Department of Mathematical Sciences University of South Carolina Aiken Aiken, SC 29801

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1 RESEARCH SUMMARY AND PERSPECTIVES KOFFI B. FADIMBA Department of Matematical Sciences University of Sout Carolina Aiken Aiken, SC Introduction My researc program as focused so far on investigating numerical approximations of equations obtained from modeling multi-pase flow troug porous media. Example applications include groundwater flow and pollution (single pase flow, concentration problem), and oil reservoir simulations (immiscible flow, saturation problems [1, 2, 3, 4]), but te tecniques developed can be applied to oter type of advection-diffusion equations. In addition to te strong coupling and nonlinearity, one main problem we address is te degenerate nature of many of tese problems: tat is te diffusion coefficient vanises for some values of te solutions. In tis case sufficient regularity of te solution is not guaranteed for numerical treatment [5, 6]. Tis calls for some form of regularization. For te case of two-pase immiscible displacement troug porous media, we ave (wit some simplification) te following nonlinear coupled system: (1.1) u = a(s) p + b(s) z in Ω (0, T ) div(u) = Q 1 in Ω (0, T ) u η = q 1 on Ω [0, T ] pdx = 0 for all t [0, T ] Ω (ϕs) + (f(s)u k(s) S) = Q(S) in Ω (0, T ) t k(s) S = q on Ω [0, T ] η S(x, 0) = S 0 (x) in Ω wit k(1) = k(0) = 0 and 0 S 0 (x) 1. Te main goal is to approximate accurately and efficiently te solution (p, S) of tis system, were p is te global pressure of te pases, and S te saturation of te invading fluid. Key words and prases. Multi-pase flow, nonlinear, degenerate, saturation, regularization, numerical metods. 1

2 2 KOFFI B. FADIMBA 2. Completed Work In our P.D Dissertation [7], (under te supervision of Dr. R. C. Sarpley, University of Sout Carolina, Columbia ), we ad considered te saturation equation, wit te absence of te pressure part: (2.1) S + (f(s)u) (k(s) S) = Q(S) in Ω (0, T ) t (f(s)u k(s) S) n = 0 on Ω [0, T 0 ] 1 S(x, 0) = S 0 (x) 0 on Ω were Ω is a bounded domain in R n, wit n = 1, 2, 3. We regularized tis general degenerate problem to a nondegenerate one, following te general direction as in [5, 6, 8]. Wit some control on ow te diffusion coefficient vanises, we establised a priori estimates and error estimates for te regularized problem[9], and error estimates for a Galerkin finite element approximation of te problem[10]. Tree papers ave come out of tis work[9, 10, 11]. After te dissertation a tird paper ([12]) followed wic furter improved some of te regularity estimates obtained in te dissertation. Many autors ave dealt wit tese types of problems, mainly te nondegenerate case. In [13], we ave considered te full system (1.1), and under reasonable pysical assumptions on te data, we proved existence and and establised some conditions for uniqueness for te system. Tis was done independently of [14], wic uses a different approac. In [15], we considered a standard Galerkin metod applied to te full system (instead of a mixed finite element metod for te pressure part and standard Galerkin metod for te saturation as in [16]), and establised error analysis for bot te continuous and te fully discretized metods. A sequel of te paper [13] was [17], were sufficient conditions for uniqueness for te pressure/saturation system were considered troug grapical experiments. Because te problem 2.1 (te saturation problem) is non linear, most of te teoretical scemes considered for a numerical approximation of te problem are also nonlinear and necessitate a furter linearization (Newton, Picard-type iterations: see, for instance, [18, 19, 20]). A self-assigned project considering a readily linear sceme (fully discretized) for 2.1 was started in a series of papers [21, 22, 23, 24]. We consider te problem: Find a sequence of functions (U n) 1 n N, wit U n M, (M a family of Finite Element Approximation spaces of H 1 (Ω)) satisfying te linear sceme ( U n+1 U n ) H t β(u n ), χ ({ (f H β )(U n ) + (U n+1 U n ) (f H β ) (U n )} u n+1, χ ) (2.2) + ( U n+1, χ ) = 0, χ M 0 n N 1 (2.3) P H β U 0 = P S 0 In [21], te feasibility of te sceme (2.2)-(2.3) was analyzed. At eac time step, we obtain an algebraic linear system wose matrix is positive-definite in some sense. In [22], we establised te stability and te consistency of te sceme. We

