Estimation of Storativity Ratio in a Layered Reservoir with Crossflow

Abstract

Abstract This paper presents a practical method to estimate the storativity ratio of a layered reservoir with cross-flow from pressure transient data. The method uses an analytically derived formula for the storativity ratio in terms of the separation between the two semi-log straight lines on pressure versus log-time plot. Knowing the storativity ratio from a well test, individual layer properties may be estimated if the layer flow rates are available from production logs. Demonstrations of the method to estimate the storativity ratio and individual layer properties are presented by examples. Comparison of the results with those obtained from the existing techniques is also provided to highlight the accuracy of the proposed technique. Introduction When the production performance of a reservoir is being modeled by using average properties, as in the case of primary recovery from a reservoir of continuously changing permeability, the well-test estimated properties may be of direct utility. In some other cases, especially where the heterogeneity manifests itself in the form of jump discontinuities at the boundaries of otherwise homogeneous zones, the individual zone properties may be required to characterize the fluid transfer among various zones. The characteristics of fluid transfer between the zones, in turn, dictate the depletion characteristics of the overall system. Multi-layer systems with cross flow are good examples of reservoirs that fall into this category. For layered systems with cross flow, the capability of pressure transient analysis to provide the average system properties is not sufficient for reservoir description unless the average properties lead to the estimation of the individual layer properties. It can be shown that to obtain the individual layer properties from the average properties, two parameters are required. These are the storativity ratio, defined as the ratio of the storativity of the layer with high flow capacity to the total system storativity, and the transmissivity ratio, which is the ratio of the maximum layer flow rate to the total system rate. An estimate of the transmissivity ratio may be obtained from production logs. The storativity ratio on the other hand needs to be determined from the pressure transient data or by independent means. For dual-porosity systems, such as naturally fractured systems, the storativity ratio may be determined from the separation between the two parallel straight lines on the pressure versus log-time plot.1,2 For dual-permeability systems, as in the layered reservoirs with cross flow, however, the separation between the two parallel semi-log straight lines is not only a function of the storativity ratio but also a function of the transmissivity ratio. Therefore, the objective of this study is to obtain a practical relation for the storativity ratio of layered systems with cross flow in terms of the separation between the two semi-log straight lines (on pressure versus log-time plot) and the calculated transmissivity ratio. Background and Definitions In the literature, dual-porosity and dual-permeability system definitions are usually associated with naturally fractured and layered systems, respectively. In principle, dual-porosity systems constitute a subset (a limiting case) of the dual-permeability systems and, as such, possesses many characteristics that resemble those of dual-permeability systems.3 Warren and Root1 showed that the transient pressure responses of wells in dual-porosity systems are controlled by two parameters:Interporosity flow coefficient, ?, is defined byEquation 1 The interporosity flow coefficient is a measure of the efficiency of fluid flow from the matrix to the fissure. The s symbol is the shape factor, which is a function of the size and shape of the matrix blocks, km is the permeability of the matrix system, and kf is the permeability of the fissure system.