An Analytical Model for Water/Gas Miscible Displacements

Abstract

Member SPE-AIME Abstract This paper describes the solution to a steady-state analytical model for gravitational segregation in a water-alternating-gas (WAG) miscible displacement. The model, introduced by Stone (1), provides a conceptual framework for understanding the segregation process. The analysis is extended in this paper to obtain a closed-form solution to the equations, which allows one to easily estimate the incremental recovery above waterflooding for homogeneous reservoirs. This solution is useful in the design of high-recovery WAG floods. Results from the model indicate that recovery increases with dimensionless viscous-gravity ratio (VGR), which is a measure of the volume in the reservoir required to obtain complete segregation of water and gas injected into a well. The model may be used to show how changes in fluid and operating conditions affect recovery. Recovery may be increased either by increasing injection rate r solvent viscosity, or by reducing pattern area or water-solvent density difference. The reservoir model consists of three distinct zones: a top zone in which only gas is mobile, a bottom zone in which only water is mobile, and an intermediate mixture zone in which the mobility is determined by the water-gas ratio of the injection mixture. Simple expressions are derived that define the boundaries of these zones at any location in the reservoir. These expressions are integrated to obtain ultimate recovery as a function of zone mobilities, injection water-gas ratio and viscous-gravity ratio (VGR). Recovery is shown to be a linear function of VGR for VGR less than one; for VGR greater than one, a logarithmic dependence is obtained. The same recovery expression governs both rectangular and radial geometries. The governing equations and assumptions are discussed in this paper. The equations are solved, and example calculations are presented. Comparisons are made with numerical simulation results, and the agreement is good. The equations are examined parametrically, showing the effect on recovery of changes in water-gas ratio and zone saturations. A simplified method for using the model to estimate recoveries for VGR less than one is discussed. The method used to determine zone saturations is also described. Sufficient detail is provided to permit recovery calculations for any set of reservoir data. Background and Motivation Computer simulation of miscible processes is difficult and expensive. Numerical dispersion inherent in most reservoir simulators may result in optimistic recoveries. Numerical dispersion may be reduced by reducing grid block sizes and by using two-point upstream mobility weighting, but only at a substantial increase in cost. In many cases it is desirable to obtain a simple estimate of recovery efficiency for given fluid and reservoir properties, and to examine the effect on recovery of different operating strategies and property variations. The model presented in this property variations. The model presented in this paper is therefore useful both as a first step paper is therefore useful both as a first step in estimating steady-state recovery, to determine if more detailed study is warranted, and as an inexpensive alternative to reservoir simulation for screening and sensitivity studies. It also provides a convenient conceptual framework for provides a convenient conceptual framework for interpreting simulation results. Governing Equations and Assumptions The geometry and nomenclature of the gravitational segregation model developed by Stone (1) for a horizontal, homogeneous, rectangular reservoir element are shown schematically in Fig. 1. The model applies to an injection fluid having a constant water-gas (WAG) ratio, injected uniformly across the left vertical face with total flow rate qt. The reservoir is assumed horizontal, and therefore there are no gravitational effects in this direction. As the mixture moves into the reservoir, gravitational segregation causes a gas-only mobile zone to form at the top, and a water-only mobile zone to form at the bottom.