Simulation Of Flow In Porous Media Research Articles

A five field formulation for flow simulations in porous media with fractures and barriers via an optimization based domain decomposition method

The present work deals with the numerical resolution of coupled 3D–2D problems arising from the simulation of fluid flow in fractured porous media modeled via the Discrete Fracture and Matrix (DFM) model. According to the DFM model, fractures are represented as planar interfaces immersed in a 3D porous matrix and can behave as preferential flow paths, in the case of conductive fractures, or can actually be a barrier for the flow, when, instead, the permeability in the normal-to-fracture direction is small compared to the permeability of the matrix. Consequently, the pressure solution in a DFM can be discontinuous across a barrier, as a result of the geometrical dimensional reduction operated on the fracture. The present work is aimed at developing a numerical scheme suitable for the simulation of the flow in a DFM with fractures and barriers, using a mesh for the 3D matrix non conforming to the fractures and that is ready for domain decomposition. This is achieved starting from a PDE-constrained optimization method, currently available in literature only for conductive fractures in a DFM. First, a novel formulation of the optimization problem is defined to account for non permeable fractures. These are described by a filtration-like coupling at the interface with the surrounding porous matrix. Also the extended finite element method with discontinuous enrichment functions is used to reproduce the pressure solution in the matrix around a barrier. The method is presented here in its simplest form, for clarity of exposition, i.e. considering the case of a single fracture in a 3D domain, also providing a proof of the well posedness of the resulting discrete problem. Four validation examples are proposed to show the viability and the effectiveness of the method.

Accelerating fluid flow simulations through doubly porous media using a FEM-assisted machine learning approach

This paper introduces an innovative methodology for simulating fluid flow through doubly porous media. This technique combines finite element (FE) simulations and machine learning (ML) methods. The approach involves training ML models with a dataset of FE simulation outcomes, enabling predictions of macroscopic permeability under varied conditions. This strategy addresses the limitations of conventional simulation methods, which suffer from high computational demands and lack of systematic insight. The approach's precision and efficiency are confirmed through numerical experiments, achieving a coefficient of determination of 0.998. The outcomes underline the potential of the FEM-assisted ML strategy to considerably enhance fluid flow simulations in porous media. The technique can find applications across diverse domains within civil engineering. Furthermore, developing an intuitive graphical user interface (GUI) streamlines the application of the proposed approach.

Implementation of a direct-addressing based lattice Boltzmann GPU solver for multiphase flow in porous media

GPU accelerated lattice Boltzmann (LB) simulations of multiphase flow in porous media have become a powerful tool to study fluid displacement process in porous media. For porous structures with a very low porosity, indirect-addressing memory access methods are preferred due to significantly reduction of the memory footprint despite that those methods are more difficult to implement and the resulting performance may be more sensible to the computing architectures, such as cache hierarchy and size. The direct-addressing methods are straightforward to implement and are able to archive high throughput when the porosity is large, while the methods become less efficient when the structures are too sparse. Nevertheless, the direct-addressing methods combined with multi-block grid technique are promising for many applications. In this work, we present a hybrid-way to tackle multiphase flow simulations in porous media, where the direct-addressing method is employed for the main LB evolution while the indirect-addressing method is employed for the complex boundary conditions. The CSF-based LB color-gradient multiphase model and the geometry-based wetting model are employed to increase accuracy and stability. Thanks to the utilization of AA-pattern streaming scheme, non-slip boundary condition of arbitrary orientation can be enforced without extra cost. We perform comprehensive analysis of the computational performance of the present solver on NVIDIA GPUs. For typical sandstones, our results show that the present implementation is able to achieve over 1.5 speedup compared with other direct-addressing schemes on V100 and A100, respectively and, particularly, the computational performance of the boundary kernels is greatly increased thanks to the increased L2 cache size of the latest GPUs.

