A Method to Determine the Permeability-Thickness Product for a Naturally Fractured Reservoir
Authors: Da Prat Giovanni , Ramey Henry J. , Cinco-Ley Heber 
Published: December 2007

Summary
A method to determine the permeability-thickness product for a naturally fractured reservoir is presented. The method involves a graph of log (p-p ) vs. t. The permeability-thickness product may be calculated from the slope. The mathematical theory is based on the extended Muskat analysis for a homogeneous reservoir and the Warren and Root model for a naturally fractured reservoir.
A comparison is made with the Pollard method, which involves a similar graph. It was found that both methods are mathematically related. The method presented is new and should have wide application to fractured reservoirs.

Introduction
Over the past 20 years a considerable amount of work has been devoted to the study of naturally fractured reservoirs. The physical principles of a fissured rock system were introduced by Barenblatt and Zheltov in 1960: "A porous block with a highly developed system of fissures can be represented as the superposition of two porous media with pores of different size."
The porous media are coupled because there is liquid flow between the two. In each point in space we can consider two pressures:the pressure of the liquid in the fissures andthe pressure of the liquid in the blocks.
The Barenblatt and Zheltov theory can be considered an initial step in the formulation of the mathematical theory of a naturally fractured reservoir. Later papers on the subject have presented different models to explain the behavior of a two-porosity system. The proposed models appear to differ in the assumptions. the flow geometry. and the particular reservoir under study. A recent study of the practical utility of the available models was performed by Mavor and Cinco-Ley. The authors concluded that the model proposed by Warren and Root in 1963 is the most useful for practical applications. The Warren and Root model is considered the forerunner of modem interpretation of two-porosity systems and has been the subject of study by many authors. Recently, Da Prat et al. extended the Warren and Root model to the study of decline curve analysis in both infinite and finite systems. Many studies of pressure buildup analysis for naturally fractured systems have appeared recently. For homogeneous systems, such techniques have been applied with success for both infinite and finite systems, as pointed out by Earlougher. For two-porosity systems, the interpretation of buildup tests has been the subject of controversy. The methods most used in the interpretation of pressure buildup data for two-porosity systems arethe Horner methods using the Warren and Root model andthe Pollard method presented in 1959.
The application and limitations of the Homer method have been discussed by Crawford et al. The Pollard method was based on a graph of log (p-p ) vs. Delta t. According to the literature, the Pollard method is not valid because of errors in the mathematical model. The Pollard method was based on an approximate solution wherein fluid was assumed to flow from the tight matrix to the fractures under pseudo steady-state conditions. We present a new method based on the extended Muskat analysis for homogeneous systems with the Warren and Root model as the basis for modeling a naturally fractured reservoir.

Mathematical Theory
The mathematical foundation for our study is based on the extended Muskat analysis. This method employs a graph of log (p-pws) vs. At. Using basic pressure buildup equations and a volumetric balance for a closed reservoir of area A, we can use the following equation to study a Muskat graph .
JPT
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