Adaptive Fully Implicit Multi-scale Meshless Multi-point Flux Method for Fluid Flow in Heterogeneous Porous Media

Flow prediction of heterogeneous nanoporous media based on physical information neural network

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

The multiphase fluid flow in porous media is one of the most fundamental phenomena in various physical processes, such as oil/gas flow in reservoir, subsurface contamination dispersion, chemical separation, etc. Due to its importance, the efficient and accurate solution and prediction of multiphase flow in porous media is highly required in engineering applications and mechanism studies, which has been a research hot spot with increasing interest in recent years. However, the strong nonlinearity implicated in the multiphase flow model has brought great challenges for the computation and analysis. In addition, the permeability in Darcy-type pressure equation is always represented as a high contrast coefficient, which further complicates the simulation difficulty especially for high fidelity prediction and decision-making process. In this study, we attempt to apply one popular global model reduction method, namely Proper Orthogonal Decomposition (POD), to accelerate the simulation of incompressible multiphase flow in porous media. The cornerstone of POD is processing information from a sequence of snapshots (or instantaneous solutions) to identify a set of basis functions to construct a low-dimensional space, the most energetic part of these basis functions is then selected to represent the solution of the original system. In our work, one key point is how to establish the accurate POD reduced order model (ROM) for incompressible mul-tiphase porous flow problems. According to Li et al. [1], the traditional Galerkin projection method (directly project the original governing equation into the low-dimensional space spanned by basis functions) is incapable of introducing the fluid saturation and permeability located on the cell face in POD-ROM for incompressible multiphase flow whether in homogeneous or heterogeneous porous media, thus we adopt the matrix operation method (project the discrete governing equation instead of the original governing equation into the low-dimensional space spanned by basis functions) to build the POD-ROM in this study. Based on the proposed POD-ROM framework, we use the implicit method to solve the (non-) wetting-phase pressure and saturation equations. However, in the implicit solution of the pressure and saturation equations, the standard POD-ROM alone sometimes cannot yield an expected acceleration owing to two main reasons: (1) the nonlinearity between pressure and saturation is sensitive, thus the saturation would change largely even with slight pressure change. And the small error of pressure results would lead to a large deviation of saturation, which in turn exerts negative impacts on the computation of pressure; (2) the complexity for computing a projected nonlinear term in POD-ROM still depends on the dimension of the original full-order system, it is unavoidable in conventional POD-ROM. Therefore, more POD modes are needed to be used in order to achieve the prescribed numerical accuracy, which could worsen the speed up efficiency of proposed POD-ROM. To further reduce the computational complexity of POD-ROM to speed up the implicit solution of pressure and saturation equations, another model reduction method called Discrete Empirical Interpolation Method (DEIM) [2] is applied to treat the projected nonlinear term in the proposed incompressible multiphase porous flow POD-ROM. Different from previous studies, we develop an adaptive DEIM (ADEIM) in this work to achieve a more flexible selection of the interpolation points. Combined the POD-ROM with ADEIM, a POD-ADEIM-ROM for incompressible multiphase flow in porous media is established. The computational efficiency and numerical accuracy of the proposed model is validated through several representative numerical examples. Numerical results indicate that an outstanding acceleration (2~3 orders of magnitude faster) without sacrificing numerical accuracy significantly is obtained from the proposed model when comparing to the traditional solution method that without any acceleration technique. In addition, the effects of POD mode number, the distribution of permeability field, and the mesh density on the overall performances of the proposed model are investigated in detail.

SummaryNumerical modeling of large deformations in fluid‐infiltrated porous media must accurately describe not only geometrically nonlinear kinematics but also fluid flow in heterogeneously deforming pore structure. Accurate simulation of fluid flow in heterogeneous porous media often requires a numerical method that features the local (elementwise) conservation property. Here, we introduce a new finite element framework for locally mass conservative solution of coupled poromechanical problems at large strains. At the core of our approach is the enriched Galerkin discretization of the fluid mass balance equation, whereby elementwise constant functions are augmented to the standard continuous Galerkin discretization. The resulting numerical method provides local mass conservation by construction with a usually affordable cost added to the continuous Galerkin counterpart. Two equivalent formulations are developed using total Lagrangian and updated Lagrangian approaches. The local mass conservation property of the proposed method is verified through numerical examples involving saturated and unsaturated flow in porous media at finite strains. The numerical examples also demonstrate that local mass conservation can be a critical element of accurate simulation of both fluid flow and large deformation in porous media.

