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Streamline method for resolving sharp fronts for complex two-phase flow in porous media

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Abstract

In this paper, we present a fast streamline-based numerical method for the two-phase flow equations in high-rate flooding scenarios for incompressible fluids in heterogeneous and anisotropic porous media. A fractional flow formulation is adopted and a discontinuous Galerkin method (DG) is employed to solve the pressure equation. Capillary effects can be neglected in high-rate flooding scenarios. This allows us to present an improved streamline approach in combination with the one-dimensional front tracking method to solve the transport equation. To handle the high computational costs of the DG approximation, domain decomposition is applied combined with an algebraic multigrid preconditioner to solve the linear system. Special care at the interior interfaces is required and the streamline tracer has to include a dynamic communication strategy. The method is validated in various two- and three-dimensional tests, where comparisons of the solutions in terms of approximation of flow front propagation with standard fully implicit finite-volume methods are provided.

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Funding

This work was partially supported by the DFG grant (WO/671 11-1).

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Authors and Affiliations

  1. Lehrstuhl für Numerische Mathematik, Technische Universität München, Boltzmannstraße 3, 85748, Garching, Germany

    Ettore Vidotto & Barbara Wohlmuth

  2. Department of Hydromechanics and Modelling of Hydrosystems, Universität Stuttgart, Pfaffenwaldring 61, 70569, Stuttgart, Germany

    Rainer Helmig & Martin Schneider

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  1. Ettore Vidotto
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  2. Rainer Helmig
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Correspondence to Ettore Vidotto.

Appendix: Implementation of streamlines on decomposed domains

Appendix: Implementation of streamlines on decomposed domains

Using the parallel features of DUNE, the pressure field can be computed using an overlapping domain decomposition approach. Therefore, the module for streamline computations has to be parallelized accordingly. A typical situation is depicted in Fig. 11, where the computational domain is decomposed into two overlapping subdomains.

Fig. 11
figure 11

Overlapping domain decomposition using two processes

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For each element, streamlines are launched from its center and are distributed among different processes. Each process starts the computation of its own local set of streamlines independently. If a streamline reaches the boundary of the process where it started, its computation is stopped. Once each process is done with its own set of streamlines, a communication between processes is required to continue the streamlines that have been interrupted at the process boundary. In the following, we present the algorithm employed to track the streamline between different processes. The communication is achieved using Message Passing Interface (MPI). Considering the situation presented in Fig. 11, each process contains a subdomain, which overlaps with the other process. Between an overlap element in one process and the corresponding interior element in the other process, data can be communicated using the DUNE class:

figure a
figure b

For ease of presentation, let us consider again the simplified problem involving only two processes. We start a streamline from an element \(E\in \mathcal {E}_{h}\) in the first process, as depicted in Fig. 12. Let us denote by \(E_{n}^{P_{2}}\) the element where the streamline ends after Δt seconds. To be determined are the elements crossed by the streamline, the corresponding crossing times and, eventually, the different saturation values. Let the velocity field \(\bar {\boldsymbol {v}}\) be given. In Algorithm 1, a simplified version of the algorithm is presented, where the procedure for tracking the streamlines over a time Δt is provided for the case depicted in Fig. 12. The extension to more processes follows the same concept. Furthermore, we point out that the presented algorithm is independent of the dimension d of the problem.

Fig. 12
figure 12

Streamline crossing the process border

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Vidotto, E., Helmig, R., Schneider, M. et al. Streamline method for resolving sharp fronts for complex two-phase flow in porous media. Comput Geosci 22, 1487–1502 (2018). https://doi.org/10.1007/s10596-018-9767-z

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