Multiscale Gradient Computation for Multiphase Flow in Porous Media

Abstract

A multiscale gradient computation method for multiphase flow in heterogeneous porous media is developed. The method constructs multiscale primal and dual coarse grids, imposed on the given fine-scale computational grid. Local multiscale basis functions are computed on (dual-) coarse blocks, constructing an accurate map (prolongation operator) between coarse-and fine-scale systems. While the expensive operations involved in computing the gradients are performed at the coarse scale, sensitivities with respect to uncertain parameters (e.g., grid block permeabilities) are expressed in the fine scale via the partial derivatives of the prolongation operator. Hence, the method allows for updating of the geological model, rather than the dynamic model only, avoiding upscaling and the inevitable loss of information. The formulation and implementation are based on automatic differentiation (AD), allowing for convenient extensions to complex physics. An IMPES coupling strategy for flow and transport is followed, in the forward simulation. The flow equation is computed using a multiscale finite volume (MSFV) formulation and the transport equation is computed at the fine scale, after reconstruction of mass conservative velocity field. To assess the performance of the method, a synthetic multiphase flow test case is considered. The multiscale gradients are compared against those obtained from a fine-scale reference strategy. Apart from its computational efficiency, the benefits of the method include flexibility to accommodate variables expressed at different scales, specially in multiscale data assimilation and reservoir management studies.

Multiscale Sampling for the Inverse Modeling of Partial Differential Equations

“…There are also Hessian-based procedures [7]. In the case of porous media flows gradient calculations can be computationally very expensive (see [8] and references therein).…”

A multiscale method for data assimilation

“…An adjoint-based multiscale finite volume method for computation of sensitivities has been presented in [20] and later extended to time-dependent [19] singlephase flow in porous media. More recently, a general framework for the computation of multiscale gradients has been introduced in [42], with an extension to multiphase flows [41]. The latter two are based on a general framework for derivative computation, whose algebraic nature does not rely on any assumption regarding the nature of the parameters, observations, or objective function type.…”

A Multiscale Method For Data Assimilation

“…The most efficient data assimilation methods are gradient-based, with gradients obtained with the adjoint method Oliver et al (2008). Krogstad et al (2011) and Moraes et al (2017a) present multiscale gradient computation strategies for multiphase flow, the former applied to a production optimization problem, and the latter to data assimilation problems. Also, the latter is based on the general framework for MS gradient computation presented by Moraes et al (2017b), whose algebraic nature does not rely on any assumption regarding the nature of the parameters, observations, or objective function type.…”