Use of Simple Mathematical Models for Predicting Reservoir Behavior
Authors: Richardson J.G. , Blackwell R.J. 
Published: December 2007

Here is an approach for solving reservoir flow problems where behavior is dominated by a rate-limiting step. Simple models are developed for gravity drainage where vertical flow occurs, for water underrunning of viscous oils, for gravity segregation of water banks in gas caps, and for control of coning by injection of oil.

Introduction
Forecasting the behavior of a reservoir is one of the more important but complicated tasks of engineers in the oil industry. Knowledge of reserves remaining in a reservoir is vital to planning optimum depletion of a field. Unfortunately, the engineer assigned the task of predicting reserves often faces a difficult choice. For the most accurate answer, he can use a computer program that takes into account all of the pertinent factors, but this approach is usually pertinent factors, but this approach is usually expensive and time consuming, and requires a detailed knowledge of the reservoir. On the other hand, he can use conventional one-dimensional displacement calculations that are easily applied but that in some cases do not adequately describe the reservoir flow system. The purpose of this paper is to describe a middle-ground approach that in special situations has many of the advantages of the above methods with out their more serious drawbacks. This approach uses mathematical models that describe the principal flow mechanisms and can be quickly applied by hand calculations. The technology of predicting reservoir behavior has grown steadily since the pioneering work of Muskat and of Buckleys and Leverett. Muskat's tank-type or zero-dimensional method of predicting behavior in dissolved gas drive reservoirs has been invaluable to the industry. Another milestone was reached with the Buckley-Leverett method of predicting linear displacement of oil by water or gas when flow was principally along the bedding plane. The classic work of Hurst, Muskat, and van Everdingen and Hursts laid a firm foundation for problems involving unsteady-state flow of fluids. Later, progress was made by Welge in solving one-dimensional progress was made by Welge in solving one-dimensional displacement equations more easily. The advent of digital computers led to the development of methods of solving problems of greater and greater complexity. Indicating the progress being made with computers, Douglas et al. in one paper, included the effect of capillary pressure in one-dimensional flow, and in another paper dealt with the flow of two phases in two dimensions. The utility of computers in predicting reservoir behavior has continued to grow as programs have become more user-oriented and as computers have become faster and more economical to employ. But even today, the time, effort, and money required to use computers to solve reservoir problems cannot always be justified. Thus, other tools are needed. An excellent example of another approach is given in a paper by Joslin. In analyzing a gas injection project in a large Venezuelan reservoir, Joslin realized project in a large Venezuelan reservoir, Joslin realized that gas had overridden the entire oil sand because production was above the critical rate. However, the production was above the critical rate. However, the presence of pancake-like shale members penetrated presence of pancake-like shale members penetrated by the wells prevented coning of gas into perforations located below the shale near the base of the sand. Oil recovery was predicted by assuming that gas displaced oil vertically downward throughout the producing area. Another paper demonstrating the practical use of simple mathematical models is that of Matthews and Lefkovits for predicting producing rates for wells in depletion-type reservoirs.
JPT
P. 1145