A generic workflow of projection-based embedded discrete fracture model for flow simulation in porous media

A generic workflow of projection-based embedded discrete fracture model for flow simulation in porous media

Stream and Potential Functions for Transient Flow Simulations in Porous Media with Pressure-Controlled Well Systems

Gaussian solutions of the diffusion equation can be applied to visualize the flow paths in subsurface reservoirs due to the spatial advance of the pressure gradient caused by engineering interventions (vertical wells, horizontal wells) in subsurface reservoirs for the extraction of natural resources (e.g., water, oil, gas, and geothermal fluids). Having solved the temporal and spatial changes in the pressure field caused by the lowered pressure of a well’s production system, the Gaussian method is extended and applied to compute and visualize velocity magnitude contours, streamlines, and other relevant flow attributes in the vicinity of well systems that are depleting the pressure in a reservoir. We derive stream function and potential function solutions that allow instantaneous modeling of flow paths and pressure contour solutions for transient flows. Such analytical solutions for transient flows have not been derived before without time-stepping. The new closed-form solutions avoid the computational complexity of time-stepping, required when time-dependent flows are modeled by superposing steady-state solutions using complex analysis methods.

A data-driven approach for a macroscopic conductivity model utilizing finite element approximation

Macroscopic modeling is useful in many application areas, such as flow simulation in porous media and reduced order approximation for fast solvers and multi-physics simulations. The focus of this work is to propose an algorithm for macroscopic modeling for elliptic problems with coefficients of random and high variations, which can often be used to describe microscopic structures in porous media applications, composite materials, and medical imaging applications. The proposed method approximates a general tensor type macroscopic conductivity with a deep neural network and the parameters in the deep neural network are then optimized using the measured data on the boundary of a microscopic model. Numerical results for various test examples are presented and they show that the proposed scheme is promising for macroscopic modeling.

Residual-based adaptivity for two-phase flow simulation in porous media using Physics-informed Neural Networks

This paper aims to provide a machine learning framework to simulate two-phase flow in porous media. The proposed algorithm is based on Physics-informed neural networks (PINN). A novel residual-based adaptive PINN is developed and compared with the residual-based adaptive refinement (RAR) method and with PINN with fixed collocation points. The proposed algorithm is expected to have great potential to be applied to different fields where adaptivity is needed. In this paper, we focus on the two-phase flow in porous media problem. We provide two numerical examples to show the effectiveness of the new algorithm. It is found that adaptivity is essential to capture moving flow fronts. We show how the results obtained through this approach are more accurate than using RAR method or PINN with fixed collocation points, while having a comparable computational cost.

The Application of REV-LBM Double Mesh Local Refinement Algorithm in Porous Media Flow Simulation

REV-scale LBM (representative elementary volume scale lattice Boltzmann method) provides a possible way for the unified microscopic simulation of transscale seepage combined with pore-scale LBM. However, shown by numerical results, when simulated fluid flows in tight porous media with discrete microfractures or dispersed dissolution pore, the nonphysical oscillation phenomenon will occur in the region of relatively greater velocity gradient, due to the insufficient computational node caused by computational capability constraints. And for that reason, the increase in simulation scale under certain computational capability will be restricted. In order to solve this problem, this paper extends the application of the original LBM local refinement algorithm into REV-LBM and corrects the original REV-LBM velocity processing method, such that REV-scale LBM has a more reasonable local refinement algorithm. Through conventional computation example, this paper proves that the local refinement algorithm can maintain well computational accuracy while at least double the time efficiency and could save more than 30% of CPU time in the practical application. This algorithm has a great potential of application in simulating flows in larger size porous media with complex micro-nano structures and can provide a new idea to transscale simulation in porous media.

Numerical simulation of gas-liquid transport in porous media using 3D color-gradient lattice Boltzmann method: trapped air and oxygen diffusion coefficient analysis

ABSTRACT In non-aqueous Li–air batteries, the liquid electrolytes penetrate the porous media such as carbon nanotube (CNT) paper structure, transport dissolved substances such as oxygen, and play a role in generating reactants on the surface of the porous media. Although the trapped air generated during the electrolyte penetration process could affect the oxygen transport and performance of the battery, this issue has not been sufficiently investigated. Therefore, in this study, the patterns of electrolyte penetration and air entrapment in porous media were investigated through numerical analysis. A multi-relaxation time color-gradient lattice Boltzmann method was employed for modeling. Based on a two-phase flow simulation in porous media, electrolyte penetration and trapped-air saturation were analyzed in terms of porosity, wettability, and viscosity ratio. The porosity and viscosity ratio did not considerably affect the trapped-air saturation, whereas wettability had a significant effect on the aforementioned parameter. In addition, for each variable, an increase in the effective diffusive coefficient corresponded to increased porosity and hydrophilicity, as well as an improved viscosity ratio.

