Pore-Scale Level Set Simulations of Capillary-Controlled Displacement with Adaptive Mesh Refinement

Abstract

Multiphase flow simulations on imaged porous rock structures require numerical methods that are accurate and robust when applied on complex geometries. A key element in this context is to investigate how simulations behave under grid refinement. In this work, we couple an existing software for structured adaptive mesh refinement (AMR) and parallelism to a previously developed level set method for capillary-controlled displacement on the pore scale. The level set method accounts for wettability by using different evolution velocities in pore and solid space rather than implementing it as a boundary condition on the pore walls. We perform simulations with up to three nested refinement levels on idealized pore geometries to validate the AMR technology. Based on simulations, we identify suitable cell refinement criteria for our applications. We demonstrate effects of AMR in simulations relevant for drainage in porous media, such as in the computation of capillary entry pressures in pore throats with different shapes and wetting states, and obtain excellent agreement with analytic results. Finally, we investigate the effects of AMR during simulation of quasi-static drainage on sandstone. The comparison of capillary equilibrium fluid configurations, capillary pressure, and specific fluid/fluid interfacial area–saturation relationships for drainage with and without AMR shows differences that diminish for less water-wet states. The general behavior is that the capillary pressure and interfacial area for a given saturation increase in simulations with AMR. The largest deviation occurs for small water saturations, suggesting AMR can be an important component in simulation tools to describe more accurately capillary behavior in the low water saturation regime where interface curvature is high.

Appendices

Appendix A: Scaling results

We performed a strong scaling test to demonstrate the capabilities of our LS code in massively parallel environments using the compute resource MareNostrum 4 at Barcelona Supercomputing Center (BSC, Spain). We consider a benchmark simulation on a domain containing $750 \times 750 \times 752$ grid cells. The simulation consists of two initial solves of the reinitialization equation to construct the signed distance functions for both the pore geometry and the initial fluid configuration. This is followed by 20 iteration–time steps with the main LS evolution equation, where a reinitialization occurs after the 10th and 20th iteration–time step to ensure that the fluid LS function maintains a signed distance function throughout the interface evolution. All the reinitialization solves use 10 iteration steps each. Thus, overall the simulation constitutes 60 iterations, of which 20 are main LS iteration–time steps and 40 are reinitialization iterations. These fractions are representative for a typical usage of our code. To determine the strong scaling behavior, we ran the simulation four times using an increasing number of cores, ranging from 264 to 2112. Figure 18 shows that the strong scaling test provides an excellent speedup for an increased number of cores.

Fig. 18

Speedup normalized to 264 cores versus number of cores from the strong scaling tests

Appendix B: Curvature data in sphere pack pore throats

See Tables 1, 2, 3 and 4.

Table 1 Analytic and simulated data in a sphere pack pore throat with $\gamma =31^{\circ }$

Table 2 Analytic and simulated data in a sphere pack pore throat with $\gamma =35^{\circ }$

Table 3 Analytic and simulated data in a sphere pack pore throat with $\gamma =40^{\circ }$

Table 4 Analytic and simulated data in a sphere pack pore throat with $\gamma =45^{\circ }$