Modern system of multiphase flow in porous media and its development trend

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

Understanding fluid flow in porous media is essential with complex and multiphase fluid flow. We demonstrate that high-resolution in-line density measurements are a valuable tool in this regard. An in-line densitometer is used in fluid flow in porous media applications to quantify fluid production and obtain quantitative and qualitative information such as breakthrough times, emulsion/foam generation, and steam condensation. In order to determine the potential applications for in-line densitometry for fluid flow in porous media, a series of sand pack floods were performed with a densitometer placed at the outlet of a sand pack. All fluids passed through the measurement cell at experiential temperatures and pressures. An algorithm was developed and applied to the density data to provide a quantitative determination of oil and water production. The second series of tests were performed at high temperature and pressure, with a densitometer placed at the inlet and outlet of a sand pack, for steam applications. In both series of experiments, data acquisition was collected at 1 hertz and the analyzed density data was compared to results from the conventional effluent analysis, including Dean-Stark, toluene separations, magnetic susceptibility measurement, and flash calculations where applicable. The high-resolution monitoring of effluent from a flow experiment through porous media in a system with two phases of known densities enables two-phase production to be accurately quantified in the case of both light and heavy oil. The frequency of measurements results in a high-resolution history of breakthrough times and fluid behavior. In the case of monitoring steam injection processes, reliable laboratory tests show that in-line density measurements enable the determination of steam quality at the inlet and outlet of a sand pack and qualitative determination of steam condensation monitoring The use of in-line densitometry provides insight on the monitoring of complex fluid flow in porous media, which typical bulk effluent analysis is not able to do. The ability to measure produced fluids at high resolution and extreme temperatures reduces mass balance error associated with the effluent collection and broadens our understanding of complex fluid flow in porous media.

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.

Chapter 10 - Multiphase Fluid and Heat Flow in Porous Media

A sequential fully implicit multi-scale meshless multi-point flux method (MS-MMPFA) for nonlinear hyperbolic partial differential equations of fluid flow in heterogeneous porous media is described in this paper. The method extends the recently proposed the meshless multi-point flux approximation (MMPFA) for general fluid flow in porous media [Lukyanov, “Meshless Upscaling Method and its Application to a Fluid Flow in Porous Media”, Proceeding ECMOR XII, 2010] by utilizing advantages of the existing multi-scale finite volume (MSFV) schemes. The MMPFA is based on a gradient approximation commonly used in meshless method and combined with the mixed corrections which ensure linear completeness. In corrected meshless method, the domain boundaries and field variables at the boundaries are approximated with the default accuracy of the method. The MMPFA method was successfully tested for a number of problems where it was clearly shown that the MMPFA gives a good agreement with analytical solutions for a given number of particles. However, the level of detail and range of property variability included in reservoir characterization models leads to a large number of particles to be considered in MMPFA method. In this paper this problem is resolved using a sequential fully implicit MS-MMPFA method. The results are presented, discussed.

A mixed pressure-velocity formulation to model flow in heterogeneous porous media with physics-informed neural networks

Multiphase fluid flow in porous media is important to a wide variety of processes of fundamental scientific and practical importance. Developing a model for the pore space of porous media represents the first step for simulating such flows. With rapid increase in the computation power and advances in instrumentation and imaging processes, it has become feasible to carry out simulation of multiphase flow in two- and three-dimensional images of porous media, hence dispensing with development of models of pore space that are based on approximating their morphology. Image-based simulations are, however, very time consuming. We describe an approach for speeding-up image-based simulation of multiphase flow in porous media based on curvelet transformations, which are specifically designed for processing of images that contain complex curved surfaces. Most porous media contain correlations in their morphology and, therefore, their images carry redundant information that, in the curvelet transform space, can be removed efficiently and accurately in order to obtain a coarser image with which the computations are far less intensive. We utilize the methodology to simulate two-phase flow of oil and water in two-dimensional digital images of sandstone and carbonate samples, and demonstrate that while the results with the curvelet-processed images are as accurate as those with the original ones, the computations are speeded up by a factor of 110–150. Thus, the methodology opens the way toward achieving the ultimate goal of simulation of multiphase flow in porous media, namely, making image-based computations a standard practice.

