Project 5

A stochastic mixed finite element heterogeneous multiscale method for flow in porous media

Flow through porous media is ubiquitous, occurring from large geological scales down to microscopic scales. Several critical engineering phenomena like contaminant spread, nuclear waste disposal and oil recovery rely on accurate analysis and prediction of these multiscale phenomena. Such analysis is complicated by heterogeneities at various length scales as well as inherent uncertainties. For these reasons in order to predict the flow and transport in stochastic porous media, some type of stochastic upscaling or coarsening is needed for computational efficiency by solving these problems on a coarse grid. However, most of the existing multiscale methods are realization based, i.e. they can only solve a deterministic problem for a single realization of the stochastic permeability. This is not sufficient for uncertainty quantification since we are mostly interested in the statistics of the flow behavior, such as mean and standard deviation. In this project, a computational methodology is developed to efficiently perform uncertainty quantification for fluid transport in porous media in the presence of both stochastic permeability and multiple scales. In order to capture the small scale heterogeneity, a new mixed multiscale finite element method is developed within the framework of the heterogeneous multiscale method (HMM) in the spatial domain. Adaptive high dimensional stochastic model representation technique (HDMR) which is developed in Project 3 is used in the stochastic space. The goal of the multiscale method is to coarsen the flow equation spatially whereas HDMR is used to address the curse of dimensionality in high dimensional stochastic space.

Our problem of interest is the following stochastic single-phase porous media problem:

The random permeability field exhibits both multiscale feature in physical space and random feature in stochastic space. We develop a new mixed multiscale finite element (MxHMM) method which is base on the HMM framework. The subgrid problem is defined at each quadrature point within each coarse element:

Fig 1. Left: The schematic of the fine- and coarse-scale grid; Right: The proposed MxHMM method where the sampling domain is same as the coarse element.

In order to better reflect the heterogeneous structure across the coarse element boundary, which is often important in guaranteeing the continuity of flux on the interface, we also propose a modified boundary condition:

Fig 2. The modified boundary condition where the flux is scaled according to the fine-scale transmissibilities.

To verify the developed multiscale method, we first solve a problem with deterministic permeability in a realistic two-dimensional reservoir. The results are shown as follows:

Fig 3. Left: Logarithm of the permeability field from the top layer of the 10th SPE model, which is defined on 60 x 220 fine scale grid. Right: Contour plots of saturation at 0.4 PVI for various coarse meshes.

Fig 4. Water cut curves for various coarse grids.

The complete schematic of the stochastic multiscale method for porous media flow is illustrated in the following figure:

Fig 5. Schematic of the developed stochastic multiscale method for porous media flow.

In the second example, we investigate the statistical properties of the transport phenomena in random heterogenous porous media. The domain of interest is the unit square and the flow is induced from left-to-right. The random log-permeability used is the same as in Project 3. In this problem, the fine-scale permeability is defined on 64 x 64 grid and the coarse grid is taken as 8 x 8. For comparison, the reference solution is taken from 1 million MC samples, where each direct problem is solved using the fine-scale solver. The stochastic problem is solved using HDMR, where the solution of each deterministic problem at the collocation points is from the multiscale solver. In this way, the accuracy of both multiscale solver and HDMR can be verified.

Fig 6. Mean and standard deviation of saturation at 0.4 PVI for isotropic random field. Top: mean (a) and standard deviation (b) from HDMR. Bottom: comparison of mean (c) and standard deviation (d) between MC and HDMR near the saturation front.

In Fig 6, the mean saturation is the same as the solution with homogeneous mean permeability. This behavior is called "heterogeneity-induced dispersion" where the heterogeity smoothes the water saturation field. The figure also indicates that higher water saturation variation are concentrated near displacement fronts, which are areas of steep saturation gradients. Next, we demonstrate the interpolatory properties of HDMR method. One of the advantages of HDMR is that it can serve as a surrogate model for the original problem. Realization of the saturation for arbitrary random input can be obtained through HDMR. To verify this property, we randomly generate one input vector and reconstruct the result from HDMR. At the same time, we run a deterministic problem with the fine-scale model and the same realization of the random input vector. The comparison of the results are shown in Fig 7.

Fig 7. Prediction of the saturation profile using HDMR and the solution of the deterministic fine-scale problem with the sample input for isotropic random field. Left: saturation at 0.4 PVI from direct simulation. Right: saturation reconstructed from HDMR.

Fig 8. PDF (a) and CDF (b) of the saturation at point (0.4, 0) and 0.4 PVI.

In Fig 8, we also plot the probability density function (PDF) and cumulative distribution function (CDF) at point (0.4) where the highest standard deviation happens. These results indicate that the corresponding HDMR approximation are indeed very accurate. Therefore, we can obtain any statistics from this stochastic model, which is an advantage of the current method over the MC method.

The paper and PowerPoint presentation can be downloaded from here.