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.

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.

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.

These three statements regarding personal ancestry were made to me by villagers during life history interviews I conducted a few months into my research in northeastern Madagascar. Each statement is an admission of slave ancestry, and I highlight them to introduce this paper for three reasons. First, such statements are not uncommon in these villages. Many villagers told stories of lost or stolen ancestors, forced labor for “nobles,” and slave ancestry. Second, much of the recent scholarship addressing slavery elsewhere in Madagascar has suggested that slavery is not easily discussed among contemporary residents of this Indian Ocean island. Thus, the fact that the people among whom I studied readily acknowledged their own slave descent by referring to their “lost” or “stolen” grandparents or to their own Makoa identity prompts further comparative inquiry. What factors explain acceptance of slave ancestry among some Malagasy and its concurrent stigmatization among others? Third, examination of variations in Madagascar's responses to slavery can lead us to new insights into the forms of identity and opportunity in other post-slave societies.