Predicting the Performance of Injection Operations In Stratified or Pseudo-Stratified Formations
Authors: Caudle B.H. 
Published: December 2007

Abstract
Nearly all petroleum reservoirs are either truly separated into strata of different flow properties, or have heterogeneities which can be treated as such stratification. In pattern injection operations this can lead to a quite different production history than would be computed for a homogeneous reservoir. The usual way to consider such stratification has been typified by the Stiles or Dykstra-Parsons approaches which consider all well injection patterns to result in linear flow. This paper details a general method for using either computed or measured (on scale models) individual stratum production histories and conductivity ratios to obtain a combined production history for the stratified systems mentioned above. The method considers the proper well geometry and the flow and saturation characteristics of the individual strata.
Introduction
Almost without exception, petroleum reservoirs are non-homogeneous in rock characteristics such as permeability and porosity. Most of our oil recovery predictions are based on the concept of homogeneous rocks. We must therefore try to adjust or adapt these if we wish to obtain realistic recovery predictions. One of the most common types of reservoir heterogeneity is permeability and porosity stratification. This is typified by a reservoir consisting of two or more blanket-like oil bearing strata which can be traced from well to well in the field. Each of these strata can often be considered homogeneous in permeability and porosity. The strata can be separated by impermeable layers (such as shale). In this case they will act as separate reservoirs with common boundary conditions. The strata can also be contiguous so that fluids may flow freely from one to the other at any point of contact. In this case, the flow of fluids is more difficult to predict. Another common type of reservoir heterogeneity is a variation in rock properties that do not occur in any regular manner. Recovery predictions are usually made by equating this complex system to a system of discrete strata of individually uniform properties. Several statistically- based methods for converting core sample properties into a pseudo-stratified reservoir are in common usage. These end by plotting permeabilities (in ascending order) vs reservoir height, and then dividing this into the most plausible system of individual strata. To be most effective, these pseudo-strata should not only have individual permeabilities but also appropriate values for porosity, initial oil in place, recoverable oil, relative permeabilities, etc. Once resolved in pseudo-strata, this type of reservoir can be treated in the same way as the real stratified systems except that there is no possibility of impermeable layers between the pseudo-strata.
Published Methods for Stratified Reservoirs
Computational methods for predicting secondary recovery results in stratified systems generally assume that fluids cannot flow from one stratum into another even though crossflow between the strata is possible. This assumption is probably reasonable as long as the mobility ratio is unfavorable (M &gt;1).* Few secondary recovery operations have favorable mobility ratios except for the oil bank fill-up period. One of the first methods to be published and widely used was that of Stiles. This method assumed that the reservoir was made up of linear strata of different permeabilities and that the mobility ratio between the injected fluid and the produced fluid was 1. All other properties of the strata were considered equal, Dykstra and Parsons enlarged this method to include the effects of unequal fluid mobilities.',' However, the assumptions of a linear flow system and identical stratum properties (except specific permeability) were kept. Neither method treated the case of a waterflood in a reservoir containing a flowable gas phase. The Dykstra-Parsons method did introduce the concept of changing fluid velocity in each stratum as the injection continued. Two more recent publications use a stratum combination method that can be applied whenever volumetric sweep and fluid conductivity data for the appropriate homogeneous system are either available or can be computed. This method and some of its logical extensions will be further discussed.
JPT