A method of FE modeling multiphase compressible flow in hydrocarbon reservoirs

We propose a model and numerical method for multiphase multicomponent flow simulation in porous media for solving problems of hydrocarbon reservoir engineering. The method takes into account the complex structure of reservoirs, a large number of operating wells, various methods of specifying the dependences of the properties of phases on pressure and mass fractions of their components, the transition of components from one phase to another, as well as compressibility and gravity. The method provides a sufficiently high computational efficiency, local conservation of mass for each component and protects the requirement of constraint of phases saturations from violation due to the following aspects. To calculate the phase flows we use a combination of continuous Galerkin finite element method (CG FEM) with a new balancing method. This combination provides high accuracy of flows calculated from the pressure field on rather coarse meshes. We also propose to use the grouping of finite elements, which allows us to remove strict restrictions on the time step when calculating the phase flows between cells. Verification of the proposed method is performed by comparing with the results of SPE tests. The effectiveness of the method is demonstrated both on simple model problems and on problems of modeling a complex reservoir. For a real field of high-viscosity oil in Tatarstan (Russia), a good agreement between the computed and observed data is obtained for all high-rate wells over a fairly long period of field development.

Minimising the impact of sub-resolution features on fluid flow simulation in porous media

Resolution of micro-computed tomography (micro-CT) images of rocks affects flow simulation results, especially on low resolution images where the pore-grain boundary is not well resolved. However, there are two occasions where conducting flow simulations on low resolution images may be beneficial. The first case is when computation costs need to be reduced, and second is when high-resolution scanned images are unavailable and segmentation on available low-resolution images is difficult due to uncertainty at the pore boundary. A novel analytical formulation, the concentric pipes method, is introduced in this paper to include the effect of sub-resolution features in flow simulation. For the first case, we compare porosity, permeability, pore connectivity, and the velocity field obtained from applying different approaches to solve for flow on images with sub-resolution features. We find that the permeabilities obtained from downsampled images using the concentric pipes method have 13.2% errors on average compared to results from high-resolution images, while computation time decreases by at least 80%. For the second case, three phase segmentation can be applied to low resolution images and the concentric pipes method can be applied to the uncertain region which allows flow in the system to be solved. Permeability estimations in this case are within 0.5% error for a sandstone image and 11% error for carbonate images. We also simulate transport of particles on downsampled images and compare the transport properties with the high-resolution images.

A novel block non-symmetric preconditioner for mixed-hybrid finite-element-based Darcy flow simulations

In this work, we propose a novel block preconditioner, labeled Explicit Decoupling Factor Approximation (EDFA), to accelerate the convergence of Krylov subspace solvers used to address the sequence of non-symmetric systems of linear equations originating from flow simulations in porous media. The flow model is discretized blending the Mixed hybrid finite element method for Darcy's equation with the Finite volume scheme for the mass conservation. The EDFA preconditioner is characterized by two features: the exploitation of the system matrix decoupling factors to recast the Schur complement and their inexact fully-parallel computation by means of restriction operators. We introduce two adaptive techniques aimed at building the restriction operators according to the properties of the system at hand. The proposed block preconditioner has been tested through extensive experimentation on both synthetic and real-case applications, pointing out its robustness and computational efficiency.

ML-LBM: Predicting and Accelerating Steady State Flow Simulation in Porous Media with Convolutional Neural Networks

Fluid mechanics simulation of steady state flow in complex geometries has many applications, from the micro-scale (cell membranes, filters, rocks) to macro-scale (groundwater, hydrocarbon reservoirs, and geothermal) and beyond. Direct simulation of steady state flow in such porous media requires significant computational resources to solve within reasonable timeframes. This study outlines an integrated method combining predictions of fluid flow (fast, limited accuracy) with direct flow simulation (slow, high accuracy) is outlined that reduces computation time by an order of magnitude without loss of accuracy. A convolutional neural network (CNNs) is trained with various configurations on simulations in 2D and 3D porous media to estimate steady state velocity fields. Permeability estimation (as an average of the field) is accurate, but the velocity fields themselves are error prone, unsuitable for further transport studies. This estimate can either be used as an indicative prediction, or as initial conditions in direct simulation to reach a fully accurate result in a fraction of the compute time. Using Deep Learning predictions (or potentially any other approximation method) to accelerate flow simulation to steady state in complex structures shows promise as a technique to push the boundaries fluid flow modelling.

