Two-Dimensional Method For Predicting Hot Waterflood Recovery Behavior
Authors: Spillette A.G. , Nielsen R.L. 
Published: December 2007

Abstract
The purpose of this paper is to further the understanding of reservoir response to hot-water injection by describing a two-dimensional, mathematical model of the process. Key assumptions are that no gas phase is present, and that the injected fluid reaches thermal equilibrium instantaneously with the reservoir fluids and sand. The resulting system of three partial differential equations is solved simultaneously through the use of a "leap-frog" application of standard alternating direction implicit methods for the solution of the mass-balance equations and the method of characteristics for solution of the energy-balance equation.
The utility of the mathematical model is demonstrated by comparing numerical and analytic temperature distributions for hot-water bank injection and by comparing calculated with observed field behavior. Additional calculations show that hot waterflooding can recover significantly more oil than cold waterflooding, and that a hot-water bank recovers, with less energy input, nearly as much oil as continuous injection.

Introduction
Consistent field success with hot fluid injection processes requires good reservoir description and a thorough understanding of recovery mechanisms. The latter is fostered by a combination of well designed experimental studies and physically sound mathematical modeling. The purpose of this paper is to further the understanding of reservoir response to hot-water injection by describing a two- dimensional mathematical technique, indicating its validity, and demonstrating its utility for studying the effects of reservoir and operating parameters.
Published experimental studies of hot fluid injection processes are few. Even if extensive data were available two considerations discourage exclusive reliance on laboratory or field work. First, because of the severity of scaling requirements, laboratory results must be interpreted with care. Second, because of the one-shot nature of field experiments - injectivity/productivity tests and pilot floods - results seldom are available over the desired range of operating conditions. These factors emphasize the need for devoting attention to mathematical modeling to complement laboratory and field work. Through the use of mathematical models, scaling uncertainties can be bridged and response can be predicted for unique combinations of reservoir and operating conditions.
Numerous mathematical models developed during recent years enable us to calculate temperature distributions, thermal efficiency (or conversely, fraction of heat lost), and oil recovery behavior for hot fluid injection. Ramey and Spillette recently have provided reviews of these methods. Most models have concentrated on predicting temperature distributions and thermal efficiency; few have been directed at predicting oil recovery. Although these techniques are useful for estimating effects of reservoir parameters and operating conditions on process performance, results are limited by the assumptions made and by the methods of coupling independently solved fluid-flow and energy-balance equations.
A computer-based method is presented for predicting total reservoir response to hot-water injection that obviates most simplifying assumptions. The model simulates fluid flow and heat transfer in two dimensions within a vertical cross-section spanning the oil sand and adjacent unproductive strata. Known numerical procedures are used to solve the governing partial differential equations. The mathematical model handles the effects of reservoir heterogeneity, gravity, capillarity, relative permeability and temperature-dependent fluid properties. In addition, a wide range of operating conditions can be modeled, including hot-water followed by cold-water injection.