3 3 ten derived convergence estimates for te sceme in L 2 (0, T 0 ; L 2 (Ω)) ([23]), and in L (0, T 0 ; L 2 (Ω)) and L 2 (0, T 0 ; H 1 (Ω)) ([24]). 3. Perspectives and Oter Directions 3.1. Work in Progress. Te analyses in [21] [24] are still teoretical. Te next pase of te project is to produce numerical results tat support (or infirm) te teoretical estimates establised in tese papers. Te numerical results could also suggest improvements to make for te sceme analyzed. We are presently writing a proposal requesting funds for equipments (computer, software, course release, etc.) tat will elp start tis pase. In [21], its was sown tat te sceme translate to a linear system of equations at eac time step wose unknowns are U n at te spatial grids points. Tis brings us to numerical linear algebra were will ave to look at various metods (direct (LU, QR decompositions) or iterated) wit or witout te software in te market, still keeping in mind te degenerate nature of our problem. We are also investigating te sceme proposed in [15], wic is decoupled and partially linearized. Te project consists in using te popular mixed finite element metod on te pressure equation and a standard finite element metod (conforming, Discontinuous Galerkin, or oter) to te saturation equation, or using a mixed finite element metod to bot te pressure and te saturation equation. In te latter case, tis supposes an adequate splitting of te saturation equation Oter Directions. Anoter promising metod is te Eulerian Lagrangian Localized Adjoint Metod (ELLAM) [25, 26]. Tis metod as been well developed for linear advection-diffusion equations (see [27], for instance). We will study a ybrid metod wic applies ELLAM to te saturation equation (2.1) and te mixed finite element metod to te pressure. Unfortunately, tis is a nonlinear advection-diffusion equation and tere are several possibilities to consider, for example: (1) Linearize (2.1), by te Picard iteration, for instance; (2) Develop te ELLAM teory for nonlinear equations; Te last consideration is a joint researc topic at te University of Sout Carolina, Columbia, and te Institute for Scientific Computation, Texas A & M University, College Station. Oter interests: (1) More general PDE (linear or nonlinear): How do tey work, for instance, wit Wavelets? [28] (2) Population Dynamics Equations (3) Atmosperic Dynamics. (4) Open for collaboration wit researc groups working on same topics or different topics

