A Numerical Study of Waterflood Performance In a Stratified System With Crossflow
Authors: Goddin C.S. , Craig F.F. , Wilkes J.O. , Tek M.R. 
Published: December 2007

Abstract
The waterflood performance of a water-wet, stratified system with crossflow is computed by a finite difference procedure. The effects of five dimensionless parameters on the oil displacement efficiency, water saturation contours and crossflow rates are evaluated in the absence of gravity forces. Crossflow due to viscous and capillary forces is shown to exert a significant erect off oil recovery in a field-scale model of a two-layered, water-wet sandstone reservoir. The crossflow is at a maximum in the vicinity of the front advancing in the more permeable layer. Under favorable mobility ratio conditions, the computed oil recovery with crossflow always is intermediate between that predicted for a uniform reservoir and that for a layered reservoir with no crossflow.
Introduction
The important effects of reservoir heterogeneity on waterflood performance are commanding increased attention in the technical literature. Much of this attention is centered on two categories of layered reservoirs: those in which layers are non-communicating and those in which crossflow of fluids occurs between the layers. In the first category, the reservoir is assumed to consist of discrete layers, each uniform within itself and differing from the others only in such properties as thickness, porosity and absolute permeability. The performance within each layer is calculated by one-dimensional flow theory, and the performance of the total reservoir is obtained by summing individual layer performances. Capillary and gravity effects usually are not considered. Representative publications dealing with this type of reservoir are those of Stiles, Dykstra and Parsons, Hiatt, Warren and Cosgrove and Higgins and Leighton. Prediction of performance for reservoirs in the second category is considerably more difficult since viscous, capillary and gravitational forces all play important roles in causing crossflow between layers. A number of authors have investigated the simpler problem of two-dimensional displacement flow in a stratified system with a mobility ratio of unity and negligible capillary and gravity effects. Others have considered two-dimensional, nonsteady-state flow of a single, slightly compressible fluid in a stratified reservoir.
A limited number of laboratory oil displacement tests in layered models with crossflow have been reported. Miscible floods (with resultant zero capillary forces) in layered five-spot models were conducted by Dyes and Braun, who studied the effect of mobility ratio with zero gravity forces, and by Craig et al. who studied the effect of gravity forces at constant mobility ratio. Waterfloods in layered five-spot models (with crossflow due to capillary, viscous and gravity forces) were conducted by Gaucher and Lindley, who showed the effect of gravity forces in causing underrunning of the injected water and by Carpenter, Bail and Bobek, who demonstrated the reliability of Rapoport's dimensionless parameters for scaling layered systems. Waterfloods in rectangular layered models were conducted by Richardson and Perkins, who investigated the effect of velocity at constant mobility ratio and with zero gravity forces, and by Hutchinson, who studied the effects of varying mobility, layer permeability and layer thickness ratios. The differential equations which rigorously describe waterflooding in a heterogenous porous medium are non-linear and do not facilitate analytical solution. By using finite difference approximations it is possible to obtain a solution to any desired degree of accuracy. Such a solution, using an alternating direction implicit procedure (ADIP), is described by Douglas, Peaceman and Rachford. In the present study, a computer program using ADIP explores systematically the effects of important parameters on waterflood performance of a two-dimensional, two-layered, field-scale model of a water-wet sandstone system. Particular attention is given to evaluation of the water saturation contours and crossflow rates at the interface between layers to gain improved understanding of the crossflow mechanism.
PROCEDURE
BASIC FLOW EQUATIONS
The basic flow equations for two-dimensional, two-phase, immiscible, incompressible flow in a porous medium are:
........................................(1-a)
........................................(1-b)
JPT
P. 765ˆ