Revisiting the Dual-Porosity/Dual-Permeability Modeling of Unconventional Reservoirs: The Induced-Interporosity Flow Field
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Copyright 2015 Society of Petroleum Engineers. This work presents a new approach to account for variable matrix-block size in the well performance of unconventional shale reservoirs. In contrast to the standard models that consider either a fixed matrix-block size or a stochastic distribution of sizes with no particular dependence on the spatial location, in this model we consider the case when the characteristic length of the blocks depends on the distance from the main hydraulic-fracture plane. In particular, we assume that, as a result of the hydraulic-fracturing treatment, the density of microfractures (natural and induced) is high near the hydraulic-fracture face, but gradually decreases away from it. In other words, the matrix-block size is an increasing function of the distance from the fracture face. We show that the boost of the contact area between the matrix blocks and the microfractures can be the determinant feature of hydrocarbon production from shale reservoirs, even when the natural and induced microfractures do not have uniform density throughout the whole stimulated reservoir volume. In addition, the elongated linear flow (the consensus characteristic signature in the well performance of shale systems) may be the result of an induced interporosity flow when the matrix-/fracture-permeability ratio is small. The duration of this flow regime increases if the effective size of the matrix-block distribution increases. On the other hand, the linear flow can be the result of the total-system response when the matrix-/fracture-permeability ratio is high; in this case, the beginning of the linear flow is strongly affected by the small matrix blocks near the hydraulic-fracture faces. The resulting mathematical model in Laplace space has a fundamentally different structure compared with standard dual-porosity models because of the spatial dependence of the parameters characterizing the interporosity flow. In this work, we develop the Airy-spline scheme, a new technique to calculate the pressure solution in an efficient way. Closed-form approximate formulae in the time domain are also provided, revealing that during the induced linear-interporosity-flow period, the production behavior is controlled by the logarithmic mean of the minimum and maximum matrix-block size. A field example from the Barnett shale is presented to illustrate the use of the new model.
