Estimation of a Permeability Field within the Two-Phase Porous Media Flow Using Nonlinear Multigrid Method

Estimation of spatially varying permeability within the two-phase porous media flow plays an important role in reservoir simulation. Usually, one needs to estimate a large number of permeability values from a limited number of observations, so the computational cost is very high even for a single field-model. This paper applies a nonlinear multigrid method to estimate the permeability field within the two-phase porous media flow. Numerical examples are provided to illustrate the feasibility and effectiveness of the proposed estimation method. In comparison with other existing methods, the most outstanding advantage of this method is the computational efficiency, computational accuracy, and antinoise ability. The proposed method has a potential applicability to a variety of parameter estimation problems.

Nonlinear Multigrid for Reservoir Simulation

Summary A feasibility study is presented on the effectiveness of applying nonlinear multigrid methods for efficient reservoir simulation of subsurface flow in porous media. A conventional strategy modeled after global linearization by means of Newton’s method is compared with an alternative strategy modeled after local linearization, leading to a nonlinear multigrid method in the form of the full-approximation scheme (FAS). It is demonstrated through numerical experiments that, without loss of robustness, the FAS method can outperform the conventional techniques in terms of algorithmic and numerical efficiency for a black-oil model. Furthermore, the use of the FAS method enables a significant reduction in memory usage compared with conventional techniques, which suggests new possibilities for improved large-scale reservoir simulation and numerical efficiency. Last, nonlinear multilevel preconditioning in the form of a hybrid-FAS/Newton strategy is demonstrated to increase robustness and efficiency.

A nonlinear multigrid method for inversion of two-dimensional acoustic wave equation

We investigate the problem of estimating the velocity in a two-dimensional acoustic wave equation, which plays an important role in geological survey. The forward problem is discretized using finite-difference methods and the estimation is formulated as a least-square minimization problem with a regularization term. To reduce the computational burden, a nonlinear multigrid method is applied to solve this inverse problem. In the multigrid inversion process, in order to make the objective functionals at different scales compatible, they are dynamically adjusted. In this way, the necessary condition of “the optimal solution should be the fixed point of multigrid inversion” can be met. The stable and fast regularized Gauss–Newton method is applied to each grid. The results of numerical simulations indicate that the proposed method can effectively reduce the required computation, improve the inversion results, and have the anti-noise ability.

An Efficient, Energy Stable Scheme for the Cahn-Hilliard-Brinkman System

We present an unconditionally energy stable and uniquely solvable finite difference scheme for the Cahn-Hilliard-Brinkman (CHB) system, which is comprised of a Cahn-Hilliard-type diffusion equation and a generalized Brinkman equation mod-eling fluid flow. The CHB system is a generalization of the Cahn-Hilliard-Stokes model and describes two phase very viscous flows in porous media. The scheme is based on a convex splitting of the discrete CH energy and is semi-implicit. The equations at the implicit time level are nonlinear, but we prove that they represent the gradient of a strictly convex functional and are therefore uniquely solvable, regardless of time step size. Owing to energy stability, we show that the scheme is stable in the time and space discrete and norms. We also present an efficient, practical nonlinear multigrid method . comprised of a standard FAS method for the Cahn-Hilliard part, and a method based on the Vanka smoothing strategy for the Brinkman part . for solving these equations. In particular, we provide evidence that the solver has nearly optimal complexity in typical situations. The solver is applied to simulate spinodal decomposition of a viscous fluid in a porous medium, as well as to the more general problems of buoyancy- and boundary-driven flows.