Analytic Solutions for Determining Naturally Fractured Reservoir Properties by Well Testing
Authors: de Swaan O. A. 
Published: December 2007

Abstract
A completely unsteady-state theory is developed that describes the well pressure response of infinite, naturally fractured reservoirs of the type composed by high-permeability fractures and tight matrix blocks.
The theory exclusively involves flow properties and dimensions of the fractures and the matrix blocks; no extra adjusting parameters are needed for predicting the pressure response of a reservoir with known properties. Comparison of computed pressure-drawdown theoretical curves with results pressure-drawdown theoretical curves with results of a numerical model applied to idealized reservoirs shows a good fit.
It is concluded that an evaluation of the following reservoir properties is possible from well-test pressure plots: the fractures' kh product and the pressure plots: the fractures' kh product and the average product of the matrix porosity by a characteristic dimension of the matrix blocks. Thus, the model allows comparison of well-test pressure response with the formation measurements. pressure response with the formation measurements

Introduction
Barenblatt and Zheltov were first to present the formulation of the problem of radial flow of a slightly compressible fluid in a naturally fractured porous medium. They assumed that flow occurs only in the fracture porous medium, in which matrix blocks of contrasting physical properties deliver their fluid contents. That is, the matrix blocks act as a uniformly distributed source in a fracture medium. In their paper, Barenblatt and Zheltov assume this source's paper, Barenblatt and Zheltov assume this source's strength is proportional to the local pressure loss at the fracture and to an average internal block pressure. An approximate solution to this problem was presented by Warren and Root, resulting in a characterization of the fractured reservoir by two parameters ambiguously related to the actual shape, dimensions, and fluid flow properties of the reservoir. Furthermore, these two properties of the reservoir. Furthermore, these two parameters have been found of "limited value" for parameters have been found of "limited value" for determining some reservoir properties.
Warren and Root give a comprehensive bibliography of the instrumental evaluation of fractured reservoirs. A promising method to this end, using sonic and resistivity logs, has been presented by Aguilera. The need to directly relate the fractured reservoir properties to the well-test plots, just as in the properties to the well-test plots, just as in the homogeneous reservoirs, is obvious.

THEORY
The theory, detailed in the Appendix, involves the following assumed mechanisms of the fluid flow and their corresponding mathematical expressions.At early times of the well test, the flow takes place only in the fractures and is described by the place only in the fractures and is described by the approximate solution of the radial infinite reservoir as applied to the fracture medium:q w 4 f pf = ------------- ln (--------t)..............(1) 4 hf kf y' r2wThis equation accounts for the first part of be fractured-reservoir typical pressure response that, in a semilog plot of pressure vs time, starts as a straight line and then shifts to another parallel straight line (see Figs. 1 and 2).As mentioned above, the matrix blocks act as a uniformly distributed source in the fractured medium. The effect of this source is evident after a certain time lag, since the response of the matrix blocks is slower than the fracture-medium response because of the blocks' "tight" nature.Assuming that the shape of the matrix blocks may be approximated by regular solids, their internal pressure distribution, and hence the flow through pressure distribution, and hence the flow through their surfaces, is a known function found in the theory of heat flow in solids (see Ref. 6). These solutions are given for solids with unitary pressure (temperature) loss at their surfaces as a boundary condition.Considering that the pressure in the fractures surrounding the regular matrix blocks is variable the outflow from the blocks is described through a convolution:
SPEJ
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