Glenn Brown takes issue with my statement, “It is hoped that Maxwell's contribution to the foundations of fluids in porous media will receive due attention, and that his novel approach will lead to new insights.” He considers that, because Maxwell did not explicitly develop his theory for fluid flow in porous media, his ideas should not be treated as a contribution in that area. Brown contends that doing so is a disservice to Darcy, and is revisionist.Brown and I differ in the way we perceive science. He looks at the material I have presented from an ideological perspective of upholding Darcy's position in history. On the other hand, I do not question Darcy's valid contribution. Rather, I presented some of Maxwell's fascinating ideas that are relevant to the study of fluid flow in porous media, published in the same year Darcy published his seminal work. I have shown that the relevance of Maxwell's ideas to flow in porous media has gone unnoticed in the literature. Scientists are fallible human beings, and important ideas and thoughts are occasionally overlooked.When, on a rare occasion, we chance upon such an oversight, it is part of our scientific enterprise to bring the finding to the attention of the scientific community. It is up to the community to judge the historical significance of the new information.

Grid coarsening, simulation of transport processes in, and scale-up of heterogeneous media: Application of multiresolution wavelet transformations

PurposeThe material point method (MPM)is a particle-based numerical method suitable for solid–liquid simulation and large deformation problems. However, MPM is generally used in solid deformation at present, to develop a multi-physics coupling MPM; the purpose of this study is to extend the MPM to simulate the heat and fluid flow and address the thermal-hydrological (TH) coupling problems.Design/methodology/approachThe porous medium was discretized into two sets of Lagrangian points, and the motion of fluid points follows the Darcy’s law. Two sets of heat transport equations were established for the heat conduction and heat exchange in the pore fluid and solid skeleton. Fractures were considered by adding the porosity gradient term in the governing equations; also a transition function was introduced to smoothen the fracture boundary.FindingsFour cases of heat and fluid flow in porous medium and fractures were presented to verify the feasibility of the proposed method. And the effects of fractures on heat and fluid flow were investigated. Additionally, a case of geothermal extraction was solved and the importance of the interstitial convective heat transfer coefficient was analyzed.Originality/valueThe proposed method extends the conventional MPM, using two sets of material points and two sets of heat transport equations to simulate the heat and fluid flow and address the TH coupling problems, which can be applied in both porous medium and fractures.

10. NMR Imaging of Fluids and Flow in Porous Media

Non-Newtonian fluid flow in porous media results in spatially varying viscosity, driven by flow–pore–geometry interactions, potentially leading to non-monotonic dispersion. In this work, using high-resolution micro-particle image velocimetry, we present a direct experimental observation of shear-viscosity-distribution-dependent transport with non-Newtonian fluid flows in porous media. We experimentally investigate dispersion in porous media in a microfluidic chip featuring a physical rock geometry, comparing a shear-thinning, non-Newtonian fluid with its Newtonian analogue at various Péclet numbers. We demonstrate that, in the absence of advective fluxes driven by elastic instabilities, non-Newtonian fluid flows at either extreme of the shear-dependent viscosity ( $\eta _0,\eta _{\infty }$ ) converge to the Newtonian analogue. In contrast, for flows between these extremes, the non-Newtonian velocity fields are broadly distributed along the streamline curvature, leading to a larger enhancement in dispersion.

Currently, fluid flow in porous media is mostly calculated by utilizing the well known diffusivity equation based on Darcy’s law. This diffusivity equation is the core fluid flow equation in all modern reservoir simulators used to predict flow behaviors. Inaccurate predictions of reservoir simulators have been reported nevertheless, history matching has been achieved. This dilemma led to questioning the adequacy of the basic governing equation of fluid flow behavior in porous media. This paper suggesting a new governing equation that includes Darcy’s viscous term, Forchheimer’s inertial term and Brinkman’s viscous term all in one model called the Modified Brinkman Model "MBM". MBM proven to accurately describe fluid flow in porous media in both Darcian and non-Darcian domains and can be used in both oil and gas reservoirs for both matrix and fracture systems. A genuine mathematical solution "Adomian decomposition technique" has been successfully employed to solve the partial differential model with great deal of accuracy and ease. The proposed MBM is expected to have wide applications in oil, gas and underground water reservoirs.

