Moving Elements for Transport Processes

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Abstract
Summary. We consider a class of convection-dominated tows fundamental to reservoir simulation and their approximate solutions by use of finite-element techniques and moving meshes. Both the solution field and the coordinate functions enter as primary unknowns in the resulting problem statement. Hence the mesh evolves with the solution, and the method is particularly well-suited to those problems where tracking of sharp solution particularly well-suited to those problems where tracking of sharp solution fronts is important. Examples of particular interest related to petroleum reservoir simulation are the model convection/diffusion, Burgers, and Buckley-Leverett equations. which are considered in the numerical studies. We consider effective solution techniques and system integration, as well as questions related to the use of constraints for limiting the permissible distortion of the mesh. Introduction Finite differencing with fixed grids is the standard technique for approximate solution of reservoir problems, including those problems in which convection is important. In recent years. some attempts to apply finite-element methods to the solution of reservoir problems have had mixed results. Part of the difficulty is that problems have had mixed results. Part of the difficulty is that many of the flow problems of interest have strong convective effects, and until the recent development of the Petrov-Galerkin finite-clement methods, these convective Petrov-Galerkin finite-clement methods, these convective terms were not appropriately treated in finite-element analyses on coarse meshes. The alternative of solving on a fine mesh is computationally prohibitive, especially in situations where the solution fronts propagate through the reservoir domain. as in EOR procedures, The Petrov-Galerkin finite-element methods use upwind-biased test Petrov-Galerkin finite-element methods use upwind-biased test functions to produce. in effect, a backward-difference approximation for the convective terms, and suffer from the same detractions as backward-differencing in the finitedifference models. Namely, they are very dissipative schemes and tend to smear the fronts. Particularly in EOR applications, where chemical slugs are injected at a well and propagate into the flow domain with little physical diffusion (dispersion), it is important that the integrity of the front be preserved and that the solution near the front be approximated accurately. If artificial dispersion is not introduced into the problem through backward-differencing, then the approximate solutions to these problems on typical meshes are oscillatory; i.e., while the problems on typical meshes are oscillatory; i.e., while the location of the front center may be approximated reasonably well, there are fictitious oscillations in the approximate solution both in front of and behind the true concentration profile. In multiphase, multicomponent flows, these oscillations are carried through the mass-balance relationships into the other components and phases. Some attempts have been made to exploit the almost hyperbolic nature of the convection-dominated flow problem in both finite-difference and finite-element problem in both finite-difference and finite-element calculations by appealing to the method of characteristics. If the physical dissipative term is negligible, then the problem physical dissipative term is negligible, then the problem degenerates to a hyperbolic system of equations and the solution fronts propagate without dispersion through the domain. For a linear scalar field problem, it is relatively straightforward to compute the characteristic coordinates arid velocities and to allow the grid to move in a fixed frame with the characteristic velocity. Hence a graded mesh that is dense near an initial front will propagate with the front and maintain the accuracy of the solution. This approach is not applicable if the diffusion in the system is significant and may be very difficult to implement for nonlinear multiphase, multicomponent problems. It does suggest, however, that a possible strategy for maintaining accuracy and stability in the calculations is to formulate the problem in a transformed variable that permits the mesh to move and to maintain the accuracy of the approximate solution. Naiki used a method-of-characteristics moving-grid approach in a finite-difference computation for polymer flooding, and Douglas and Russell have considered finite elements in a characteristic frame. Brackbill has used moving meshes in finite-difference computation for shock-wave propagation problems along lines similar to those in Yanenko et al. Jensen and Finlayson applied a moving-coordinate system to a finite-element reservoir simulator. This method tracked the motion of a displacement front by moving all nodes at the same velocity as the front. The ideas developed in the present study are an extension of those introduced in Refs. 7 and 8. In Miller's moving-finite-element method, the node velocities are determined by the same variational principle that governs the evolution of the solution values. Some features of the methodology are also described in Carey and Mueller and Thrasher and Sepehrnoori. A more general description of some aspects of the work is also given in Mueller and Carey. SPERE p. 401

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