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Abstract Using extensive numerical simulation of the Navier–Stokes equations, we study the transition from the Darcy’s law for slow flow of fluids through a disordered porous medium to the nonlinear flow regime in which the effect of inertia cannot be neglected. The porous medium is represented by two-dimensional slices of a three-dimensional image of a sandstone. We study the problem over wide ranges of porosity and the Reynolds number, as well as two types of boundary conditions, and compute essential features of fluid flow, namely, the strength of the vorticity, the effective permeability of the pore space, the frictional drag, and the relationship between the macroscopic pressure gradient$${\varvec{\nabla }}P$$ $\nabla P$and the fluid velocityv. The results indicate that when the Reynolds number Re is low enough that the Darcy’s law holds, the magnitude$$\omega _z$$ ${\omega }_z$of the vorticity is nearly zero. As Re increases, however, so also does$$\omega _z$$ ${\omega }_z$, and its rise from nearly zero begins at the same Re at which the Darcy’s law breaks down. We also show that a nonlinear relation between the macroscopic pressure gradient and the fluid velocityv, given by,$$-{\varvec{\nabla }}P=(\mu /K_e)\textbf{v}+\beta _n\rho |\textbf{v}|^2\textbf{v}$$ $-\nabla P=(\mu /K_e)v+{\beta }_n\rho {\left|v\right|}^{2}v$, provides accurate representation of the numerical data, where$$\mu$$ $\mu$and$$\rho$$ $\rho$are the fluid’s viscosity and density,$$K_e$$ $K_e$is the effective Darcy permeability in the linear regime, and$$\beta _n$$ ${\beta }_n$is a generalized nonlinear resistance. Theoretical justification for the relation is presented, and its predictions are also compared with those of the Forchheimer’s equation. 

Predicting geomechanical properties of rock and other types of porous media is essential to accurate modeling of many important processes, such as wave propagations, seismic events, and underground gas storage, and CO₂sequestration, all of which involve deformation of the pore space. We propose a model to predict the porosity dependence of the Young's and bulk moduli in heterogeneous porous media by combining the universal power law, predicted by percolation theory that describes the behavior of elastic moduli near the percolation threshold of the solid skeletons, and the effective‐medium approximation (EMA) for elastic materials that is accurate away from the threshold. The parameters of the model have unambiguous physical meanings, and can, in principle, be measured. We estimate the parameters ‐ the percolation threshold , crossover point between the EMA and percolation power law, the average particle coordination number , and the elastic moduli of the solid skeleton by using experimental data or numerical simulations for a wide variety of porous media in both two and three dimensions. Whenever data are available, the predictions are consistent with them. We then predict the elastic moduli for another 10 porous media using the proposed model and the estimated parameters without adjusting any new parameter. The predictions are in most cases in agreement with the data, hence indicating the accuracy of the approach. 

Percolation theory and the associated conductance networks have provided deep insights into the flow and transport properties of a vast number of heterogeneous materials and media. In practically all cases, however, the conductance of the networks’ bonds remains constant throughout the entire process. There are, however, many important problems in which the conductance of the bonds evolves over time and does not remain constant. Examples include clogging, dissolution and precipitation, and catalytic processes in porous materials, as well as the deformation of a porous medium by applying an external pressure or stress to it that reduces the size of its pores. We introduce two percolation models to study the evolution of the conductivity of such networks. The two models are related to natural and industrial processes involving clogging, precipitation, and dissolution processes in porous media and materials. The effective conductivity of the models is shown to follow known power laws near the percolation threshold, despite radically different behavior both away from and even close to the percolation threshold. The behavior of the networks close to the percolation threshold is described by critical exponents, yielding bounds for traditional percolation exponents. We show that one of the two models belongs to the traditional universality class of percolation conductivity, while the second model yields nonuniversal scaling exponents. 

Abstract Fluid flow in heterogeneous porous media arises in many systems, from biological tissues to composite materials, soil, wood, and paper. With advances in instrumentations, high-resolution images of porous media can be obtained and used directly in the simulation of fluid flow. The computations are, however, highly intensive. Although machine learning (ML) algorithms have been used for predicting flow properties of porous media, they lack a rigorous, physics-based foundation and rely on correlations. We introduce an ML approach that incorporates mass conservation and the Navier–Stokes equations in its learning process. By training the algorithm to relatively limited data obtained from the solutions of the equations over a time interval, we show that the approach provides highly accurate predictions for the flow properties of porous media at all other times and spatial locations, while reducing the computation time. We also show that when the network is used for a different porous medium, it again provides very accurate predictions. 

Multiphase fluid flow in porous media is relevant to many fundamental scientific problems as well as numerous practical applications. With advances in instrumentations, it has become possible to obtain high-resolution three-dimensional (3D) images of complex porous media and use them directly in the simulation of multiphase flows. A prime method for carrying out such simulations is the color-fluid lattice Boltzmann method with multi-relaxation time (CFLB-MRT) collision operator. The simulations are, however, time consuming and intensive. We propose a method to accelerate image-based computations with the CFLB-MRT method, in which the 3D image is preprocessed by curvelet transforming it and eliminating those details that do not contribute significantly to multiphase flow. The coarsening is done by thresholding the image. After inverting the coarser image back to the real space, it is utilized in the simulation of multiphase flow by the CFLB-MRT approach. As the test of the method, we carry out simulation of a two-phase flow problem in which the porous media are initially saturated by brine or water, which is then displaced by CO2 or oil, injected into the pore space. The simulations are carried out with two types of sandstone. We show that the method accelerates the computations significantly by a factor of up to 35.