The analysis of mass and heat transfer in magnetohydrodynamic (MHD) flows has significant applications in heat exchangers, cooling nuclear reactors, designing energy systems and casting and injection processes of different types of fluids. On the other hand, extraction of crude oil, the flow of human or animal blood, as well as other polymer fluids or liquid crystals are just some examples of micropolar fluid flows. Due to the broad application spectrum of the theory of micropolar fluid flows, and the significance the impact the external magnetic field has on the flow of these fluids, this paper considers the stationary flow of a micropolar fluid between two plates under the influence of an external magnetic field which is perpendicular to the direction of the flow. Stationary plates are maintained at constant and different temperatures, while the whole problem is considered in the non-inductive approximation. The equation system used to define the physical problem under consideration is reduced to the system of differential equations that have been solved analytically and the solutions of which are of general nature. In addition to the solutions for velocity, microrotation and temperature, the paper gives solutions for shear stress at plates, the Nusselt number and flow rate. The provided solutions have been applied in order to reach some general conclusions about the influence of the magnetic field and physical characteristics of a micropolar fluid and the characteristics of porous media on the nature of micropolar fluid flows in porous media by means of chart analysis. General conclusions, obtained in the result analysis in this paper, give us the opportunity to understand the flows of micropolar fluids and highlight their significance.

This article studies the combined effect of spatial heterogeneity and capillary pressure on the saturation of two fluids during the injection of immiscible nanoparticles. Various literature review exhibited that the nanoparticles are helpful in enhancing the oil recovery by varying several mechanisms, like wettability alteration, interfacial tension, disjoining pressure and mobility control. Multiphase modelling of fluids in porous media comprise balance equation formulation, and constitutive relations for both interphase mass transfer and pressure saturation curves. A classical equation of advection-dispersion is normally used to simulate the fluid flow in porous media, but this equation is unable to simulate nanoparticles flow due to the adsorption effect which happens. Several modifications on computational fluid dynamics (CFD) have been made to increase the number of unknown variables. The simulation results indicated the successful transportation of nanoparticles in two phase fluid flow in porous medium which helps in decreasing the wettability of rocks and hence increasing the oil recovery. The saturation, permeability and capillary pressure curves show that the wettability of the rocks increases with the increasing saturation of wetting phase (brine).

Mobility changes from region to region can occur in a reservoir due to inhomogeneities in permeability and due to change in saturating fluid. In either case interfaces are present across which mobility changes abruptly. The problem of fluid flow in porous media in the presence of such interfaces is solved in this paper by using complex velocity potentials. On the interface tangential component of efflux vector is discontinuous. A vortex sheet appears on the interface and it affects the flow. Complex velocity potential in such a case can be expressed as a linear superposition of the potentials due to sources (injection wells), sinks (production wells) and due to vortex sheets along the interfaces of mobility discontinuity. Vorticity distribution on interfaces and the velocity field are inter-related. Therefore we get an integral equation for the vorticity distribution. The solution of the integral equation enables us to define the complex velocity potential completely and hence the velocity field in the flow region. It is shown that in the general case of several wells and interfaces the integral equation has a unique solution and leads to a unique flow pattern. An exact solution of the integral equation for a circular interface is obtained. An impervious no-flow boundary is an extreme case of mobility discontinuity. It is shown that its effect on the flow can be considered by an appropriate vortex sheet along it. Thus a formal solution in the form of an integral equation is obtained for instantaneous velocity distribution for arbitrary well pattern and mobility distribution.

Derivation of macroscopic filtration law for transient linear viscoelastic fluid flow in porous media

Implicit Hybrid Upwinding for two-phase flow in heterogeneous porous media with buoyancy and capillarity