An efficient two-stage algebraic multiscale solver (TAMS) that converges to the fine-scale solution is described. The first (global) stage is a multiscale solution obtained algebraically for the given fine-scale problem. In the second stage, a local preconditioner, such as the Block ILU (BILU) or the Additive Schwarz (AS) method, is used. Spectral analysis shows that the multiscale solution step captures the low-frequency parts of the error spectrum quite well, while the local preconditioner represents the high-frequency components accurately. Combining the two stages in an iterative scheme results in efficient treatment of all the error components associated with the fine-scale problem. TAMS is shown to converge to the reference fine-scale solution. Moreover, the eigenvalues of the TAMS iteration matrix show significant clustering, which is favorable for Krylov-based methods. Accurate solution of the nonlinear saturation equations (i.e., transport problem) requires having locally conservative velocity fields. TAMS guarantees local mass conservation by concluding the iterations with a multiscale finite-volume step. We demonstrate the performance of TAMS using several test cases with strong permeability heterogeneity and large-grid aspect ratios. Different choices in the TAMS algorithm are investigated, including the Galerkin and finite-volume restriction operators, as well as the BILU and AS preconditioners for the second stage. TAMS for the elliptic flow problem is comparable to state-of-the-art algebraic multigrid methods, which are in wide use. Moreover, the computational time of TAMS grows nearly linearly with problem size.
Recent advances in multiscale methods have shown great promise in modeling multiphase flow in highly detailed heterogeneous domains. Existing multiscale methods, however, solve for the flow field (pressure and total velocity) only. Once the fine-scale flow field is reconstructed, the saturation equations are solved on the fine scale. With the efficiency in dealing with the flow equations greatly improved by multiscale formulations, solving the saturation equations on the fine scale becomes the relatively more expensive part. In this paper, we describe an adaptive multiscale finite-volume (MSFV) formulation for nonlinear transport (saturation) equations. A general algebraic multiscale formulation consistent with the operator-based framework proposed by Zhou and Tchelepi (SPE Journal, June 2008, pages 267-273) is presented. Thus, the flow and transport equations are solved in a unified multiscale framework. Two types of multiscale operators--namely, restriction and prolongation--are used to construct the multiscale saturation solution. The restriction operator is defined as the sum of the fine-scale transport equations in a coarse gridblock. Three adaptive prolongation operators are defined according to the local saturation history at a particular coarse block. The three operators have different computational complexities, and they are used adaptively in the course of a simulation run. When properly used, they yield excellent computational efficiency while preserving accuracy. This adaptive multiscale formulation has been tested using several challenging problems with strong heterogeneity, large buoyancy effects, and changes in the well operating conditions (e.g., switching injectors and producers during simulation). The results demonstrate that adaptive multiscale transport calculations are in excellent agreement with fine-scale reference solutions, but at a much lower computational cost.
A continuing challenge in hydraulic fracturing of tight gas formations is associated with remediation of formation damage caused by fluid invasion into the porous media. Numerous studies documenting the use of complex nanofluids and surfactants to remediate formation damage have been reported. Recent publications have demonstrated that complex nanofluid additives resulted in lower pressures to displace injected frac fluids over conventional surfactants, and led to greater enhancement of gas and water production. These findings were also confirmed by several recent statistical analyses that took into consideration differences in the properties of treated wells. Many field case studies and supplementary laboratory data have illustrated benefits of complex nanofluid treatment over conventional surfactants. While these publications describe the successes of complex nanofluid treatment, the influence that the formulation composition has on its performance has not been fully investigated. In the present study, we prepared complex nanofluids with different chemical compositions and examined their performance in fluid recovery tests using columns packed with sand, ceramic proppant, and shale, as well as their ability to enhance permeability of sandstone cores to gas. We have established that performance of complex nanofluids in these applications was dependent on the amount of microemulsifed solvent in the original formulations and that optimal performance across all applications was achieved with a complex nanofluid formulation with a near-balanced composition.