Summary Fractures are common features of many well‐known reservoirs. Naturally fractured reservoirs contain fractures in igneous, metamorphic, and sedimentary formations. Faults in many naturally fractured carbonate reservoirs often have high‐permeability zones, and are connected to many fractures with varying conductivities. Furthermore, in many naturally fractured reservoirs, faults and fractures can be discrete (i.e., not a connected‐network fracture system). New semianalytical solutions are used to understand the pressure behavior of naturally fractured reservoirs containing a network of discrete and/or connected (continuous) finite‐ and infinite‐conductivity fractures. We present an extensive literature review of the pressure‐transient behavior of fractured reservoirs. First, we show that the Warren and Root (1963) dual‐porosity model is a fictitious homogeneous porous medium because it does not contain any fractures. Second, by use of the new solutions, we show that for most naturally fractured reservoirs, the Warren and Root (1963) dual‐porosity model is inappropriate and fundamentally incomplete for the interpretation of pressure‐transient well tests because it does not capture the behavior of these reservoirs. We examined many field well tests published in the literature. With few exceptions, none of them shows the behavior of the Warren and Root (1963) dual‐porosity model. These examples exhibit very diverse pressure behaviors of discretely and continuously fractured reservoirs. Unlike the single derivative shape of the Warren and Root (1963) model, the derivatives of these examples exhibit many different flow regimes depending on fracture distribution and on their intensity and conductivity. We show these flow regimes with our new model for discretely and continuously fractured reservoirs. Most well tests published in the literature do not exhibit the Warren and Root (1963) dual‐porosity reservoir‐model behavior. If we interpret them by use of this dual‐porosity model, then the estimated permeability, skin factor, interporosity flow coefficient (λ), and storativity ratio (ω) will not represent the actual reservoir parameters.

Summary. The objective of this theoretical study is to delineate the characteristics of a well producing at a constant pressure in a naturally fractured reservoir. We consider flow to a well located at the center of a closed circular reservoir. New solutions not documented previously are presented. Five flow regimes are identified and information that can be extracted from each regime is documented. Methods to determine formation parameters and to predict well deliverability are discussed.

Introduction

This paper is a theoretical study of the performance of a well located in a fractured reservoir and produced at a constant pressure. As is well known, a fractured reservoir is usually considered to be a system in which the conducting properties of the rock are a result of the fracture system and the storage capacity of the reservoir is due to the matrix system. All mathematical models that describe the performance of fractured reservoirs assume that the well response is a combination of a rapid response that is a result of the fracture system and a slower induced response that reflects the contribution of the matrix system. The principal difference between the models suggested in the literature involves the interaction of the matrix system and the fracture system. Warren and Root and Odeh assume quasisteady- or pseudosteady-state flow in the matrix system, whereas Kazemi, deSwaan- O., Najurieta, and Kucuk and Sawyer assume unsteady-state flow in the matrix system. Mavor and Cinco-Ley and DaPrat et al. used the Warren and Root model to examine the response of a well flowing at a constant pressure. Their objective was to predict well deliverability when the reservoir parameters were determined independently. In this work, we assume that the fractured reservoir may be represented by the rectangular slabmodel idealization used in Refs. 3 and 4 (see Fig. 1). Our first objective is to discuss procedures whereby early-time rate data may be analyzed at a well to determine the properties of the fracture and the matrix systems. The second objective is to present methods to predict future performance or well deliverability.

In this work, we identify five possible flow regimes. Three of these flow regimes may exist if the well response is unaffected by boundary effects. Following the terminology of Ref. 9, we identify the flow regimes as Flow Regimes 1 through 3. Two other flow regimes may be evident after the outer boundary begins to dominate the well response. We refer to these flow regimes as Flow Regimes 4 and 5.

In the following, we delineate conditions under which Flow Regimes 1 through 5 will exist and discuss the information that can be extracted from each. In particular, we show that for specific values of reservoir properties, not all flow regimes will be evident. We also show that reservoir size determines the flow regime that will govern the well response during boundary-dominated flow.

Mathematical Model

We consider the flow of a slightly compressible fluid of constant viscosity in a cylindrical reservoir in which the outer boundary is closed. The well is located at the center of the cylinder and fluid is produced at a constant pressure. Initially, the pressure is uniform throughout the reservoir. Gravitational effects are assumed to be negligible. We consider a skin region and assume that it is "infinitesimally thin." The wellbore storage effect is not considered.

As already mentioned, we assume that the naturally fractured reservoir be described by the slab model suggested by Kazemi and deSwaan-O. (Fig. 1). Individually, the matrix system and the fracture system are assumed to be uniform, isotropic porous media. All production is by way of the fracture system, and flow from the matrix system to the fracture system is onedimensional (1D) (normal to the fracture system).

For modeling purposes, the mathematical problem may be formulated by examination of flow in only one of the repetitive elements shown in Fig. 1. The mathematical formulation is discussed in the Appendix. The solution in Laplace space (for infinite-acting and bounded systems) and the relevant asymptotic approximations are given in the Appendix. All results presented in this study were obtained by inversion of the rigorous analytic solution given in the Appendix numerically, and not from the asymptotic expansions. The asymptotic expansions are useful primarily in identifying the structure of the solutions.