4 4 KOFFI B. FADIMBA References [1] J. Bear and A. Verruijt. Modeling Groundwater Flow and Pollution. D. Reidel Publication Company, Dodreic, Holland, [2] G. Cavent and J. Jaffre. Matematical Models and Finite Element for Reservoir Simulation: Single pase, multipase and multicomponent flows troug Porous Media. Nort-Holland, New York, [3] R. E. Ewing. Problems arising in te modeling of processes for ydrocarbon recovery. In R. E. Ewing, editor, Te Matematics of Reservoir Simulation, pages 3 34, Piladelpia, S.I.A.M. [4] Z. Cen, G. Huan, and Y Ma. Computational Metods for Multipase Flows in Porous Media. SIAM Computational Science and Engineering, Piladelpia, [5] M. E. Rose. Numerical metods for flow troug porous media I. Mat. Comp., 40: , [6] M. E. Rose. Numerical metods for flow troug porous media-ii. Comput. Mat. wit Appls, 6:99 122, [7] K. B. Fadimba. Regularization and Numerical Metods for a Class of Porous Medium Equations. PD tesis, University of Sout Carolina, Columbia, [8] D. L. Smylie. A Near Optimal Order Approximation to a Class of Two-sided Nonlinear Parabolic Partial differential Equations. PD tesis, University of Wyoming, Laramie, [9] K. B. Fadimba and R. C. Sarpley. A priori estimates and regularization for a class of porous medium equations. Nonlin. World, 2:13 41, [10] K. B. Fadimba and R. C. Sarpley. Galerkin finite element metod for a class of porous medium equations. Nonlinear Analysis: Real World Applications, 5: , [11] K. B. Fadimba. Sur l oprateur de rsolution de poisson et son analogue discret. Journal de la Recerce Scientifique de l Universite de Lome, 4.2: , [12] K. B. Fadimba. Error estimates for a regularization of a class of porous medium equations. Imotep, J. Mat Pures Appl., 3:77 96, [13] K. B. Fadimba. On existence and uniqueness for a coupled system modeling immiscible flow troug a porous medium. J. Mat. Anal. Appl., 328: , [14] Z. Cen. Degenerate two-pase incompressible flow-i: Existence, uniqueness, and regularity of a weak solution. J. Differential Equations, 171: , [15] K. B. Fadimba. Error analysis for a galerkin finite element metod applied to a coupled nonlinear degenerate system of advection-diffusion equations 6.1 (2006): Comput. Metods Appl. Mat., 6(1):3 30, [16] Z. Cen and R. E. Ewing. Degenerate two-pase incompressible flow. III: Sarp error estimates. Numer. Mat., 90.2: , [17] K. B. Fadimba. Pressure/saturation system for immiscible two-pase flow: Uniqueness revisited. Applied Matematics, 2: , [18] M.A. Celia and al. A general mass-conservative numerical solution for te unsarurated flow equation. Water Resources Researc, 26, No. 7: , [19] F. A. Radu and I. S. Pop. Newton metod for reactive solute transport wit equilibrium sorption in porous media. J. Comput. Appl. Mat., 234: , [20] F. A. Radu and I. S. Pop. Mixed finite element discretization and Newton iteration for a reactive contaminant transport model wit nonequilibrium sorption: convergence analysis and error estimates. Comput. Geosci., 15: , [21] K. B. Fadimba. A linearization of a backward euler sceme for a class of degenerate nonlinear advection-diffusion equations. Nonlinear Analysis, 63/5 7:e1097 e1106, [22] K. B. Fadimba. A linear backward euler sceme for te saturation equation: Regularity results and consistency. Journal of Computational and Applied Matematics, 234: , [23] K. B. Fadimba. Convergence estimates for a linear backward euler sceme for te saturation equation. Te Open Applied Pysics Journal, 5:41 53, [24] K. B. Fadimba. A linear backward euler sceme for a class of degenerate advectiondiffusion equations: A matematical analysisof te convergence in L (0, T 0 ; L 2 (Ω)) and in L 2 (0, T 0 ; H 1 (Ω)). Analysis and Applications, To appear, [25] J. E. Vag, H. Wang, and H. K. Dale. Eulerian-Lagrangian Localized Adjoint Metods for systems of nonlinear advection-diffusion-reaction equations. Adv. Water Resourc., 19: , 2000.

5 5 [26] H. Wang et al. An ellam simulator for igly compressible flow in porous media wit multiple wells. fluid flow and transport in porous media: matematical and numerical treatment. In Contemp. Mat. (295), editor, Sout Hadley, MA, 2001, page , Providence, RI, Amer. Mat. Soc. [27] H. Wang, H. K. Dale, R. E. Ewing, M. S. Espedal, R. C. Sarpley, and S. Man. An ELLAM sceme for advection-diffusion in two dimensions. SIAM J. Sci. Comput., 20: [28] R. E. Ewing et al. Adaptive wavelet metods for advection-reaction equations. In Contemp. Mat. (329), editor, Current trends in scientific computing (Xi an, 2002), page , Providence, RI, Amer. Mat. Soc. Department of Matematical Sciences, University of Sout Carolina Aiken, Aiken, SC address: