Most of the existing methods for the analysis of well tests taken from vertically fractured wells are based on assuming a constant production rate. However, actual rates decline rapidly during the testing of low permeability reservoirs. In such cases, the appropriate procedure involves a multi-rate modification of the constant rate solution. In radial flow analysis the dependence of pressure on the reservoir parameters and time can be separated. Therefore a straight line plot of the pressure against a time group can be used for data analysis. However, this is not possible in fracture flow analysis where pressure dependence on the reservoir parameters and time cannot be separated. In order to implement the multi-rate modification into fracture flow analysis, an iterative plotting procedure was devised. With the aid of an interactive computer program this procedure is very convenient to use. Synthetic and field examples are given to illustrate the utility and limitations of this method. Fracture half-length and fracture conductivity are determined from the post-fracture buildup when formation permeability or initial pressure is known.
The theory of fluid flow into a vertically fractured well has been analyzed by a number of authors.1–6 In the planar flow approximation1–2, the solutions are simple, and the analysis procedures require plotting the pressure against functions of time. The slopes of the resultant straight lines can be used to estimate the product of the formation permeability, k, and the square of the fracture half-length, xf2. With no other source of information, it is not possible to determine k and xf separately.
Solutions that do not use the planar flow assumption are more complicated3–6. There is no simple pressure-time relation to allow a straight line data plotting analysis procedure. Initially most authors suggested the use of type curve matching. Since these curves were generated under the constant rate assumption, an extra assumption of the total flow time being much longer than the shut-in time was required to analyze buildup data. This assumption is not valid for most post-fracture buildup tests.
Agarwal7 modified the type curve matching technique by plotting pressure against a time group instead of the shut-in time. Since this time group was obtained by considering the radial flow theory, there can be no theoretical justification in applying it to analyze fracture flow.
Jaggernauth et al.8 applied a regression technique for data analysis. In general regression is best applied to systems with a known analytical solution. Otherwise the computer cost can be excessive. Since a solution to the finite fracture conductivity system can be obtained only numerically4–6, a regression technique cannot be used.
The objective of this paper is to present an interactive plotting procedure for buildup analysis. In this method both the flow time and the variable rates are accounted for by the multi-rate theory. This method requires only the numerical values of the solution for the system of interest. Such values can be calculated from an analytical expression, or they can be interpolated from a table.