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.