Application of the Pseudolinear-Flow Model to the Pressure-Transient Analysis of Fractured Wells

Abstract

Summary The theoretical basis for the pseudolinear-flow model is established. It is demonstrated by use of an analytical model that the linear-flow graph (p vs. root of t) can be extended to the analysis of pressure data of fractured wells intersected by an intermediate- or pressure data of fractured wells intersected by an intermediate- or high-conductivity fracture (CfD greater than 5 pi). It appears that the fracture-conductivity effect during the pseudolinear-flow period can be handled as pseudoskin pressure drop that adds to the pressure drop caused by fluid-loss damage. The combination of the pseudolinear-flow analysis with other interpretation techniques is illustrated through examples of field cases. Introduction During the last 2 decades, many analysis methods to interpret pressure data to estimate both formation and hydraulic fracture pressure data to estimate both formation and hydraulic fracture parameters have been proposed. In addition to type-curve analysis, parameters have been proposed. In addition to type-curve analysis, three of the most widely used graphical methods of interpretation are the linear-flow graph (p vs. root of t), the bilinear-flow graph (p vs, t 1/4), and the pseudoradial-flow graph (p vs. log t). It cannot be overemphasized that each technique applies to a specific flow regime. Unfortunately, the general pressure behavior of a fractured well includes not only these flow periods but also several intermediate transition flow periods, as shown in Fig. 1. The only current graphical analysis method available to interpret pressure data falling in the transition periods is the type-curve-matching technique. The linear-flow model has been applied in the past to early time/pressure data of infinite-conductivity fractures (for practical purposes, a finite-conductivity fracture can be considered as an purposes, a finite-conductivity fracture can be considered as an infinite-conductivity-fracture case whenever the pressure drop along the fracture is negligible--i.e., Cf/kxf greater than or equal to 300). This model assumes uniform flux in the formation and negligible storage capacity within the fracture. The application of the linear-flow graph was extended empirically to cases where the fracture has low or intermediate values of conductivity. This requires either knowledge of the formation permeability from prefracture testing or a trial-and-error procedure. permeability from prefracture testing or a trial-and-error procedure. Hanley and Bandyopadhyay used a uniform-flux analytical model to show that the pressure behavior of a finite-conductivity vertical fracture at intermediate values of time can be described by the pressure behavior of the linear-flow model plus an extra pressure drop that is a function of the fracture conductivity. They also pressure drop that is a function of the fracture conductivity. They also assumed that the fracture penetrates the formation completely in the horizontal direction (xe/xf = 1). The purpose of this work is to show through the analysis of an analytical solution for finite-conductivity fractures that the linear-flow model may be extended to cases of high fracture conductivity without assuming a uniform flux along the fracture. Furthermore, analysis of field cases illustrates the application of the pseudolinear-flow model and its combination with other interpretation pseudolinear-flow model and its combination with other interpretation techniques to obtain reliable estimates for fracture and reservoir parameters. parameters. Pseudolinear-Flow Model Pseudolinear-Flow Model This study considers the model presented in previous works; i.e.. an infinite homogeneous reservoir is produced through a well intersected by a symmetrical finite-conductivity vertical fracture of constant properties, as indicated in Fig. 2. Several authors have demonstrated that the pressure behavior of a fractured well can be expressed in terms of three parameters: dimensionless pressure drop, delta pwD, dimensionless time, parameters: dimensionless pressure drop, delta pwD, dimensionless time, txfD, and dimensionless fracture conductivity, CfD, as illustrated in Fig. 1. So (1) For the period including the bilinear flow, Transition Zone 1, and the linear flow, Eq. 1 can be expressed in the Laplace space as (2) where (pwD) is the Laplace transform of the dimensionless wellbore pressure and s is the Laplace space parameter. The hyperbolic cotangent function may he written as an infinite series (3) (4) At large values of time, s goes to 0 and the series in Eq. 4 can be approximated by use of two terms (5) Laplace inversion of this equation yields (6) The first term on the right side corresponds to the classic linear-flow expression discussed extensively in the literature. The second term is a result of the finite conductivity of the fracture. Hanley and Bandyopadhyay presented a similar equation, the only difference is that their numerical constant in the second term is unity instead of the pi/3 constant. It is obvious that this equation also includes the infinite-conductivity case for which the second term on the right side goes to zero. By multiplying Eq. 6 by CfD, we obtain (7) This equation indicates that a graph of pwD(CfD) vs. must give a single straight line with an intercept equal to pi/3 for different values of CfD. Fig. 3 shows this type of graph for the pressure behavior of a finite-conductivity vertical fracture. It is evident that a portion of the curves for CfD greater than 5 pi falls on the straight line defined by Eq. 7. Hence, an interesting conclusion is evident: the pressure behavior of a well intersected by a fracture of conductivity greater than 5 pi is identical to the behavior of an infinite-conductivity fracture with a "skin" that depends on the fracture conductivity. This condition is valid for a certain time range, txfD(CfD)2 greater than 1. Agarwal et al. found this type of behavior and provided a graph for the relationship between fracture conductivity and the intercept in the linear-flow graph; however, no analytical proof was presented. In addition, Lee and Holditch extended the use of the presented. In addition, Lee and Holditch extended the use of the linear-flow graph to finite-conductivity fracture cases, they presented empirical correlations of both slope and intercept as a function of fracture conductivity. SPEFE P. 438