A Compositional Formulation of the Pressure and Saturation Equations

Abstract

Watts, J.W., SPE, Exxon Production Research Co. Summary This paper describes a sequential implicit formulation of the compositional reservoir flow equations. This procedure offers improved stability over implicit pressure, explicit saturation (IMPES) methods but, in general, is not as expensive computationally as fully implicit methods. It also possesses several other attractive features. It permits an efficient combination of compositional and black-oil capabilities in a single program. Important to this is the fact that it lends itself to a modular program design. Finally, it has a natural physical interpretation that can be very helpful in the understanding of model behavior. Introduction In black-oil reservoir simulation, there are currently three common approaches. The simplest is the IMPES procedure, in which an implicit pressure calculation is made procedure, in which an implicit pressure calculation is made at each timestep, followed by an explicit saturation calculation. The IMPES procedure is the fastest approach on a per-timestep basis, but it can have stability problems that restrict timestep size. The most stable procedure is the coupled implicit method, in which the nonlinear functions in the interblock flow terms are evaluated at the end of each timestep. This method is the slowest approach on a per-timestep basis, particularly on large models. The sequential implicit procedure combines the attractive features of the other two. It uses the same pressure calculation as IMPES but follows it with an implicit saturation calculation. It seems to be the best method for large models in which stability is a consideration. Coats discusses these methods in more detail. Young and Stephenson proposed two categories of methods for compositional simulation-Newton-Raphson and non-Newton-Raphson. The difference in the methods lies in the way the pressure equations are formed. In Newton-Raphson methods, the pressure equation is an out-growth of the iterative technique. In non-Newton-Raphson methods, the pressure equation is based on certain physical principles. In general, Newton-Raphson methods physical principles. In general, Newton-Raphson methods include compositional effects that are neglected in non-Newton-Raphson methods, and, as a result, require fewer iterations per timestep . The method of Acts et al. is an exception to this statement, however. It is a non-Newton-Raphson, IMPES-type procedure in which the pressure equation includes the compositional effects pressure equation includes the compositional effects mentioned above. Young and Stephenson's categorization does not address the question of implicitness of interblock flow calculations. Most of the methods published (for example, those of Young and Stephenson, Acts et al., and Kazemi et al .) use explicit relative permeabilities in these calculations. An exception is the Newton-Raphson-type model described by Coats. This paper describes a sequential implicit compositional formulation. Basically, it combines and extends the ideas of Acts et al. and Spillette et al. The procedure is as follows.Construct a set of pressure equations and solve them for pressures at the new time level.Compute interblock total fluid velocities with these pressures. pressures.Use interblock total velocities and theBuckley-Leverett phase velocity relationship to construct a set of saturation equations having, in general, implicit treatment of relative permeabilities and capillary pressures. Solve these for saturations at the new time level.Use the saturations to compute interblock phasevelocities.Use the phase velocities to compute interblockcomponent transport and the amount of each component in each block at the end of the timestep.Make the required fluid property and functionevaluation calculations, and proceed to the next timestep. A general-purpose simulator based on this procedure is described elsewhere. This paper presents the formulation in some detail. Particular attention is paid to demonstrating that the compositional formulation reduces to the common black-oil formulation when it is used with black-oil fluid properties. This is a very significant advantage, because it makes solving both types of problems in a single program practical. The Pressure Equation First, the compositional pressure equation is derived. Then it is shown to have a useful physical interpretation, which, in turn, can be used to write the equation directly. The derivation seems overly long given the simplicity of the result. Unfortunately, no way to shorten it significantly has been found. Derivation. The starting point for the compositional formulation is the molar continuity equation for Component m . (1) SPERE p. 243