Waterflooding Under Bottom-water Conditions-An Analytical Model for Two-layer Reservoirs
Authors: Shirif E. , Ali S.M. Farouq 
Published: March 2010

Abstract
Many reservoirs in Alberta and Saskatchewan contain a high water saturation zone (bottom-water) underlying the oil zone. Waterflooding under such conditions is typically ineffective because of channeling of water through the bottom-water zone. However, in some cases, waterflooding such reservoirs may still be feasible and economically viable. While there is no doubt that some of the injected water bypasses the oil zone through the bottom-water zone, most of the injected water may still displace the oil, depending on the reservoir conditions. Therefore, a mechanistic understanding of oil displacement by a waterflood in the presence of a bottom-water zone is the basis for predicting recovery performance, and there is a need for developing a mathematical model to describe water channeling under bottomwater conditions. Given the reservoir description, the mathematical model should be able to describe the amount of water channeling into the bottom-water, together with a prediction of oil recovery. In this paper, an analytical model was developed to predict waterflood performance when a water zone is present. Results of the mathematical predictions are compared with experimental and simulation results, showing good agreement.

Introduction
Among the factors influencing fluid flow in layered permeable media is flow from one layer to another in a direction perpendicular to bulk flow. This crossflow may be the result of any or all of the four forces that cause fluid to flow in a permeable medium: viscous forces, capillarity, gravity, and concentration. These driving forces interact with each other in experimental displacements(1,2), making it difficult to credit the observed crossflow to the correct mechanism. This is true to a large extent in simulated displacements(3), but it is possible to simulate displacements with only one effect present(4).
Yeung(5) developed a mathematical model for a two-layered reservoir, the lower layer being a water zone to account for crossflow, based on the major assumption that crossflow did not alter the mobility in either layer. It was concluded that crossflow occurred near the injection end and waterflood performance was independent of the point of injection and the injection rate. Hassan(6) modified Yeung's model but used the same assumption. He also developed a semi-analytical model to predict oil recovery performance and calculate the frontal movement of the two flood fronts for bottom-water reservoirs.
In view of the foregoing, when a stratified reservoir is being studied for waterflooding, failure to account for crossflow can lead to large errors in oil recovery prediction; however, under bottom-water conditions, this effect is aggravated due to the presence of the mobile water phase.
The objective of this chapter is to describe theoretically the effects of the crossflow caused by viscous forces on displacements in a two-layer reservoir, the lower layer being a water zone. This can be accomplished in an unconventional way by solving the flow equations for maximum crossflow, which utilizes the idea of vertical equilibrium solution techniques.
The vertical equilibrium concept has been used extensively in the petroleum literature(7-11), mainly as a way to collapse simulations to a lower dimension.