In this paper, our aim is to present (1) an embedded fracture model (EFM) for coupled flow and mechanics problem based on the dual continuum approach on the fine grid and (2) an upscaled model for the resulting fine grid equations. The mathematical model is described by the coupled system of equation for displacement, fracture and matrix pressures. For a fine grid approximation, we use the finite volume method for flow problem and finite element method for mechanics. Due to the complexity of fractures, solutions have a variety of scales, and fine grid approximation results in a large discrete system. Our second focus in the construction of the upscaled coarse grid poroelasticity model for fractured media.Our upscaled approach is based on the nonlocal multicontinuum (NLMC) upscaling for coupled flow and mechanics problem, which involves computations of local basis functions via an energy minimization principle. This concept allows a systematic upscaling for processes in the fractured porous media, and provides an effective coarse scale model whose degrees of freedoms have physical meaning. We obtain a fast and accurate solver for the poroelasticity problem on a coarse grid and, at the same time, derive a novel upscaled model. We present numerical results for the two dimensional model problem.

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Summary We investigate spatial stability with various numerical discretizations in displacement and pressure fields for poroelasticity. We study 2 sources of the early time instability: discontinuity of pressure and violation of the inf‐sup condition. We consider both compressible and incompressible fluids by employing the monolithic, stabilized monolithic, and fixed‐stress sequential methods. Four different spatial discretization schemes are used: Q1Q1, Q2Q1, Q1P0, and Q2P0. From mathematic analysis and numerical tests, the piecewise constant finite volume method for flow provides stability at the early time for the case of the pressure discontinuity. On the other hand, a piecewise continuous (or higher‐order) interpolation of pressure shows spatial oscillation, having lower limits of time step size, although lower approximations of pressure than displacement can alleviate the oscillation. For an incompressible fluid, Q2Q1 can be better than Q1P0, because Q1P0 might not satisfy the inf‐sup condition. However, regardless of fluid compressibility and the pressure discontinuity, the fixed‐stress method can effectively stabilize the oscillation without an artificial stabilizer. Even when Q1P0 and Q1Q1 with the monolithic method cannot satisfy the inf‐sup condition, the fixed‐stress method can yield the full‐rank linear system, providing stability. Thus, the fixed‐stress method with Q1P0 can effectively circumvent the aforementioned 2 types of instability.

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We investigate high‐order accuracy in time integration by examining two operator splitting methods for poroelastic problems: the two‐pass and the spectral deferred correction (SDC) methods. To enhance the order of accuracy, the two‐pass method partitions a coupled operator symmetrically, whereas the SDC method corrects truncation errors by establishing an error equation. These high‐order methods are applied to underlying solution strategies, that is, monolithic, fixed‐stress sequential, and undrained sequential methods. We observe that semi‐discretized systems from spatial discretization have forms similar to those of index‐1 differential algebraic equations (DAEs), causing order reduction against the two‐pass method when it is used in conjunction with either the monolithic or sequential method. On the other hand, the SDC in conjunction with the monolithic method exhibits the desired second‐order accuracy in poroelastic problems while increasing the order of accuracy for index‐1 DAEs. However, the SDC in conjunction with either of the two sequential methods does not achieve the desired order of accuracy, and maintains first order because the flow equation for poroelasticity has an additional approximation associated with the volumetric strain rate term, which does not yield exactly the same forms as those of conventional DAEs. Thus, the monolithic SDC method can achieve higher‐order accuracy, but may require higher computational costs because it involves solving matrix systems larger than those for the sequential methods.