A general model which approximates the effect of cross-flow has been developed to give a practical method for predicting the water flood behavior of a stratified reservoir. The model is based on a modification of Dietz's theory and allows for variations in both the permeability and hydrocarbon pore volume.in the particular cases considered, it is assumed that the permeability can be characterized by a log- normal distribution and the hydrocarbon pore volume by a normal distribution. A simple graphical method which enables the practicing engineer to predict the behavior of a stratified system is presented The results obtained by the proposed method are compared with those obtained by the Dykstra-Parsons method As a result of this study the following conclusions have been drawn:
The effect of cross-flow in a stratified system can be appreciable, particularly at very favorable or very unfavorable mobility ratios;
Under normal conditions, the effect of variations in the hydrocarbon pore volume can be neglected; and
The failure to use all of the available permeability data can lead to large errors in the prediction of the behavior of a stratified reservoir.
Various methods have been proposed to characterize and to predict the waterflood behavior of a stratified reservoir. Most of the methods assume that the reservoir is composed of discrete homogeneous continuous layers. With this model, the degree of stratification can be measured by several parameters based on core-analysis data. Among these are the Lorenz coefficient and the variation of a log-normal permeability distribution. The behavior of stratified systems is usually predicted by the Stiles, Dykstra-Parsons method or some modification of these. In both of these methods the reservoir is divided into discrete homogeneous layers with no cross-flow between layers. In the Stiles method the mobility ratio is assumed to be equal to unity while in the Dykstra-Parsons method it is allowed to take on any value. Other predictive methods have been proposed; Hiatt's method allows for cross-flow between the beds and a method by Schmalz and Rahme correlates the recovery directly with the Lorenz coefficient.In this new approach, which is essentially a continuous analog of Hiatt's method, the effects of both mobility ratio and cross-flow between the beds have been included. It is assumed that the permeability can be represented by a log-normal distribution and the hydrocarbon pore volume or porosity by a normal distribution. These types of distributions have been observed by several authors, and most field data seem to confirm their observations; e.g., if these distributions are assumed and there is a 1:1 correspondence between porosity and permeability samples, the commonly encountered linear relationship between porosity and the log of permeability is obtained.
(1)
Since, in many field cases the permeability and porosity data are truncated or cut-off at predetermined upper and/or lower values, the effect of discarding part of the data was investigated with the proposed method. A better technique for truncating the data is suggested.
In the derivation of the method to be described in this paper the following assumptions are made:
Capillary forces are negligible per se; their effects are only manifest in the relative permeability curves.
The fluids are immiscible, incompressible and homogeneous.
The reservoir is horizontal and uniformly thick; it is initially liquid saturated i.e., connate water and oil.
SPEJ
P. 149^
