This paper presents a new iterative technique for interpreting early-time pressure-buildup data for hydraulically fractured wells. The technique uses a modified square-root-of-time analysis with permeability and fracture-length correction curves generated from Cinco-Ley et al.'s solution to flow equations for finite-conductivity fractures. The method is applicable to low-permeability reservoirs where conventional methods of analysis are difficult to apply.
Interest in exploitation of low-permeability gas reservoirs has increased in recent years. Frequently gas is encountered in low-permeability reservoirs, and to have profitable production, these reservoirs often are stimulated by hydraulic fractures. A careful study of reservoir and fracture properties is necessary to optimize production.
Several investigators have developed analytical and numerical models to analyze pressure-buildup data for vertically fractured wells, but each is restricted by different assumptions. Russell and Truitt first presented a technique to analyze reservoir and fracture properties for a vertically fractured gas well. Permeability is calculated by analyzing a Homer plot of pressure-buildup data. The major limitation of this method is that correct values for reservoir and fracture properties cannot be obtained unless pseudoradial flow is reached during the buildup test. In most cases, this flow regime is not attained within a reasonable amount of time because of low reservoir permeability or long hydraulic fractures. Also, this method assumes infinite fracture conductivity, which is invalid in many actual cases.
Cinco-Ley et al. developed type curves for finite-conductivity fractures. This method requires matching buildup data with one of their type curves. The matches, however, tend to be nonunique; thus, several estimates for fracture half-length and formation permeability may be obtained from a given test. Millheim and Cichowicz developed a method based on linear flow in the formation that exists long before pseudoradial flow is reached. In this situation, when pressure is plotted vs. the square root of time on Cartesian coordinate paper, a straight line is obtained. The slope is inversely proportional to the product of fracture length and the square root of formation permeability. The limitation of this method is that the formation permeability has to be known to calculate fracture length.
Holditch et al. developed correction factors for permeability estimated from the final slope of buildup test data on a Homer graph. These correction factors were based on semilog plots of Cinco-Ley et al.'s solutions for finite-conductivity fractures. Using these correction factors, they proposed an iterative method in which reservoir and fracture parameters converge to accurate values. This method requires that the data be on a linear trend when plotted on a graph of pressure vs. square root of time that we will call the "pseudolinear" flow regime. This region is illustrated in Fig. 1. Unfortunately, in many low-permeability gas wells, insufficient data are obtained during the pseudolinear flow regime. If a proper straight line is not achieved on a square-root-of-time graph, both the Millheim-Cichowicz and the Holditch et at. iterative schemes have proved to be inaccurate.
In low-permeability reservoirs, impractically long buildup test times are required to reach either pseudolinear or pseudoradial flow. The regime that dominates during such tests is bilinear flow. With conventional semilog or type-curve methods, these data cannot be analyzed. To date, no published method allows us to estimate permeability or fracture half-length with data from the bilinear flow regime only. Fracture conductivity can be estimated from these data, given an independent estimate of permeability.
In this paper, we demonstrate how buildup tests dominated almost totally by bilinear flow can be rigorously evaluated with a computer model. The model provides estimates of fracture conductivity, fracture length, and formation permeability from data collected before pseudolinear flow is achieved. The computer time required to run the program is negligible compared with a finite-difference history-matching simulator, the other major alternative for analyzing these test data. Our program is short and simple and can even be used with a programmable desk calculator.
We have generated correction curves using slope and intercept values based on Cinco-Ley et al.'s finite-conductivity-fracture solutions. Two different cot action curves are used in the model. One was generated for use with a semilog plot of pressure and time. The other was generated for use with a plot of pressure and square root of time. The technique assumes no wellbore storage effects because there were none in Cinco-Ley et al.'s solution. In addition, a careful initial estimate of slope is required for the method to converge.
Iterative Scheme Development
The proposed technique combines the Horner method for permeability calculation and the Millheim and Cichowicz method for fracture length and conductivity calculations. These calculations are modified with permeability correction factors that were generated from semilog plots of the finite-conductivity-fracture solutions and the permeability correction factors generated in this study that use square-root-of-time plots of Cinco-Ley el al.'s solution.