An Analysis of Upstream Differencing

Abstract

Summary. A complete convergence analysis is carried out for upstream differencing as applied to one-dimensional (1D) tilted reservoirs experiencing gravity-segregation effects and possible countercurrent flows. It is shown that the standard upstream method converges to the correct physical solution of this problem, provided that an appropriate stability physical solution of this problem, provided that an appropriate stability condition holds. Results from the theory of monotone-difference schemes are used. it is concluded that in addition to the practical advantages of the upstream formulation, there is also a strong theoretical justification for use of upstream techniques in multidimensional reservoir simulation. Introduction Interest has been shown recently in the application of computational methods drawn from the study of conservation laws for the simulation of multiphase reservoir flow. Such methods are often based on schemes for advancing saturation profiles in a 1D reservoir. When coupled with auxiliary techniques, such as operator splitting, the ability to advance multidimensional saturation profiles is recovered. Then, when used with implicit-pressure/ profiles is recovered. Then, when used with implicit-pressure/ explicit-saturation (IMPES) techniques and an associated method for calculating pressure and total fluid velocities, a complete reservoir simulation model results. These techniques are somewhat complicated, especially when such effects as gravity or modeling of sources and sinks must be included. Difficulties often arise even in the solution of the appropriate 1D problems, as evidenced by the complications encountered by Gudunov's method or the random-choice methods as applied to the general nonconvex, nonmonotonic flux functions that describe simple two-phase problems with gravity effects. Complicated schemes for 1D problems are used in an attempt to control front smearing introduced by numerical diffusion. It is hoped that reduction of numerical diffusion will produce multidimensional fronts with little smearing when such schemes are incorporated into multidimensional IMPES simulators. Of course, because the fluid velocities in a 1D problem are easily determined, calculation of diffusion-free solutions for 1D flow is a realistic goal. However, because the velocities and saturation distributions can interact in multidimensional adverse-mobility-ratio flow problems to give physical instabilities (fingering), which must be damped by diffusive effects, complete removal of diffusion effects (numerical or otherwise) is not realistic in multidimensional flow. For instance, grid-orientation difficulties, which result from the uncontrolled growth of small numerical errors, are typically controlled by introducing rotationally invariant diffusion. Thus, the case for diffusion-free 1D schemes as components of multidimensional simulaters can be overstated. Standard upstream techniques, although somewhat prone to front smearing, are almost universally applied in multiphase, mul-tidimensional simulators. Formulations involving gravity osource/sink modeling offer no difficulties, and the schemes are mass conservative and physically reasonable. They can be easily applied physically reasonable. They can be easily applied to both implicit and IMPES simulators. This paper shows that when applied in an IMPES fashion as a method for simulating 1D reservoirs undergoing gravity segregation and possible countercurrent flows, standard upstream techniques are simple to compute, stable with no over/undershoot, and converge to the correct physical solution under a reasonable IMPES throughput condition. The analysis ignores compressibility effects because an oil/water situation is envisioned and relies on the theory of monotone-difference schemes. Thus, it follows that there is as much mathematical justification for using standard upstream techniques as for using more complicated schemes. It should be noted that Crandall and Majda's analysis applies not only to 1D schemes, but also to multidimensional schemes and to operator-split schemes in particular. Unfortunately, their analysis is based on conservation-law techniques and requires knowledge of the underlying velocity field (more or less constant). Under these conditions, diffusion-free solutions can be discussed. In multidimensional reservoir situations, however, a possibility exists fo real velocity-field-saturation distribution interactions that could result in unstable phenomena, such as fingering. Thus, such conservation-law analysis attacks only some of the problems inherent in multi-dimensional reservoir simulation. The next section describes the standard upstream-difference scheme and some of its properties. The description of the throughput condition is given in the Appendix. Convergence Properties of Upstreamed Schemes As noted before, a 1D reservoir situation will be envisaged involving two incompressible, immiscible fluids: oil and water. No capillary effects will be considered. For convenience, constant porosity, phi, and permeability, k, wig be used throughout the reservoir, porosity, phi, and permeability, k, wig be used throughout the reservoir, where × = 0 marks the inlet end and the reservoir extends in the positive × direction. The reservoir is assumed to have a constant crosssectional area, A, and to be undergoing a flood from the inlet (x = 0) end at a (positive) volumetric rate, q. Because the pressure level for such a problem is indeterminate, a single pressure value can be specified. This situation can be modeled with a uniform computational grid with cells of volume V=alpha A, where the ith cell is centered at × = i alpha × and i is any integer. Fluid injection is specified by attaching an infinitely long source of invading fluid associated with Cells i where i less than 0. Thus, the saturations S, in all Cells i at time level n=0 forj=o, w are assumed known. Two IMPES mass-conservation equations for S, j=o, w, in the ith cell can now be written (1) where S + S =1 must hold. In Eq. 1, n and n + 1 denote the time level, t =m alpha t; p denotes the (oil) pressure in Cell i; and p and mu i denote the (constant) density and viscosity of Fluidj, p and mu i denote the (constant) density and viscosity of Fluidj, re-spectivlely. The parameter g represents the component of the ac celeration caused by gravity in the positive × direction: g = 0 if the reservoir is flat; g less than 0 if the × = 0 injection end is the low end; and g is greater than 0 otherwise. It will be assumed, for convenience, that the fluids have been labeled so that g(p -p ) less than 0. Thus, if g less than 0, the heavier fluid should be labeled "water" and if g >0, the lighter fluid should be similarly labeled. This naming convention covers the case of greatest interest: injection of (heavier) water from an aquifer (or well) downdip in the reservoir. The functions 0 less than k (Sj) less than 1 are the given (increasing) relative permeability functions. The superscripts L and R indicate that the correct upstream direction must be used for the relative permeability evaluation. (This L/R-superscript convention will also apply later to the total mobility, fractional flow, and flux functions built from the relative permeability functions.) permeability functions.) SPERE P. 1053