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We describe the underlying mathematics, validation, and applications of a novel Helmholtz free-energy—minimizing phase-field model solved within the framework of the lattice Boltzmann method (LBM) for efficiently simulating two-phase pore-scale flow directly on large 3D images of real rocks obtained from micro-computed tomography (micro-CT) scanning. The code implementation of the technique, coined as the eLBM (energy-based LBM), is performed in CUDA programming language to take maximum advantage of accelerated computing by use of multinode general-purpose graphics processing units (GPGPUs). eLBM’s momentum-balance solver is based on the multiple-relaxation-time (MRT) model. The Boltzmann equation is discretized in space, velocity (momentum), and time coordinates using a 3D 19-velocity grid (D3Q19 scheme), which provides the best compromise between accuracy and computational efficiency. The benefits of the MRT model over the conventional single-relaxation-time Bhatnagar-Gross-Krook (BGK) model are (I) enhanced numerical stability, (II) independent bulk and shear viscosities, and (III) viscosity-independent, nonslip boundary conditions. The drawback of the MRT model is that it is slightly more computationally demanding compared to the BGK model. This minor hurdle is easily overcome through a GPGPU implementation of the MRT model for eLBM. eLBM is, to our knowledge, the first industrial grade–distributed parallel implementation of an energy-based LBM taking advantage of multiple GPGPU nodes. The Cahn-Hilliard equation that governs the order-parameter distribution is fully integrated into the LBM framework that accelerates the pore-scale simulation on real systems significantly. While individual components of the eLBM simulator can be separately found in various references, our novel contributions are (1) integrating all computational and high-performance computing components together into a unified implementation and (2) providing comprehensive and definitive quantitative validation results with eLBM in terms of robustness and accuracy for a variety of flow domains including various types of real rock images. We successfully validate and apply the eLBM on several transient two-phase flow problems of gradually increasing complexity. Investigated problems include the following: (1) snap-off in constricted capillary tubes; (2) Haines jumps on a micromodel (during drainage), Ketton limestone image, and Fontainebleau and Castlegate sandstone images (during drainage and subsequent imbibition); and (3) capillary desaturation simulations on a Berea sandstone image including a comparison of numerically computed residual non-wetting-phase saturations (as a function of the capillary number) to data reported in the literature. Extensive physical validation tests and applications on large 3D rock images demonstrate the reliability, robustness, and efficacy of the eLBM as a direct visco-capillary pore-scale two-phase flow simulator for digital rock physics workflows.

Digital rock physics (DRP) is a rapidly evolving technology targeting fast turnaround times for repeatable core analysis and multi-physics simulation of rock properties. We develop and validate a rapid and scalable distributed-parallel single-phase pore-scale flow simulator for permeability estimation on real 3D pore-scale micro-CT images using a novel variant of the lattice Boltzmann method (LBM). The LBM code implementation is designed to take maximum advantage of distributed computing on multiple general-purpose graphics processing units (GPGPUs). We describe and extensively test the distributed parallel implementation of an innovative LBM algorithm for simulating flow in pore-scale media based on the multiple-relaxation-time (MRT) model that utilizes a precise treatment of body force. While the individual components of the resulting simulator can be separately found in various references, our novel contributions are (1) the integration of all of the mathematical and high-performance computing components together with a highly optimized code implementation and (2) the delivery of quantitative results with the simulator in terms of robustness, accuracy, and computational efficiency for a variety of flow geometries including various types of real rock images. We report on extensive validations of the simulator in terms of accuracy and provide near-ideal distributed parallel scalability results on large pore-scale image volumes that were largely computationally inaccessible prior to our implementation. We validate the accuracy of the MRT-LBM simulator on model geometries with analytical solutions. Permeability estimation results are then provided on large 3D binary microstructures including a sphere pack and rocks from various sandstone and carbonate formations. We quantify the scalability behavior of the distributed parallel implementation of MRT-LBM as a function of model type/size and the number of utilized GPGPUs for a panoply of permeability estimation problems.

