The Application of Variational Methods to Waterflooding Problems

Abstract

ABSTRACT A numerical method based on the calculus of variations is proposed for thesolution of displacement problems. The mathematical development is given inconsiderable detail and two numerical examples are presented. Displacementsleading to smooth saturation profiles are approximated with very high accuracy.Displacements leading to very steep saturation profiles are approximated withconsider ably greater accuracy by the Galerkin method of this paper than byfinite differences for the same amount of calculation. The sharp front islocated with a small error. INTRODUCTION DISPLACEMENT OF ONE FLUID BY ANOTHER from the pores of rock has been studiedextensively. The most widely used approach for obtaining predictionsengineering purposes has been the use of finite for difference procedures. Insome cases, this approach has been most satisfactory. However, displacementproblems are characterized by the movement through the reservoir of a "front,"a zone of rapid change in saturation, and adequate prediction of frontalprofiles by finite difference methods usually requires an un economicrefinement of the grid. The problem and alternative approaches are discussed in [8, 12, 14].Reference [8] recognizes the almost hyperbolic nature of the problem, andproposes a hyperbolic method, namely the method of characteristics, formiscible displacement. In its implementation, a number of moving points aredefined to treat convection (i.e., transport by flow), and dispersion istreated by approximating the small amount of transport due to mixing by afinite difference technique applied to a fixed grid through which the pointsmove. In [14], the idea is generalized to the immiscible case, and it waspointed out there that the point density (i.e. the number of points per cell ofthe grid must be increased as the grid is refined.) This increase must be madein exactly the same way that the precision of the computations (i.e., wordlength of the machine) must be increased as the grid is refined, because thetruncation error dependence upon point density enters in much the same way asthe rounding error dependence on the precision of computation. In [12], an experimental study of convergence rate (i.e., the rate at whichthe computed solution approaches the exact solution of the problem as the gridis refined) using a constant point density indeed verified the expected lack ofconvergence, while revealing that with a fixed density the errors remain smallbut bounded. In [12], there was also discussed an alternative approach based onvariational methods for which the experimental convergence studies on "hard"problems, (i.e., ones with sharp fronts) are most encouraging. This workexamined the Galerkin formulation for one-dimensional miscible displacement andconcluded that it offered far superior accuracy for the computing work involvedwhen compared with finite difference techniques. This same conclusion may bevalid for the comparison with the method of characteristics. as the pointdensity must theoretically be increased linearly with the number of gridintervals. This, however, remains an open question. The purpose of this paper is to study experimentally the value of Galerkinprocedures applied to a significantly more difficult displacement problem, namely waterflooding in which a substantial Buckley-Leverett "shock" develops.In the physical problem in which capillarity plays a role, such shocks are notdiscontinuities, but degenerate into regions of very steep