Although reservoir permeability variations are normal, many of our presently used predictive models are based on assumptions of homogeneity. The permeability variations usually exist as differences in both the horizontal and vertical planes of the reservoir. This paper is concerned with the variations in permeability paper is concerned with the variations in permeability of the different layers that are created when sedementary rocks are formed (permeability stratification). This type of permeability heterogeneity usually shows the greatest variation and has, by far, the greatest effect on secondary recovery operations.
Three specific computer models have been developed for use in predicting hydrocarbon recovery from secondary recovery projects in stratified reservoirs. These three models represent the following types of permeability stratification. permeability stratification.
Continuous strata with the same vertical permeability profile at each point in the reservoir with no vertical communication between the strata.
Strata which are locally continuous but allowing different vertical permeability profiles in the various localities, with no vertical communication between the strata.
Like #1 (continuous strata) but allowing for some vertical communication between strata.
In all of the models the flow system of the reservoir is defined using a two dimensional streamline generator. The generalized equations of this generator are shown in Appendix A. Basically, the stream-lines are traced for any given system of production and injection wells by starting the fluid particles for a given production well and a set number of streamlines evenly around the production well radius RI. Once this is done the particles are moved along the streamlines using the equations shown in Appendix A. Thus, any number of streamlines can be traced for any given production-injection system. The small distance used in the time calculation is assumed to be the well radius, RI, and it has been shown that so long a RI is less than or equal to two percent of the distance between wells this provides an accurate flow description. provides an accurate flow description. Once the flow system in a reservoir has been described the streamlines are retraced going now from the injection to the production wells and the recovery history for the reservoir is calculated. During this part of the model the reservoir is assumed homogeneous using average field data and oil, gas, and water production, water injection, and conductivity ratio are calculated. As water injection continues, fill up is observed along with the development and movement of an oil bank followed by a water front in the reservoir. During this part of the model variable reservoir thickness is also considered utilizing data in the form of a thickness grid over the reservoir area. This information is used to adjust the fluid velocity, both for the water and the oil fronts as they move through the reservoir.
Since most petroleum reservoirs are confined or bounded these models would be of little use if they could not be bounded also. The method that is used to bound the flow system consists of placing bounding wells around the outside of the reservoir. These imaginary wells can then be used to mathematically create a no flow boundary at the reservoir edges. The rates at which the boundary are calculated by solving simultaneously a set of equations developed from the LaPlace line source and sink equation. These equations are presented in Appendix B.
Using the basic model previously described as a base, the first stratified model can be used with reservoirs having continuous stratification with strata mobility ratios that do not vary greatly from each other or with reservoirs for which a minimum amount of data exist.