A finite-conductivity vertical fracture intersecting a well produced at a constant rate or at a constant pressure is considered. The pressure (or rate) response is obtained from a numerical model. Two aspects of this problem are considered:variable fracture conductivity andunequal fracture wing lengths. The first part of this paper examines the influence of fracture conductivity on the well response. In practice, the fracture conductivity is a decreasing function of distance from the wellbore. If the fracture conductivity decreases monotonically with distance from the wellbore, then at late times the variable fracture conductivity solutions behave like a constant-conductivity fracture, with conductivity equal to the arithmetic average of the conductivity. At early times the response is identical to that of a constant-conductivity fracture, corresponding to the highest conductivity of the fracture. For the variable fracture conductivity case, the bilinear flow period characterized by a one-quarter slope line may be obscured. Thus, analysis of short-time data can be difficult. We also consider situations where the fracture conductivity does not decrease monotonically with distance. The response for these cases is discussed in detail. The second part of the paper examines the effect of unequal wing lengths on the pressure response. We delineate conditions under which the effect of wing length on the response will become dominant. We discuss the influence of wing length on both early- and long-time data. Introduction To our knowledge no quantitative information is available in the literature regarding the effect of nonuniform fracture conductivity on the transient behavior of a fractured well. (In this work, nonuniform or variable fracture conductivity refers to the situation where the fracture conductivity is a function of distance from the wellbore.) The fracture is usually considered to be of constant width and constant permeability. However, in virtually all situations, fractures are designed such that the fracture conductivity is not uniform. The principal thrust of the first part of this paper concerns the influence of nonuniform (variable) fracture conductivity on the transient behavior of the well. We evaluate the consequences of neglecting variations in fracture conductivity when constant (uniform) conductivity solutions are used to determine the fracture half-length. We found that the effect of variable conductivity on well performance could be quantified in general terms if the conductivity decreases uniformly from the wellbore to the tip of the fracture. The behavior of a fractured well that is plugged is examined. The effect of a damaged zone (within the fracture) adjacent to the wellbore is also examined. Virtually all studies on transient pressure response assume that the well is located at the center of the fracture-i.e., the fracture wing lengths are equal. SPEJ P. 219^
Summary Although even a perfunctory survey of the literature suggests that considerable information is available on the response of finite-conductivity fractures in single-layer systems, the influence of the settling of propping agents and the effect of fracture height on the well response need to be examined. These topics are examined in this paper. We suggest methods to analyze well performance when the fracture paper. We suggest methods to analyze well performance when the fracture conductivity is a function of fracture height and fracture length. The performance of wells with fracture height greater than the formation performance of wells with fracture height greater than the formation thickness is documented. The consequences of being unable to contain the fracture within the pay zone are also examined. Although incidental to this study, we found that solutions presented by various authors are not in agreement for all time ranges. In this paper, we discuss a systematic procedure to obtain a grid (mesh) so that paper, we discuss a systematic procedure to obtain a grid (mesh) so that accurate results are obtained by a finite-difference model. This procedure can be used for both two-dimensional (2D) and three-dimensional (3D) problems. problems. Introduction This paper examines the performance of wells intercepting finite-conductivity vertical fractures. Although much work has been presented in this area of pressure analysis, several aspects of well behavior have yet to be examined. We examine some of these topics. In this work we examine the influence of vertical variations in fracture conductivity on well performance. Concerns regarding the effect of the settling of propping agents on well productivity addressed in this paper complement our work on the influence of lateral variations in fracture conductivity. We also examine situations where the fracture extends below and/or above the productive interval. We consider two possibilities:the fracture length is assumed to be fixed and the fracture height is variable (volume of fracture treatment is variable); andthe fracture volume is assumed to be fixed (the product of the fracture half-length and fracture height is assumed to be constant). The latter case is of interest if one is not able to contain the fracture within the pay zone of interest. In the former case, we show that this approach provides a means to increase the effective fracture conductivity. These topics have not previously been examined in the literature. Verification of the finite-difference model used in this study consumed a significant portion of the time spent in this study. Although some works have reported problems in obtaining accurate solutions, no guidelines for choosing a grid (mesh) for this problem are available. We give empirical guidelines for systematically choosing a grid to obtain accurate solutions. We believe that these guidelines will significantly reduce the time spent by researchers in developing their own models and consider it an important contribution. We also present methods to modify grids developed for a given set of conditions if the fracture and/or reservoir dimensions are changed.
TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractIn recent years, HSE concerns have been a driving force pushing towards emission-free well testing. Several methods have been proposed in order to attain that objective, the two most known being closed-chamber testing and downhole production/reinjection. With closed-chamber testing being of limited interest because of the reduced radius of investigation it offers, and harmonic testing being impractical due to time requirements, downhole production/reinjection appears as the only way of fully replacing conventional well testing.Planning such a test requires greater care than for a conventional test because the selection of an adequate injection interval is of major importance to the success of such an operation. The injected fluids must be contained within the disposal zone so as to avoid surface leakage but also so that the test data are not affected by an interference effect due to the injection process. There is also a great concern regarding the quality of the data because of hard downhole conditions. But if these challenges can be overcome, downhole production/reinjection can provide the same information as a conventional well test but with improved safety, no direct emission of pollutants and at reduced cost and test duration.
New analytical solutions for the response at a well intercepting a layered reservoir are derived. The well is assumed to produce at a constant rate or a constant pressure. We examine reservoir systems without interlayer communication and document the usefulness of these solutions, which enable us to obtain increased physical understanding of the performance of fractured wells in layered reservoirs. The influence of vertical variations in fracture conductivity is also considered. Example" applications of the approximations derived here are also presented.
SUMMARY The response of a fractured well in a multilayered reservoir is the primary subject of this study. Both analytical approximations and numerical results are presented. The analytical solutions served three important functions:they enabled us to verify the numerical solutions used in this study;they provided information on the structure of the solution and thus increased physical understanding; andmost importantly, they suggested a method whereby we were able to correlate multilayer solutions with single-layer solutions. We introduce the concept of dimensionless reservoir conductivity and show that in most realistic cases this parameter can be used to correlate commingled-reservoir solutions with the single-layer solutions available in the literature throughout the infinite-acting period. This concept and its utility in correlating solutions are the main contributions of this work. We also consider the analysis of buildup data following a short producing time. As in the drawdown case, we show that the multilayer producing time. As in the drawdown case, we show that the multilayer buildup solutions can be correlated with single-layer buildup solutions through the concept of dimensionless reservoir conductivity. Introduction This work examines the performance of vertically fractured wells in reservoirs without interlayer communication. To the best of our knowledge, the results presented in this paper are not available in the literature. All results presented in this work were obtained by a finite- difference model. Numerical methods, however, are not useful in understanding the structure of the solutions. Analytical solutions, either exact or approximate, have an important role in this regard: they provide information on the structure of the solution. Thus they increase physical understanding and suggest procedures to combine results. The discovery of the bilinear-flow regime bears eloquent testimony to this observation. This flow regime was discovered mainly (perhaps only) because an approximate analytical solution was derived; the numerical solutions had preceded the analytical solutions by 2 to 3 years. preceded the analytical solutions by 2 to 3 years. New analytical solutions for a fractured well intercepting a layered reservoir were derived during the course of this study. The analytical solutions are summarized in the Appendix, and complete documentation is given in Ref 2. In addition to the advantages discussed previously, the analytical solutions served two important functions:they allowed us to verify the accuracy of the finite-difference solutions (no solutions are available in the literature); andthey suggested a method to correlate multilayer-reservoir solutions with the solutions for single-layer systems provided that boundary effects are negligible. From the viewpoint of provided that boundary effects are negligible. From the viewpoint of analyzing data, this result is the most important contribution of our study. Had the analytical solutions been unavailable, it is doubtful that we would have realized how to correlate the solutions for layered systems with the solutions for single-layer systems. Most of our results examine only two-layer reservoirs. We do demonstrate, however, the applicability of the results given here to systems containing more than two layers. The analytical solutions presented in Ref. 2 are valid for any number of layers. We examine the presented in Ref. 2 are valid for any number of layers. We examine the influence of the contrasts in layer properties (permeability, porosity, and compressibility) and show that the ratio of reservoir thickness to fracture height may be important under certain circumstances. Our primary aim is to show that multilayer solutions can be correlated with single-layer solutions by means of the dimensionless reservoir conductivity during the transient-flow period. It should be noted that numerical simulations of the pressure responses for commingled reservoirs producing from fractured wells are given in Refs. 3 and 4. Their objective was to examine the ability of an engineer to history match pressure data using a numerical model. Procedures to generalize the results given in these works are discussed by Procedures to generalize the results given in these works are discussed by Camacho et al. Mathematical Formulation We consider a two-layer reservoir in the form of a rectangular drainage region with the well located in the center of the drainage area (Fig. 1). However, all results given here assume that the influence of the boundaries is negligible. SPEFE p. 259
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