Numerical solution of reservoir flow models based on large time step operator splitting algorithms

A phase-field lattice Boltzmann model for simulating multiphase flows in porous media: Application and comparison to experiments of CO2 sequestration at pore scale

Traditionally reservoir simulation has been accomplished by solution of reservoir flow models using variations of the methods of finite differences and less frequently by finite elements. The accuracy of these methods depends heavily on the size of grid blocks and time steps employed for the numerical solution. Consequently, computational time and memory requirements may be overwhelming in many cases. In the present study, a recently proposed method called the finite analytic method is shown to alleviate the difficulties associated with the conventional methods. The finite analytic method is operationally and principally different than the conventional finite difference and finite element methods in that a general analytical solution of the simulation model is used over mesh elements to obtain a numerical scheme. Therefore, for comparable accuracy, finite analytic method can lead to higher order approximations than those achieved by the finite difference and finite element techniques. Finite analytic schemes are derived from the reservoir flow models and therefore they are naturally problem dependent. This is the reason for the efficiency of the finite analytic method. By this method problems such as numerical diffusion, numerical instability, slow convergence, as well as requirements of large number of time steps and grid blocks are significantly reduced. In this study, the methodology of formulating and calculating with this method is illustrated by means of a typical two-phase immiscible flow problem described by the Buckley-Leverett equation. The results are compared with the analytic solution of Buckley-Leverett problem and solutions obtained by finite difference and finite element methods. The present study demonstrates that the finite analytic method is capable of producing accurate flow behavior in porous media even when relatively large time steps and grid blocks are used.

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

This paper presents a numerical model that intends to simulate efficiently the surface instability that arise in multiphase flows, typically liquid-gas, both for laminar or turbulent regimes. The model is developed on the in-house computing platform TermoFluids, and operates the finite-volume, direct numerical simulation (DNS) of multiphase flows by means of a conservative level-set method for the interface-capturing. The mesh size is optimized by means of an adaptive mesh refinement (AMR) strategy, that allows the dynamic re-concentration of the mesh in the vicinity of the interfaces between fluids, in order to correctly represent the diverse structures (as ligaments and droplets) that may rise from unstable phenomena. In addition, special attention is given to the discretization of the various terms of the momentum equations, to ensure stability of the flow and correct representation of turbulent vortices. As shown, the method is capable of truthfully simulate the interface phenomena as the Kelvin-Helmholtz instability and the Plateau-Rayleigh instability, both in the case of 2-D and 3-D configurations. Therefore it is suitable for the simulation of complex phenomena such as simulation of air-blast atomization, with several important application in the field of automotive and aerospace engines. A prove is given by our preliminary study of the 3-D coaxial liquid-gas jet.

The first paper of this two‐part series [Abdin and Kaluarachchi, this issue] developed a spectral/perturbation analysis to predict the variability of individual and capillary pressures of three‐phase (water, oil, and air) flow in heterogeneous porous media. The work in this manuscript is an extension of that work to validate the applicability of the perturbation analysis and to study the stochastic behavior of three‐phase flow. The input stochastic variables were soil grain size distribution index, α, and log k where k is the permeability. In this work a one‐dimensional numerical solution was developed to solve the three coupled nonlinear flow equations. The results using three different combinations of input stochastic variables showed good agreement between perturbation and numerical analyses over a wide range of flow conditions. In addition, the results showed a large influence of subsurface heterogeneity on the predicted capillary pressure variability. In general, this variability was highest with the uncorrelated α and log k case followed by deterministic α (stochastic log k) and correlated cases. The results suggest that subsurface heterogeneity is important in predicting three‐phase flow in heterogeneous porous media. Finally, the results of this work can be used as a preliminary step for stochastic analysis of three‐fluid phase flow.

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.

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.

Radial flows in heterogeneous porous media with a linear injection scheme

Upscaling of 1D two‐phase flows in heterogeneous porous media is important in interpretation of laboratory coreflood data, streamline quasi 3D modeling, and numerical reservoir simulation. In 1D heterogeneous media with properties varying along the flow direction, phase permeabilities are coordinate‐dependent. This yields the Buckley‐Leverett equation with coordinate‐dependent fractional flow f = f(s, x), which reflects the heterogeneity. So, an x‐dependency is considered to reflect microscale heterogeneity and averaging over x—upscaling. This work aims to average or upscale the heterogeneous system to obtain the homogenized media with such fractional flow function F(S) that provides the same water‐cut history at the reservoir outlet, x = 1. Thus, F(S) is an equivalent property of the medium. So far, the exact upscaling for 1D micro heterogeneous systems has not been derived. With the x‐dependency of fractional flow, the Riemann invariant is flux f, which yields exact integration of 1D flow problems. The novel exact solutions are derived for flows with continuous saturation profile, transition of shock into continuous wave, transition of continuous wave into shock, and transport in heterogeneous piecewise‐uniform rocks. The exact procedure of upscaling from f = f(s, x) to F(S) is as follows: the inverse function to the upscaled F(S) is equal to the averaged saturation over x of the inverse microscale function s = f −1(f, x). It was found that the Welge's method as applied to heterogeneous cores provides the upscaled F(S). For characteristic finite‐difference scheme, the fluxes for microscale and upscaled‐numerical‐cell systems, coincide in all grid nodes.

Many physical processes such as multiphase flow in heterogeneous porous media are described by partial differential algebraic equations (PDAEs) as a set of partial differential, ordinary differential and algebraic equations. However, due to lack of exact analytical solution for most PDAEs, numerical methods should be used to approximate their solution. The main contribution of this article is a comparison among the features of such numerical methods for oil and gas reservoir simulation. Different gridding techniques have been included in this comparison to find the most suitable numerical scheme to simulate multiphase flow in heterogeneous porous media.

Application of the multiscale finite element method to groundwater flow in heterogeneous porous media

Discontinuous Galerkin approximation of two-phase flows in heterogeneous porous media with discontinuous capillary pressures

Finite analytic numerical method for two-dimensional fluid flow in heterogeneous porous media

A stable Alternating Direction Implicit finite difference algorithm is used to model transient fluid flow in heterogeneous porous media. In connection with the laboratory system for pressure transient testing of core permeability, the effects of permeability heterogeneities on the characteristics of pressure transient are investigated. The results show that the early portion of the pressure transient is sensitive to the heterogeneity, while the late time portion is primarily controlled by the effective permeability of the sample. As in the steady flow case, lineation in permeability distribution produces anisotropy in measured permeability. Particularly, in the caSe of lineated fractures, where the background permeability is small, such anisotropy can produce orders of magnitude difference in permeability.

In this work,we present three-dimensional numerical simulations of water-oil flow in porous media in order to analyze the influence of the heterogeneities in the porosity and permeability fields and, mainly, their relationships upon the phenomenon known in the literature as viscous fingering. For this, typical scenarios of heterogeneous reservoirs submitted to water injection (secondary recovery method) are considered. The results show that the porosity heterogeneities have a markable influence in the flow behavior when the permeability is closely related with porosity, for example, by the Kozeny-Carman (KC) relation.This kind of positive relation leads to a larger oil recovery, as the areas of high permeability(higher flow velocities) are associated with areas of high porosity (higher volume of pores), causing a delay in the breakthrough time. On the other hand, when both fields (porosity and permeability) are heterogeneous but independent of each other the influence of the porosity heterogeneities is smaller and may be negligible.