The development of a framework to support smoothed particle hydrodynamics (SPH) simulations of fluid flow and transport in porous media is described. The framework is built using the Common Component Architecture (CCA) toolkit and it supports SPH simulations using a variety of different SPH models and setup formats. The SPH simulation code is decomposed into independent components that represent self-contained units of functionality. Different physics models can be developed within the framework by re-implementing key components but no modification of other components is required. A model for defining components and developing abstract interfaces that support a high degree of modularity and minimal dependencies between components is discussed in detail.

1 - Modelling fluid flow in saturated porous media and at interfaces

In the last decade, the multi-point flux approximations have found significant interest. However, non-physical oscillations can appear in the developed multi-point flux approximations when the anisotropy is really strong. It has been found that the oscillations are closely related to the poor approximation of pressure gradient in the flux computation. In this paper, the meshless multi-point flux approximation (MMPFA) for general fluid flow in porous media is proposed. The MMPFA is based on a gradient approximation commonly used in thermal, viscous, and pressure projection problems and can be extended to include higher-order terms in the appropriate Taylor series. The proposed MMPFA is combined with the mixed corrections which ensure linear completeness. The mixed correction utilizes Shepard Functions in combination with a correction to derivative approximations. Incompleteness of the kernel support combined with the lack of consistency of the kernel interpolation in conventional meshless method results in fuzzy boundaries. In corrected meshless method, the domain boundaries and field variables at the boundaries are approximated with the default accuracy of the method. The resulting normalized and corrected MMPFA scheme not only ensures first order consistency O(h) but also alleviates the particledeficiency (kernel support incompleteness) problem. Furthermore, a number of improvements to the kernel derivative approximation are proposed. The primary attraction of the present method is that it provides a weak formulation for Darcy's law which can be of use in further development of meshless methods. The SPH model can be used to model three-dimensional miscible flow and transport in porous media with complex geometry, and for large (field) scale simulation of transport in porous media with general permeability distributions. To illustrate the performance of the MMPFA, a modelling of a single phase fluid flow in fully anisotropic porous media is presented.

A physics-constrained deep learning model for simulating multiphase flow in 3D heterogeneous porous media

An improved gray lattice Boltzmann model for simulating fluid flow in multi-scale porous media

A finite-element scheme has been formulated, which is capable of solving transient analysis of both conductive and convective heat transfers due to fluid flow in porous media. The model also includes the latent heat effect to consider the phase change aspect of a frozen medium. To test the validity of the model, it was applied to six cases for which analytical solutions are available. The test cases cover (i) single-phase fluid flow through porous media, (ii) radial conduction with and without phase change, (iii) conductive and convective heat transfer in an aquifer, and (iv) two-phase immiscible flow in porous media. In all these cases, good agreement with analytical solutions are observed validating the computational scheme. This computational scheme should be useful in solving frozen ground problems, thermal stimulation technique for natural gas recovery from hydrates, and single-phase and two-phase convective heat transfer problems in enhanced oil recovery scheme in petroleum engineering.

This research reports on the development of a two dimensional lattice gas automata (LGA) model to simulate fluid flow in porous media. In lattice gas automata simple rules of particle interactions at a lattice are used to simulate complex flow phenomena. Since the numerical operations involved are largerly bit manipulations, lattice gas automata can be potentially more efficient in memory usage than conventional methods, such as finite difference or finite element methods. There are basically two motivations for the utilisation of lattice-gas automata methods for studying fluid flow in porous media. First, the no-slip boundary condition of hydrodynamics is easily implemented as a simple bounce-back reflection at solid walls. Second, the discrete nature of the lattice-gas method makes it computationally efficient in terms of the work necessary to update a single site of the lattice. Fluid flow in porous media is generally described by Darcy's law, which linearly relates fluid velocity to pressure gradient. Other research on the use of lattice gas automata to model fluid flow in porous media primarily focuses on Stokes flow at the pore level with the intent of understanding flow at the core scale. This work concentrates on modeling Darcy flow in a heterogeneous field. Permeability variations and anisotropy are modeled by a distribution of scatter. Collision of incoming particles with scatterers is a stochastic process.

Chapter 1 - Introduction