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Most practical reservoir simulation studies are performed using the so-called black oil model, in which the phase behavior is represented using solubilities and formation volume factors. We extend the multiscale finite-volume (MSFV) method to deal with nonlinear immiscible three-phase compressible flow in the presence of gravity and capillary forces (i.e., black oil model). Consistent with the MSFV framework, flow and transport are treated separately and differently using a sequential implicit algorithm. A multiscale operator splitting strategy is used to solve the overall mass balance (i.e., the pressure equation). The black-oil pressure equation, which is nonlinear and parabolic, is decomposed into three parts. The first is a homogeneous elliptic equation, for which the original MSFV method is used to compute the dual basis functions and the coarse-scale transmissibilities. The second equation accounts for gravity and capillary effects; the third equation accounts for mass accumulation and sources/ sinks (wells). With the basis functions of the elliptic part, the coarse-scale operator can be assembled. The gravity/capillary pressure part is made up of an elliptic part and a correction term, which is computed using solutions of gravity-driven local problems. A particular solution represents accumulation and wells. The reconstructed fine-scale pressure is used to compute the finescale phase fluxes, which are then used to solve the nonlinear saturation equations. For this purpose, a Schwarz iterative scheme is used on the primal coarse grid. The framework is demonstrated using challenging black-oil examples of nonlinear compressible multiphase flow in strongly heterogeneous formations.

Gas flow through fractured nanoporous shale formations is complicated by a hierarchy of structural features (ranging from nanopores to microseismic and hydraulic fractures) and by several transport mechanisms that differ from the standard viscous flow used in reservoir modelling. In small pores, self-diffusion becomes more important than advection; also, slippage effects and Knudsen diffusion might become relevant at low densities. We derive a nonlinear effective diffusion coefficient that describes the main transport mechanisms in shale-gas production. In dimensionless form, this coefficient depends only on a geometric factor (or dimensionless permeability) and on the kinetic model that describes the gas. To simplify the description of the complex structure of fractured shales, we observe that the production rate is controlled by the flow from the shale matrix (which has the smallest diffusivity) into the fracture network, which is assumed to produce instantaneously. Therefore, we propose to model the flow in the shale matrix and estimate the production rate with a simple bundle-of-dual-tubes model (BoDTM), in which each tube is characterized by two diameters (one for transport and the other for storage). The solution of a single tube is approximately self-similar at early time, but not at late time, when the gas flux decays exponentially owing to the finite length of the tube. To construct a BoDTM, a reliable estimate of the joint statistics of the matrix-porosity parameters is required. This can be either inferred from core measurements or postulated on the basis of some a priori assumptions when information from laboratory and field measurements is scarce. By comparison with field production data from the Barnett shale-gas field, we demonstrate that BoDTM can be calibrated to estimate structural parameters of the shale formation and to predict the cumulative production of shale gas. Our framework has enough flexibility to construct models of increasing complexity that can be employed in the presence of a complex dataset or when more information is available.

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