Use of a Monte Carlo Method To Simulate Unstable Miscible and Immiscible Flow Through Porous Media

Abstract

Summary A Monte Carlo solution technique has been implemented within the framework of an implicit-pressure, explicit-saturation (IMPES) simulator for two-phase, or two-component, flow. The saturation equation is interpreted as a probability density function and solved statistically. This automatically triggers an unstable solution when the mobility ratio is unfavorable, thereby mimicking the instability produced by heterogeneities in the porous medium. Results are presented for linear and one-quarter five-spot miscible and immiscible problems. Introduction Following the application by Paterson1 of diffusion-limited aggregation (DLA) to the problem of infinite adverse mobility ratio miscible flow, interest developed in using Monte Carlo (statistical) techniques to solve the reservoir simulation equations. DeGregoria2,3 and Sherwood4 produced miscible formulations where the strict applicability of DLA to infinite mobility ratio has been relaxed to allow solutions to be obtained for finite mobility ratios. King and Scher5 extended this approach to immiscible flow. In such formulations, the statistical nature of the solution automatically triggers an unstable solution when the mobility ratio is unfavorable, thereby mimicking the instability produced by heterogeneities in the porous medium. A Monte Carlo solution technique has been implemented within the framework of a conventional simulator for two-phase, or two-component, flow. The implicit pressure equation is solved in the usual way. However, the explicit equation describing the rate of change of saturation or concentration of the invading fluid with time is solved statistically, interpreting the equation as a probability density function. This has involved a number of developments over and above those described in the literature; in particular, the formulation allows regions invaded by displacing fluid to desaturate. The formulations and implementation of the algorithm known as "fingering with an explicit statistical saturation equation" (FINESSE) are discussed below. The results of a series of both linear and one-quarter five-spot miscible and immiscible simulations are presented and discussed. Results are compared with an analytical model of unstable immiscible flow published recently by Hughes and Murphy.6 The advantages and disadvantages of the Monte Carlo approach to numerical simulation of porous medium flow and possible future developments are also discussed. Formulation and Implementation The Monte Carlo solution technique FINESSE has been programmed into a conventional two-phase IMPES simulator. Two-component, miscible problems are treated by entering linear relative permeability functions. The implicit solution of the pressure field is unaltered and will not be discussed further. However, the solution of the water saturation equation is undertaken on a statistical basis, as described below. For two-phase incompressible flow, the water saturation equation, neglecting the effects of gravity and capillarity, isEquation 2 where ?w=kkrw/µw. Oil saturation is calculated fromEquation 2 In finite-difference form, for each gridblock (see Fig. 1) the rate of change of water saturation with time isEquation 3 where the injection well is rate constrained and w represents upstream values. A positive value of (?S w)i, j, k indicates that water saturation in the gridblock will rise, and a negative value indicates that it will fall. If these rates of change of water saturation are multiplied by the timestep, ?t(=tn+1-tn), and summed over all gridblocks taking the sign into account, then the result represents the amount of water injected during the timestep. In a conventional IMPES simulation, the water saturation in each gridblock is adjusted at each timestep by an amount calculated with Eq. 3. In the Monte Carlo approach, the total saturation change over the timestep is allocated to a single gridblock in a discrete amount. The timestep size is selected to ensure that this discrete amount is equal to, or is some reciprocal integer fraction of, the maximum possible saturation change in a gridblock. The gridblock in which the discrete saturation change is placed is determined on a statistical basis.