Fast direct flow simulation in porous media by coupling with pore network and Laplace models

Permeability characterises flow in porous rocks/media for upscaling, while steady-state flow fields allow analysis of reactive transport, fines migration, and tight unconventional rocks. Fast calculation of permeability and flow fields obtained from Pore Network Models (PNM) and Laplace Semi-Analytical Solvers (SAS) deviate from computationally demanding simulation of Navier Stokes Equations (NSE) due to flow and geometry simplifications. Coupling PNM/SAS with direct simulation via Lattice Boltzmann Method (LBM) provides 5--150× speed-up without accuracy loss over 100 samples from 0.7mD to 3.5D. Permeability errors in PNM/SAS show 10-20% (up to 50-70%) error. PNM shows higher variance from geometric simplifications compared to SAS which only makes flow-based assumptions. PNM/SAS errors are eliminated by coupling with LBM at a fraction of LBM-only compute cost. Steady-state conditions with PNM/SAS-LBM are reached in <2,000 timesteps, compared to LBM-only which can require >105 timesteps in tight domains such as cemented sandstone, carbonate, coal, and shale.

Quasi-implicit treatment of velocity-dependent mobilities in underground porous media flow simulation

Quasi-implicit schemes for treating velocity-dependent mobilities in underground porous media flow simulation, occurring when modeling non-Newtonian and non-Darcy effects as well as capillary desaturation, are presented. With low-order finite-volume discretizations, the principle is to evaluate mobilities at cell edges using normal velocity components calculated implicitly, and transverse velocity components calculated explicitly (i.e., based on the previously converged time-step); the pressure gradient driving the flow is, as usual, treated implicitly. On 3D hexahedral meshes, the proposed schemes require the same 7-point stencil as that of common semi-implicit schemes where mobilities are evaluated with an entirely explicit velocity argument. When formulated appropriately, their higher level of implicitness however places them, in terms of numerical stability, closer to “real” fully implicit schemes requiring at least a 19-point stencil. A von Neumann stability analysis of these proposed schemes is performed on a simplified pressure equation, representative of both single-phase and multiphase flows, following an approach previously used by the authors to study semi-implicit schemes. Whereas the latter are subject to stability constraints which limit their usage in certain cases where the logarithmic derivative of mobility with respect to velocity is large in magnitude, the former are unconditionally stable for 1D and 2D flows, and only subject to weak restrictionsfor 3D flows.

A 6M digital twin for modeling and simulation in subsurface reservoirs

A 6M digital twin for modeling and simulation in subsurface reservoirs

Darcy and inertial fluid flow simulations in porous media using the non-orthogonal central moments lattice Boltzmann method

In this study, two-dimensional (2D) and three-dimensional (3D) lattice Boltzmann method (LBM) has been investigated using the non-orthogonal central moments collision method in order to study the Darcy and inertial (non-Darcy) fluid flow in porous media. A GPU parallel CUDA code has been developed and is employed to study the proposed method. A comparative study has been performed between the non-orthogonal central moments method and the Bhatnagar–Groos–Krook (BGK) method to measure the permeability. The effect of selecting sets of relaxation times and no-slip boundary conditions has been investigated using the 2D hexagonal and 3D body-centered cubic (BCC) structure, and an appropriate model has been presented for flow simulation in porous media. The comparison of the permeability obtained by the proposed numerical method with the analytical solutions shows that the proposed model is highly accurate and eliminates the dependence of permeability on the viscosity, which is one of the problems in the LBM. This model’s capability has been evaluated to simulate Darcy and inertial flows using 2D hexagonal geometry, 2D granular porous media with a random structure, 3D BCC geometry, 3D porous media with a random structure, and real porous media created by using computed tomography (CT) images. In addition, the apparent permeability, the non-Darcy β factor, and the onset of non-Darcy flow have been predicted to demonstrate the capability of the proposed method. The results show the high capability of the proposed model for fluid flow simulation in porous media with complex geometries; so, this model can be used as a powerful and stable tool with the capability of fluid flow simulation in porous media from the Darcy regime to large Reynolds numbers and the inertial regime.

