ n ¼ 1400 kg m ¹ 3 , m n ¼ 0·001 kg ms ¹ 1 for the non- wetting phase DNAPL, and ̺ w ¼ 1000 kg m ¹ 3 , m w ¼ 0·001 kg ms ¹ 1 for water. The discretizations used are Dz ¼ 0·01 m, and Dt ¼ 5 s. The entry pressure of P d ¼ 1324 Pa of the low permeable layer corresponds to a saturation of S p n ¼ 0·91. Thus, the physically correct solution shows a peak in the saturation above the heterogeneity of exactly this value. First we take the dp c dS w grad S w formulation which is widely used in combination with constitutive relation- ships after Brooks and Corey. 18 For this formulation FUG produces an accumulation of the non-wetting phase satura- tion at the node above the heterogeneity, but Newton’s method fails to converge when the entry pressure is approached Fig. 11 c. In contrast to FUG, SG and PG produce a numerical solution see Fig. 11 a,b; however the results do not consider the effect of the entry pressures and therefore do not correspond to the physically correct solution. Now we consider the grad p c -scheme according to eqn 17. For the Fully-Upwind Galerkin method, a damming-up of the non-wetting phase above the low per- meability lens is observed Fig. 12 c. A penetration into the second layer starts when a saturation of 0·78 is reached at the node above the heterogeneity. However, the theo- retical value of saturation S n necessary for penetration is 0·91, i.e. when the entry pressure is achieved. This value is only a limit value for an infinitesimal reference volume. When refining the discretization the nodal value of S n at which the penetration starts converges to the value of 0·91, which is required by the entry pressure condition. Thus, the numerical scheme describes the physical process in an accurate manner. At the second transition, a constant flux of the non-wetting phase occurs. The saturation value at Fig. 10. Flow through a vertical column with a low permeability layer zone 2. Fig. 11. Case A: DNAPL saturation at the interface for a Standard Galerkin, b Petrov–Galerkin, c Fully-Upwind Galerkin, using dp c dS w grad S w formulation with Brooks– Corey model. 706 R. Helmig, R. Huber the node upstream of the transition stays fixed at a relative small level of approximately 0·06. The mobility of this node saturation in conjunction with the pressure gradient over the transition element which also approaches a fixed value controls the flux Fig. 13 c. In zone 3, which has the same material properties as zone 1, the same saturation level occurs. In contrast to FUG, SG and PG are unable to capture the physical process at the heterogeneity interface as discussed in the previous section. The coupling of the gradient of w n and mobility l n does not work for these two methods. The non-wetting phase becomes mobile within the transition element for case A and the gradient of w n points in the opposite direction of phase flow. This yields a non-physical flux from the downstream node to the upstream node. Consequently, negative saturations develop on the low Fig. 12. Case A: DNAPL saturation at the interface for a Standard Galerkin, b Petrov–Galerkin, c Fully-Upwind Galerkin, using grad p c formulation with Brooks–Corey model. Fig. 13. Case B: DNAPL saturation at the interface for a Standard Galerkin, b Petrov–Galerkin, c Fully-Upwind Galerkin, using grad p c formulation with Brooks–Corey model. Comparison of Galerkin-type discretization techniques 707 permeability side of the interface. On the other side of the interface, the entry pressure is achieved within two time steps Fig. 12 a, b. The level of 0·91 is reached independently of the discretization lengths used. Negative saturations develop for both methods, SG and PG. But we are still interested in the behaviour of these two methods, which have obviously been disqualified at the first interface, i.e. for case A, at the second transition constituting case B. Here, a steep gradient of the non-wetting phase pressure causes a massive flux of non-wetting phase across this interface Fig. 13 a, b. Because mobilities are averaged over both nodes of the transition element, negative saturations develop ahead of the interface. While on the other side, the node saturation stays fixed at 0·91. This unphysical flux is a direct consequence of a non-zero not full-upstream-weighted mobility at the transition element, Fig. 14. Case A: DNAPL saturation at the interface for a Standard Galerkin, b Petrov–Galerkin, c Fully-Upwind Galerkin, using grad p c formulation with van Genuchten model. Fig. 15. Case B: DNAPL saturation at the interface for a Standard Galerkin, b Petrov–Galerkin, c Fully-Upwind Galerkin, using grad p c formulation with van Genuchten model. 708 R. Helmig, R. Huber and cannot be fixed in the case of PG through the employ- ment of the grad p c -chord-slope technique. Computing the same problem with van Genuchten parameters yields the correct solution for all three methods. The van Genuchten model gives no strict entry condition. A small amount of non-wetting phase reaches the fine sand side. Because the capillary pressure–saturation curve for sand II is considerably steeper than for sand I in the vicinity of 1 ¹ S nr , this causes a rapid increase in capillary pressure on the fine sand side. The Newton-type iterative scheme converges for all three methods and for all time-steps to solutions with monotone pressure distributions. A damming-up of the non-wetting phase at the interface up to a certain value takes place. In less technical and more mathematical terms, this phenomenon can be explained by the fact that when using the van Genuchten model the derivative of p c with respect to S n is very large near zero S n . As a consequence, diffusion terms become dominant, making this problem essentially parabolic; hence SG methods work. From Fig. 12 and Fig. 14, it is apparent that the entry pressure of the van Genuchten model and the corresponding entry pressure P d of the Brooks–Corey model result in the same accumulation of the non-wetting phase and subse- quently must be the same. A massive flux across the inter- face starts when the non-wetting phase has dammed up in front of the interface and as this entry pressure of the van Genuchten model is approached Fig. 14. When using the van Genuchten model, all three methods describe correctly the flow process at the transition from fine to coarse sand case B. The corresponding numerical results are depicted in Fig. 15.

6.3 2-D problem in a heterogeneous medium