Single-phase fluid flow velocity maps in Ketton and Estaillades carbonate rock core plugs are computed at a pore scale, using the lattice Boltzmann method (LBM) simulations performed directly on three-dimensional (3D) X-ray micro-computed tomography (µCT) images (≤ 7 µm spatial resolution) of the core plugs. The simulations are then benchmarked on a voxel-by-voxel and pore-by-pore basis to quantitative, 3D spatially resolved magnetic resonance imaging (MRI) flow velocity maps, acquired at 35 µm isotropic spatial resolution for flow of water through the same rock samples. Co-registration of the 3D experimental and simulated velocity maps and coarse-graining of the simulation to the same resolution as the experimental data allowed the data to be directly compared. First, the results are demonstrated for Ketton limestone rock, for which good qualitative and quantitative agreement was found between the simulated and experimental velocity maps. The flow-carrying microstructural features in Ketton rock are mostly larger than the spatial resolution of the µCT images, so that the segmented images are an adequate representation of the pore space. Second, the flow data are presented for Estaillades limestone, which presents a more heterogeneous case with microstructural features below the spatial resolution of the µCT images. Still, many of the complex flow patterns were qualitatively reproduced by the LBM simulation in this rock, although in some pores, noticeable differences between the LBM and MRI velocity maps were observed. It was shown that 80% of the flow (fractional summed z -velocities within pores) in the Estaillades rock sample is carried by just 10% of the number of macropores, which is an indication of the high structural heterogeneity of the rock; in the more homogeneous Ketton rock, 50% of the flow is carried by 10% of the macropores. By analysing the 3D MRI velocity map, it was found that approximately one-third of the total flow rate through the Estaillades rock is carried by microporosity—a porosity that is not captured at the spatial resolution of the µCT image.

Consistent discretization methods are a natural fit for the novel cut-cell gridding technique for reservoir simulation, which preserves the orthogonality characteristic in the lateral direction. Both uniform (global) and novel hybrid (local) variants of consistent discretization methods are implemented and tested in the vicinity of fault representations on cut-cell grids. Novel consistent discretization methods, which do not require major intrusive changes to the solver structure of industrial-grade reservoir simulators, are investigated in this work. Cell-centered methods such as multi-point flux approximation (MPFA), average multi-point flux approximation (AvgMPFA), and nonlinear two-point flux approximation (NTPFA) methods fit naturally into the framework of existing industrial-grade simulators. Thus, cut-cell compatible variants of AvgMPFA and NTPFA and their novel hybridizations with TPFA are implemented and tested. An implementation of the relatively more computationally expensive MPFA is also made to serve as accuracy reference to AvgMPFA and NTPFA. AvgMPFA and NTPFA multiphase simulation results are compared in terms of accuracy and computational performance against the ones computed with reference MPFA and TPFA methods on a set of synthetic cut-cell grid models of varying complexity including conceptual models and a field-scale model. It is observed that AvgMPFA consistently yields more accurate and computationally efficient simulations than NTPFA on cut-cell grids. Moreover, AvgMPFA-TPFA hybrids run faster than NTPFA-TPFA hybrids when compared on the same problem for the same hybridization strategy. On the other hand, the computational performance of AvgMPFA degrades more rapidly compared to NTPFA with increasing “rings” of orthogonal blocks around cut-cells. Auspiciously, only one or two “rings” of orthogonal blocks around cut cells are sufficient for AvgMPFA to deliver high accuracy.

The Galerkin finite-element discretization of the force balance equation typically leads to large linear systems for geomechanical problems with realistic dimensions. In iteratively coupled flow and geomechanics modeling, a large linear system is solved at every timestep often multiple times during coupling iterations. The iterative solution of the linear system stemming from the poroelasticity equations constitutes the most time-consuming and memory-intensive component of coupled modeling. Block Jacobi, LSOR, and Incomplete LU factorization are popular preconditioning techniques used for accelerating the iterative solution of the poroelasticity linear systems. However, the need for more effective, efficient, and robust iterative solution techniques still remains especially for large coupled modeling problems requiring the solution of the poroelasticity system for a large number of timesteps. We developed a supercoarsening multigrid method (SCMG) which can be multiplicatively combined with commonly used preconditioning techniques. SCMG has been tested on a variety of coupled flow and geomechanics problems involving single-phase depletion and multiphase displacement of in-situ hydrocarbons, CO 2 injection, and extreme material property contrasts. Our analysis indicates that the SCMG consistently improves the convergence properties of the linear systems arising from the poroelasticity equations, and thus, accelerates the coupled simulations for all cases subject to investigation. The joint utilization of the two-level SCMG with the ILU1 preconditioner emerges as the most optimal preconditioning/iterative solution strategy in a great majority of the problems evaluated in this work. The BiCGSTAB iterative solver converges more rapidly compared to PCG in a number of test cases, in which various SCMG-accelerated preconditioning strategies are applied to both iterators.