Introduction During the last few years, there has been an explostion of information in the field of well-test analysis. Because of increased physical understanding of transient fluid flow, it is possible to analyze the entire pressure history of a well test, not just long-time data as in conventional analysis.1 It is now often possible to specify the time of beginning of the correct semilog straight line and determine whether the correct straight lie has been properly identified. It is also possible to identify wellbore storage effects, and the nature of wellbore stimulation as to permeability improvement, or fracturing, and to quantitatively analyze those effects. Such accomplishments have been augmented by attempts to understand the short-time pressure data from well testing - data that were often classified as too complex for analysis. One recent study of short-time pressure behavior2 showed that it was important to specify the physical nature of the stimulation in considering the behavior of a stimulated well. That is, stating that the van Everdingen-Hurst infinitesimal skin effect was negative was not sufficient to define short-time well behavior. For instance, acidized (but not acid-fractured) and hydraulically fractured wells might not necessarily exhibit the same behavior at early times, even though they could possess the same value of negative skin effect. In the same manner, hydraulic fracturing leading to horizontal or vertical fractures could produce the same skin effect, but with possibly different short-time pressure data. This could then provide a way to determine the orientation of fractures created by this type of well stimulation. In fact, it is generally agreed that hydraulic fracturing usually results in one vertical fracture, the plane of which includes the wellbore. Most studies of the flow behavior for a fractured well consider vertical fractures only.3–11 Yet it is also agreed that horizontal fractures could occur in shallow formations. Furthermore, it would appear that notch-fracturing would lead to horizontal fractures. Surprisingly, no detailed study of the horizontal fracture case had been performed until recently.12 A solution to this problem was presented by Gringarten and Ramey.13 In the course of their study, it was found that a large variety of new transient pressure behavior solutions useful in well and reservoir analysis could be constructed from instantaneous Green's functions.14 Possibilities included a well with a single vertical fracture in an infinite reservoir, or at any location in a rectangle.

Summary This paper presents a discussion of fractured-horizontal-well performance in millidarcy permeability (conventional) and micro- to nanodarcy permeability (unconventional) reservoirs. It provides interpretations of the reasons to fracture horizontal wells in both types of formations. The objective of the paper is to highlight the special productivity features of unconventional shale reservoirs. By using a trilinear-flow model, it is shown that the drainage volume of a multiple-fractured horizontal well in a shale reservoir is limited to the inner reservoir between the fractures. Unlike conventional reservoirs, high reservoir permeability and high hydraulic-fracture conductivity may not warrant favorable productivity in shale reservoirs. An efficient way to improve the productivity of ultratight shale formations is to increase the density of natural fractures. High natural-fracture conductivities may not necessarily contribute to productivity either. Decreasing hydraulic-fracture spacing increases the productivity of the well, but the incremental production gain for each additional hydraulic fracture decreases. The trilinear-flow model presented in this work and the information derived from it should help the design and performance prediction of multiple-fractured horizontal wells in shale reservoirs.

This paper presents an analytical trilinear-flow solution to simulate the pressure-transient and production behaviors of fractured horizontal wells in unconventional shale reservoirs (Ozkan et al. 2009). The model is simple, but versatile enough to incorporate the fundamental petrophysical characteristics of shale reservoirs, including the intrinsic properties of the matrix and the natural fractures. Special characteristics of fluid exchange among various reservoir components may also be considered. Computational convenience of the trilinear-flow solution makes it a practical alternative to more rigorous but computationally intensive and time-consuming solutions. Another advantage of the trilinear-flow solution is the convenience in deriving asymptotic approximations that provide insight about potential flow regimes and the conditions leading to these flow regimes. Though linear-and bilinear-flow regimes have been noted for fractured horizontal wells in the literature on the basis of their diagnostic features, they have not been associated with particular reservoir characteristics and flow relationships. The trilinear-flow solution also provides a suitable algorithm for the regression analysis of pressure-transient tests in shale reservoirs.

The objective of this paper is to incorporate a more detailed description of flow in shale matrix to improve modeling of production from fractured shale-gas reservoirs. Currently, most modeling approaches for shale-gas and -oil production are based on the dominance of Darcy flow in both natural fractures and matrix. We improve the description of matrix flow by considering diffusive (Knudsen) flow in nanopores. In our dual-mechanism approach, when Darcy flow becomes insignificant due to nanodarcy matrix permeability, Knudsen flow takes over and contributes, substantially, to the transfer of fluids from matrix to fracture network. Furthermore, we consider stress-dependent permeability in the fracture network. Therefore, incorporating Darcy and diffusive flows in the matrix and stress-dependent permeability in the fractures, we develop a dual-mechanism dual-porosity naturally fractured reservoir formulation and derive a new transfer function for fractured shale-gas reservoirs. The dual-mechanism dualporosity formulation presented in this paper can be used for numerical or analytical modeling purposes. We use the new formulation of matrix to fracture fluid transfer with an analytical model and demonstrate the differences from the conventional formulation.

Point-source solutions are derived in the Laplace-transform domain and an extensive library of solutions is documented to obtain pressure distributions and well responses for a wide variety of wellbore pressure distributions and well responses for a wide variety of wellbore configurations:partially penetrating vertical wells, horizontal wells, and fractured wells(complete or limited entry). Wells may be located in infinite or bounded systems (rectangular or circular reservoirs). Several combinations of closed and/or constant-pressure boundary conditions are considered at the vertical and lateral reservoir boundaries. These solutions may be used to examine homogeneous or naturally fractured reservoirs.

This paper presents a discussion of fractured horizontal-well performance in conventional (milli-Darcy permeability) and unconventional (micro- to nano-Darcy permeability) reservoirs. It provides interpretations of the objective of fracturing horizontal wells in both types of formations. By using a trilinear-flow model, it is shown that the drainage volume of multiply-fractured-horizontal-wells is limited to the inner reservoir between the fractures even for relatively large matrix permeabilities. Unlike conventional reservoirs, favorable productivities are not warranted in unconventional-tight reservoirs because of high reservoir permeability and high hydraulic fracture conductivity. The most efficient mechanism to improve the productivity of unconventional-tight formations is to increase the density of natural fractures. High natural fracture permeabilities may not necessarily contribute to productivity. Decreasing fracture spacing increases the productivity of the well, but the incremental production for each additional fracture decreases. The trilinear-flow model presented in this work can be used to determine optimum hydraulic fracture properties for a multiply-fractured-horizontal-well. The model can also be used as a predictive tool. The information given in this paper should help the design of multiply-fractured-horizontal-wells and predict their performances. Introduction The objective of hydraulic fracturing in conventional-tight reservoirs (as in tight sands with permeabilities in the milli- to micro-Darcy range) has been to create a high-conductivity flow path to improve flow convergence in the reservoir. Fig. 1 provides the definition of fracture conductivity on a sketch of a hydraulic fracture intercepting a vertical well. Accordingly, a practical interpretation of the objective of fracturing a horizontal well is to create a system whose long-term performance is identical to that of a single effective (total) fracture of length equal to the spacing between the outermost fractures (Raghavan et al., 1997, and Chen and Raghavan, 1997). Fig 2 provides a sketch of this interpretation. With this interpretation, performances of fractured horizontal wells can be correlated in terms of an effective fracture conductivity and effective fracture half-length. The conductivity of this effective fracture depends on the permeability of the reservoir and the number, distance between, and conductivities of the individual hydraulic fractures as demonstrated in Fig. 3.

