LNAPL problem Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol21.Issue6.1998:

In order to avoid the extra expense of full Newton itera- tion, there are a number of possibilities. Since we are sol- ving a true time-dependent problem, then it is reasonable to suppose that it may not be necessary to use the exact Jacobian. One possibility is to simply ignore the Jacobian entries which lie outside of the sparsity pattern generated using first-order upstream weighting. The simplest and most efficient technique for constructing an approximate Jacobian is simply to use the efficient algorithm described in 26 , but using the flux-limited form for the flux functions. This method constructs the Jacobian using numerical differ- entiation. The Jacobian is constructed by columns, and, if there are neq unknowns per node, the entire Jacobian is constructed with work equivalent to neq þ 1 residual evaluations. In our case, this means that for the equilibrium case, this method requires the work of three residual evalua- tions, while for the nonequilibrium case, four residual eva- luations are required. If we simply use the algorithm described in 26 , and use a flux limiter to evaluate dissolved NAPL flux, then this method ignores the derivatives with respect to i2up in eqns 22–24Eq, 25eqn 26. Another method is to use the first-order Jacobian, but compute the residual using the high-order method 33 . This technique is commonly used in aerospace applications 25 . To be more precise, if x k is the kth iterate for the vector of primary variables at t ¼ N þ 1, then the nonlinear iteration is While not converged 29 J x k þ 1 ¹ x k ¼ ¹ r k EndWhile where J in eqn 29 is the Jacobian constructed using first- order upstream weighting, and the residual vector r k is constructed using a high-order flux-limited method for the NAPL mole fraction as in eqn 22 and eqn 26. To recapitulate, we will investigate three methods for solving the nonlinear algebraic equations: • Full Newton: the Jacobian and residual are evalu- ated using high-order flux limiter. • Approximate Jacobian: all Jacobian entries corre- sponding to derivatives with respect to i2up in eqns 22–27 are ignored. • First-order Jacobian: the Jacobian is evaluated using first-order upstream weighting. The residual vector is evaluated using a high-order flux limiter for the NAPL mole fraction. 6 CODE VERIFICATION The equilibrium model developed in this work was com- pared to the test problems described in 9 . There was good agreement with the equilibrium, multiphase compositional simulations reported in 9 . Several nonequilibrium cases with various mass transfer correlations were compared with the results in 14 , with good qualitative agreement. These tests are reported in detail in 27 . 7 COMPUTATIONAL PARAMETERS The nonlinear algebraic equations eqns 14, 15 and 18 are solved in the following using full or approximate Newton Iteration. A block incomplete LU ILU factoriza- tion iterative solver 39–42 with CGSTAB acceleration 43 was used to solve the matrix. An ILU 1 level 1 precondition- ing is used 40,41 . Reverse Cuthill-McKee RCM ordering 44 was used to order the unknowns. The tolerances for the Newton iteration are Pressure tolerance ¼ 0:01 kpa 30 Saturation tolerance ¼ 0:001 Mole fraction tolerance ¼ 10 ¹ 7 with inner iteration tolerances an order of magnitude smal- ler than the Newton iteration tolerances eqn 30. These nonlinear iteration tolerances typically resulted in a cumu- lative at the end of the run relative material balance error of less than 10 ¹ 4 . Variable timestepping was employed using a method similar to that in 45 . This method is based on selecting a desired maximum error in the solution, and predicting a timestep which will result in this error. 8 TEST PROBLEMS We consider a number of two-dimensional test problems, with different boundary conditions, constitutive data, and LNAPL less dense than water and DNAPL more dense than water contaminants.

8.1 LNAPL problem

The two-dimensional cross-section for this problem is shown in Fig. 2. The computational domain for these test consist of a 50 3 40 grid, with Dx ¼ 1.0 m and Dz ¼ 0.1 m. A unit thickness was specified in the y direction normal to the cross-section. Hydrostatic pressure boundary condi- tions are imposed on the left and right ends of the region with the water table elevations as shown in Fig. 2. Fig. 2. Two-dimensional cross-section computational domain for the LNAPL simulation. Nonlinear multiphase flow 439 The physical properties data are given in Table 3 and Table 4. The longitudinal dispersivity is typical of the Borden sand, based on matching field scale experiments 46 . The value of b in Table 3 b ¼ 1.0 is typical of that obtained by matching field-scale experiments 13 . Laboratory experiments typically have b . 0.5–0.75 16 . Note that in the following we assume that the permeability tensor is diagonal, with diagonal entries in the x and z directions k x , k z : K ¼ k x k z 31 The Courant number for this problem, based on the max- imum timestep of 10 days was . 10. The Peclet number is about . 2, for the dispersion parameters in Table 3. The LNAPL data in Table 4 was used for this example. The relative permeability and capillary pressure data are given in Table 5 and Table 6. Hysteresis effects are ignored in all simulations reported in this paper. This problem is initially fully saturated S w ¼ 1 every- where, and was run to 7 days with no NAPL injection to equilibrate the water table. LNAPL was then injected a rate 1.6 m 3 day for 0.5 days at the point shown in Fig. 2. Injec- tion of LNAPL then ceased, and the problem was run to completion. Fig. 3 shows the Log 10 normalized mole frac- tion X n X n at 11 days for this run, using a nonequilibrium model eqn 1, eqn 4, eqn 5 and eqn 7. Fig. 4 shows the saturation contours for this problem, but using the equilibrium model. Comparing the contours at 23.5 and 365 days, it is clear that a large portion of the LNAPL has been dissolved by the flowing water phase. In contrast, Fig. 5 shows the LNAPL saturation contours at 365 days using the nonequilibrium model. Comparing Fig. 5 with Fig. 4 365 days, we can see that the extent of the NAPL phase is much greater for the nonequilibrium model than for the equilibrium model. Of course, this is to be expected since the nonequilibrium model will tend to dissolve the NAPL phase slower than the equilibrium model.

8.2 DNAPL problem