by Benjamin Ganis
The enhanced velocity (EV) method is a useful way to directly construct a strongly flux-continuous velocity approximation between non-matching subdomain grids . It is a straightforward method to implement based on a two-point flux approximation on Cartesian grids. The EV method has recently been implemented in the IPARS simulator on a new type of discretization called semi-structured grids. This enables powerful local mesh refinement capabilities, as well as dynamic adaptive mesh refinement (AMR).
Until recently, three restrictions were present in the previous code framework for EV: (1) interfaces were restricted to be on the external boundaries of subdomains, (2) subdomains could not be nested within each other, and (3) the decompositions were static in time. Figure 1 shows the types of grids that the EV method was able to previously support. The left figure shows three structured subdomains with different resolutions coming together in a spatially non-conforming way in a “side-by-side” configuration. The right figure shows that when local refinements were needed in the interior of the domain, it needed to be broken up into logically structured pieces, which was somewhat cumbersome.
|Figure 1. Two examples of decompositions possible under the previous EV implementation with interfaces restricted to external subdomain boundaries. Colors represent subdomains.|
In this project, we have relaxed the above three restrictions on the code framework. More specifically, structured subdomain grids are allowed to have inactive cells, such that other subdomains may have active cells in these vacant areas. From this point of view, the subdomains are placed “on top” of each other, and each subdomain is associated with its own mesh refinement level. We define this as a semi-structured grid. This enables adaptive grid refinement in any part of the domain with a minimal number of subdomains and greater ease of use. It also allows one to accurately resolve solution features such as moving fronts, or domain features such as channels.
Figure 2 shows an example of an admissible spatial decomposition using a semi-structured grid. The grid has been refined to resolve the sharp gradient of a saturation front caused by a radial flow pattern around a fluid injection source. The first subdomain has a 10 x 10 grid, and the others are refined by a factor of 2 in each dimension, leading to the finest subdomain having a 80 × 80 grid. Each subdomain contains inactive cells (not shown) to create vacant areas where other subdomains may have their own active regions. The subdomains are polyhedral in shape, but not necessarily contiguous.
|Figure 2: An example showing how the union of four non-overlapping polygonal subdomains with different spatial resolutions forms a semi-structured grid.|
The first simulation shows how the EV method on semi-structured grids can be used for adaptive mesh refinement. We use an Equation of State compositional flow model with an indicator for adaptive mesh refinement based on the gradient of water saturation. The reservoir properties are homogeneous, and the wells are placed in a five spot pattern with 4 water injectors and one producer. Figure 3 shows the adaptive grid for a 20 day simulation (left) and the corresponding water saturation solution (right). The criterion for grid refinement allows the method to capture the evolution of a sharp front throughout the reservoir, while keeping the computational cost low by using a minimal number of elements.
|Figure 3. Adaptive mesh refinement in a homogeneous reservoir: mesh (top) and water saturation (bottom).|
The next simulation shows how the adaptive mesh refinement can work with realistic heterogeneous reservoir properties. The porosity and permeability were generated from several layers the SPE10 dataset, and then upscaled to various grid resolutions. Three pairs of injectors and producers were placed inside three high permeability channels. Figure 4 shows the adaptive grid for an 80 day simulation (left) and the corresponding water saturation solution (right). The simulation starts as a coarse mesh, and once again the indicator based on gradient of water saturation dynamically refines in the vicinity of the front, and de-refines the mesh behind it.
|Figure 4. Adaptive mesh refinement in a highly heterogeneous reservoir: mesh (top) and water saturation (bottom).|
This work on using the EV method with semi-structured grids for adaptive mesh refinement is currently a submitted technical report  and a talk was recently given on the topic at the 2017 SIAM Geosciences Conference in Erlangen, Germany. Preliminary numerical results show good agreement with fine scale solutions and a significant savings in computational runtime. This method is also being applied to a number of concurrent and future projects. In particular, the newly implemented AMR capabilities of the IPARS simulator are being used to couple these fluid models with phase field fracture propagation models. A variation of the EV method known as the local flux technique has been implemented as a way to couple non-matching distorted hexahedral grids with multipoint flux mixed finite element methods. In the near future, the EV method will be generalized to space-time, allowing local time stepping. We are also developing a posteriori error indicators for multiphase flow and transport.
 B. Ganis, G. Pencheva, M. F. Wheeler. Adaptive Mesh Refinement with an Enhanced Velocity Mixed Finite Element Method on Semi-Structured Grids using a Fully-Coupled Solver. ICES Report 18-03, The University of Texas at Austin, (2018). https://www.ices.utexas.edu/media/reports/2018/1803.pdf
 J. Wheeler, M.F. Wheeler, and I. Yotov. Enhanced velocity mixed finite element methods for flow in multiblock domains. Computational Geosciences, 6(3):315–332, 2002. https://doi.org/10.1023/A:1021270509932