Intercomparison of boundary schemes in Lattice Boltzmann method for flow simulation in porous media

AbstractThe Lattice Boltzmann method has been widely adopted to simulate flow in porous media. The choice of appropriate boundary schemes is essential to achieve simulation accuracy; however, the criteria for the most suitable boundary treatment in the simulation of flow in porous media flow remain unresolved. Here, three types of the most commonly used boundary conditions are tested: interpolation bounce back (IBB), partial saturated method (PSM), and immersed boundary method (IBM). The dimensionless drag of face‐centered cubic (FCC) sphere array and the dimensionless permeability of a random closely packed (RCP) sphere array are calculated and compared at different viscosities and resolutions. In the FCC sphere array case where spheres are not contacted, the IBB and PSM exhibit the same accuracy and both are of the second‐order convergence rate. The IBM is less accurate and is of the first‐order convergence rate. In the RCP sphere array case where the spheres are contacted, the IBB shows finer results and a second‐order convergence rate. PSM underestimates the dimensionless permeability and increases resolution only slightly improved the results of PSM. The IBM overestimates the dimensionless permeability. These results indicate that among the three methods, the IBB is the most accurate. The PSM has the same accuracy as the IBB when sediments are not contacted; however, it loses its accuracy in the simulation of flow in closely packed porous media. This work could serve as a benchmark for further research in choosing the most appropriate method in the simulation of flow in porous media.

A data-driven surrogate to image-based flow simulations in porous media

The objective for this work is to develop a data-driven surrogate to high-fidelity numerical flow simulations using digital images of porous media. The proposed model can capture the pixel-scale velocity vectors in a large verity of digital porous media created by random two-dimensional (2D) circle packs. To develop the model, images of the 2D media (binary images of solid grains and void spaces) along with their corresponding velocity vectors at the pixel level computed using lattice Boltzmann simulation runs are used to train and to predict the solutions with a high accuracy in much less computational time. The velocity vector predictions made by the surrogate models are used to compute the permeability tensor for samples that have not been used in the training. The results show high accuracy in the prediction of both velocity vectors and permeability tensors. The proposed methodology harness the enormous amount of generated data from high-fidelity flow simulations to decode the often under-utilized patterns in simulations and to accurately predict solutions to new cases. The developed model can truly capture the physics of the problem and enhance the prediction capabilities of the simulations at a much lower cost. These predictive models, in essence, do not spatially reduce the order of the problem. They, however, possess the same numerical resolutions as their Lattice Boltzmann simulations equivalents do with the great advantage that their solutions can be achieved by a significant reduction in computational costs (speed and memory).

Deep global model reduction learning in porous media flow simulation

In this paper, we combine deep learning concepts and some proper orthogonal decomposition (POD) model reduction methods for predicting flow in heterogeneous porous media. Nonlinear flow dynamics is studied, where the dynamics is regarded as a multi-layer network. The solution at the current time step is regarded as a multi-layer network of the solution at the initial time and input parameters. As for input, we consider various sources, which include source terms (well rates), permeability fields, and initial conditions. We consider the flow dynamics, where the solution is known at some locations and the data is integrated to the flow dynamics by modifying the reduced-order model. This approach allows modifying the reduced-order formulation of the problem. Because of the small problem size, limited observed data can be handled. We consider enriching the observed data using the computational data in deep learning networks. The basis functions of the global reduced-order model are selected such that the degrees of freedom represent the solution at observation points. This way, we can avoid learning basis functions, which can also be done using neural networks. We present numerical results, where we consider channelized permeability fields, where the network is constructed for various channel configurations. Our numerical results show that one can achieve a good approximation using forward feed maps based on multi-layer networks.