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P. 186^

Summary

This paper presents a new analytical model for interpreting pressure-transient tests for wells producing from dual-porosity reservoirs. This model includes unsteady-state matrix flow and incorporates the effects of wellbore storage, skin, and, for gas reservoirs, desorption. The model is applicable to bounded and infinite-acting reservoirs.

Introduction

Numerous analytical models have been presented recently to describe the transient pressure behavior of dual-porosity reservoirs. Dual-porosity or naturally fractured reservoirs are formations composed of two porous media of different porosities and permeabilities. One medium, the matrix blocks constituting the primary porosity, contains the majority of the fluid stored in the reservoir and possesses a low conductivity. The other medium, the fractured network constituting the secondary porosity, acts as the conductive medium for fluid and possesses a high flow capacity but low storativity. The storativities of the two media usually differ by several orders of magnitude: consequently, these reservoir types are referred to as dual-porosity reservoirs. These types of reservoirs are also characterized by a large permeability contrast between the two media. The basis for the study of dual-porosity media was presented by Barenblatt and Zheltov, who treated the fractured reservoir as a continuum with the fractured network superimposed on the primary porosity. Furthermore, they assumed that the flow of fluid within the matrix occurs under pseudosteady-state conditions. Warren and Root, using a formulation similar to Barenblatt and Zheitov, were the first to present analytical solutions to this model with the assumption of a pseudosteady-state matrix flow and developed a procedure for interpretation of buildup tests without wellbore storage and skin effects. Warren and Root showed that. on a semilog graph, their solution yielded two parallel straight lines with slopes related to formation flow capacity. The existence of two parallel semilog straight lines was disputed by Odeh, who used a model similar to that used by Warren and Root but who investigated different ranges of parameters. Kazemi was the first to consider the effects of unsteady-state matrix flow. He used a numerical model and assumed that the dual-porosity system can be simulated by a layered radial system. His results are similar to those of Warren and Root with the exception of a smooth unsteady-state transition zone between the two parallel semilog straight lines compared with the flat pressure profile characteristic of the pseudosteady-state transition. Later, de Swaa presented analytical unsteady-state solutions for a well producing at a constant rate in naturally fractured reservoirs. He introduced new diffusivity definitions for reservoir characterization. Kucuk and Sawyer presented a comprehensive model for gas flow in a naturally fractured reservoir of the Devonian shale. They investigated the behavior of dual-porosity gas reservoirs including the Klinkenberg effect in the tight shale matrix and the effect of gas desorption from pore surfaces of the shale matrix. Mavor and Cinco-Ley extended Warren and Root's solution to take into account the effects of wellbore storage and skin. Bourdet and Gringarten were the first to identify the existence of a semilog straight line during the transition period. They stated that this line had a slope one-half the classic parallel semilog straight lines and existed if the fracture storativity was not too large. Streltsova and Serra et al. analyzed the transition period in detail and confirmed the existence of the straight line of slope 0.5756, one-half the classic semilog straight line (1.151). Serra et al.'s solution includes unsteady-state matrix flow but not wellbore storage effects. Chen et al. presented an application of classic techniques to bounded dual-porosity systems and discussed flow regimes that may be exhibited by drawdown data. Their work, however, did not include wellbore storage, skin, or the effects of gas desorption. Cinco-Ley and Samaniego-V. presented a model based on the transient matrix flow model formulated by de Swaan-O. and demonstrated that the behavior of dual-porosity reservoirs can be correlated by use of three dimensionless parameters (i.e., w, AFD, and maD). They also established that, regardless of matrix geometry, the transition period might exhibit a straight line with a slope equal to one-half the slope of the classic parallel semilog straight lines. The purpose of this paper is to present an analytical solution for dual-porosity reservoirs, capable of modeling both pseudosteady- and unsteady-state matrix flow, for both finite and infinite-acting reservoirs. The solution includes the effect of gas desorption from the pore surfaces of shale matrix in dual-porosity gas reservoirs with sorbed gas. This is of particular application to the Devonian shale gas reservoirs and any dual-porosity "black shale" reservoir with matrix kerogen. Wellbore storage and skin effects are included in the solution. Furthermore, application of the model to analysis of field pressure-transient data with an automatic parameter-estimation technique is demonstrated.

Theoretical Formulation

Formulation of Flow Equations. The differential equations governing fluid flow in naturally fractured reservoirs are derived in a manner similar to de Swaan-O. formulation and are presented in Appendix A of the original version of this paper. The derivation is based on the following assumptions: (1) unsteady-state radial flow in an isotropic dual-porosity reservoir at uniform thickness, (2) negligible gravitational forces and small pressure gradients; (3) uniform initial reservoir pressure throughout the reservoir; (4) fluid production through the fracture network with the matrix blocks acting as a uniformly distributed source; (5) one-dimensional (ID) unsteady-state flow in the matrix blocks that are of regular shape; (6) gas-desorption source uniformly distributed within the matrix blocks; (7) gas-desorption rate linear with pressure; and (8) well producing at a constant rate in a finite reservoir with wellbore storage and skin effects. The diffusivity equations describing flow in the fracture network for both oil and gas reservoirs in dimensionless form are

............(1)

for oil reservoirs and

............(2)

for gas reservoirs with desorption. The dimensionless pressure, PfD, is defined identically for oil and gas except that adjusted pressure instead of real pressure is used for gas to linearize the gasflow equations.