We examine the productivity of a multiply-fractured well in a reservoir that is in the form ofa rectangle. The properties of the individual fractures (length, conductivity, skin, etc.) are entirely arbitrary. The fractures are assumed to be parallel to one of the sides of the rectangle and maybe either parallel or perpendicular to the horizontal well. The primary objective of this work is to outline a computationally-efficient scheme to compute well productivity. This objective yields a simple method to incorporate this completion scheme in reservoir-simulation models. It should permit an engineer to understand interactions between the fractures in the presence of boundaries. The model also permits us to compute pressure distributions under depletion conditions for single-phase oil and gas reservoirs. Further, it provides for a realistic design of this completion scheme.It is to be emphasized that the problem we consider is a nontrivial one in that flow to a single vertical fracture of fmite conductivity in a rectangular drainage area is yet to be modeled successfully. IntroductionThe Ozkan-Raghavan solution! for a point source in a rectangular parallelepiped is used to derive expressions for pressure distributions caused by a multiply-fractured horizontal well in a rectangular drainage region. The central result in the algorithm discussed here is an outline of a scheme that permits us to incorporate the transitional period from infmiteacting to boundary-dominated flow without recourse to approximations. We do not resort to the use of image wells. This is particularly important because this well configuration may result in the aspect ratio of the drainage area becoming rather severe, and, under such circumstances, the image-well concept is a dubious proposition. (Even for the simple case of a single vertical fracture at the center of a square drainage region, no simple method exists to determine the number of image wells a priori should the length of the fracture be comparable with the length of the drainage boundary.)At this time, we envisage four potential uses for the model proposed here: (i) to design of fracture-treatment programs (for example, determination of the number of fractures), (ii) to evaluate the efficacy of this completion scheme; for example as a screening tool, (iii) to analyze pres-

This paper presents an analytical trilinear flow solution to simulate the pressure transient and production behavior of fractured horizontal wells in unconventional reservoirs (Ozkan et al., 2009). The model is simple, but versatile enough to incorporate the fundamental petrophysical characteristics of unconventional reservoirs, including the intrinsic properties of the matrix and the natural fractures. Special characteristics of fluid exchange among various reservoir components may also be considered. This practical solution provides an excellent alternative to rigorous solutions, which are cumbersome to evaluate. One of the advantages of the trilinear flow solution is its convenience in deriving asymptotic approximations. Though linear and bilinear flow regimes have been noted for fractured horizontal wells in the literature based on their diagnostic features, they have not been associated with particular reservoir characteristics and flow relationships. The solutions documented in this work provide insight about potential flow regimes and the conditions leading to these flow regimes. Identification of these flow regimes is important for the characterization of unconventional reservoirs from pressure transient tests. Because of its relative simplicity, the trilinear flow solution does not require special expertise to compute and apply. Introduction Although it is possible to develop detailed analytical (Chen and Raghavan, 1997, and Raghavan et al., 1997) and numerical (Medeiros et al., 2008) models to represent transient fluid flow toward a multiply fractured horizontal well in tight, unconventional reservoirs such as shale, the downside of these models is the increased computational requirements, the implicit functional relationships of key parameters, and the inconvenience in their use in iterative applications. Despite the complex interplay of flow among matrix, natural fractures, and hydraulic fractures, the key characteristics of flow convergence toward a multiply fractured horizontal well may be preserved in a relatively simple, trilinear flow model (Ozkan et al., 2009). The basis of the trilinear flow model is the premise that the productive lives of fractured horizontal wells in tight formations are dominated by linear flow regimes (Medeiros et al., 2008). To derive a practical solution, the trilinear flow model includes some idealizations and simplifying assumptions. The model is for single-phase flow of a constant compressibility fluid and is applicable to single-phase gas flow through pseudopressure transformation. Identical hydraulic fractures are assumed to be equally spaced along the horizontal well, which is a common field practice. Dual-porosity idealization (Warren and Root, 1963, Kazemi, 1969, de Swaan-O, 1976, and Serra et al., 1983) is used to simulate the naturally fractured porous medium. The solution is derived in the Laplace-transform domain because we consider a naturally fractured inner reservoir. The results are then numerically inverted to the time domain using the algorithm proposed by Stehfest (1970). One-dimensional linear flow, akin to flow in vertical-well fractures, is assumed in the hydraulic fractures because wellbore storage normally masks the very early-time (radial) flow convergence toward the well within the hydraulic fractures (Soliman et al., 1990, Mukherjee and Economides, 1991, and Larsen and Hegre, 1991, 1994). However, the impact of radial flow convergence at the fracture-horizontal well intersection is taken into account by a flow choking skin, and the wellborestorage effect is incorporated into the model by convolution.