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P. 384

Abstract

This work is a rigorous evaluation of vertically fractured well responses producing fractured reservoirs. Detailed evaluation of plausible responses to be expected from short-term teats are provided. New diagnostic rules, analytical provided. New diagnostic rules, analytical procedure., and working curves are presented. The procedure., and working curves are presented. The effect of fracture length on flow patterns in the reservoir is documented. Practical issues that the analyst must be cognizant of are documented.

Introduction

This work, considers production via a vertically fractured well located at the center of a closed circular reservoir, accomplishes the following objectives: (i) provides a critical evaluation of flow patterns that result as a consequence of the production of a naturally fractured reservoir via a production of a naturally fractured reservoir via a hydraulic fracture, (ii) provides for a convenient methodology to analyze well responses, and (iii) provides a simple procedure to compute long-time provides a simple procedure to compute long-time (pseudoradial and pseudosteady state) responses of a well in a closed system; the basis for our method is clearly documented and ad hoc procedures are avoided.

In passing, we note that, for the systems considered here, the characteristic pressure traces of heterogeneous reservoirs or the characteristic responses of the wells produced via vertical fractures are usually not evident. Our solutions highlight the need to obtain information on the heterogeneous character of the formation prior to stimulation programs. We further note one other characteristic of fractured wells which is usually ignored. Although fractured well solutions in the literature cover a wide range of values of dimensionless fracture conductivity, the solutions are not unique once the dimensionless conductivity falls below 3 (see Ref. 1). More importantly, buildup responses for all practical purposes become insensitive to fracture half-length once the dimensionless fracture conductivity becomes less than 5.

MATHEMATICAL MODEL

We consider a well produced by a vertical fracture that is located at the center of a closed circular reservoir. The reservoir is assumed to be a naturally fractured system and, for purposes of discussion, we assume that the slab idealization of Kazemi and de Swaan-O is applicable (Fig. 1). The parameters that govern reservoir characteristics will parameters that govern reservoir characteristics will be denoted by ' and lamba' where

' = , (1)

and

' = 12 . (2)

Here, h mt is the thickness of the matrix system and h ft is the thickness of the fracture system, i.e., the total thickness, h = h mt +h ft = n (h m + h f) where n is the total number of fractures. Thus, h m is the characteristic block dimension. The subscripts m and f refer to the matrix and fracture systems respectively. As is well known, if boundary effects are ignored, then three flow regimes may manifest themselves due to the heterogeneous nature of the rock: Flow Regime 1 where the well response is unaffected by the matrix system, Flow Regime 2 wherein we have unsteady flow in the fracture and the matrix systems, and Flow Regime 3 wherein the flow in the matrix system becomes pseudosteady (center line of the matrix system affects the well response).

P. 565

Abstract Fractures are common features of many well-known reservoirs. Naturally fractured reservoirs consist of fractures in igneous, metamorphic, and sedimentary rocks (matrix) and formations. Faults in many naturally fractured carbonate reservoirs often have high permeability zones, and are connected to many fractures with varying conductivities. Furthermore, in many naturally fractured reservoirs, faults and fractures can be discrete (i.e., not a connected-network fracture system). To understand the pressure behavior of these continuously and discretely fractured reservoirs, semianalytical solutions are presented. These solutions are used for transient well test interpretation of formations containing a network of discrete and/or connected (continuous) finite- and infinite-conductivity fractures. In this paper we present an extensive literature review of the pressure transient behavior of fractured reservoirs. First, we show that the Warren and Root (1963) dual-porosity model is a fictitious homogenous porous medium because it does not contain any fractures. Second, using the new solutions, we show that for most naturally fractured reservoirs the Warren and Root (1963) dual-porosity model is inappropriate and fundamentally incomplete for the interpretation of pressure transient well tests because it does not capture the behavior of these reservoirs. We examined many field well tests published in the literature. With few exceptions, none of them shows the behavior of the Warren and Root (1963) dual-porosity model. These examples exhibit the very rich pressure behavior of discretely and continuously fractured reservoirs. Unlike the single derivative shape of the Warren and Root (1963) model, the derivatives of these examples exhibit many different flow regimes depending on fracture distribution, and on their intensity and conductivity. We show these flow regimes using our new model for discretely and continuously fractured reservoirs. These derivatives will be a valuable diagnostic tool for well test interpretation. Most well tests published in the literature do not exhibit the Warren and Root (1963) dual-porosity reservoir model behavior. If we interpret them this dual-porosity model, then the estimated permeability, skin factor, interporosity flow coefficient (λ), and storativity ratio (ω) will not represent the actual reservoir parameters.