Comparative Study of Different Techniques for Numerical Reservoir Simulation

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.

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.

Multiphase flow of immiscible fluids is studied by means of a stochastic flow path approach. This approach is based on a differential geometric formulation that replaces the partial differential equations (PDEs) of flow by a set of ordinary differential equations (ODEs) that determine the flow paths and impose conservation of flux. It is shown that flux conservation along the flow paths involves a space transform. Other formulations of the multiphase flow equations involve Jacobian mappings. Flow realizations as well as statistical flow moments can be derived by means of the stochastic flow path method. Advantages of the stochastic flow path method include: reduction of a PDE to an ODE system, independence from perturbation approximations and Green's functions, and the freedom to use random initial conditions at the boundary. Using the stochastic flow path method, closed‐form expressions are obtained for two‐phase flow in uniaxially heterogeneous media. Two‐phase flow in a heterogeneous two‐dimensional medium is also investigated using numerical simulations.

Radial flows in heterogeneous porous media with a linear injection scheme

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

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

We investigated the onset of preferential flow in heterogeneous porous media using the random walk particle tracking (RWPT) concept. The RWPT model is first used to analyze empirically the required number of particles to achieve accurate concentration distributions in two-dimensional homogeneous media and under uniform flow conditions. The analysis is then extended to randomly heterogeneous systems. By increasing the variance of log-normal hydraulic conductivity fields, the transition between homogeneous and preferential flows is observed. To analyze the degree of preferential flow in the porous media, we provide a diagram that consists of two dimensionless parameters: normalized travel time and distance. All the heterogeneous media synthetically generated show a linear relation in the diagram. The characteristic travel velocity increases with increasing heterogeneity. We found the diagram is a useful tool to analyze preferential flow.

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.

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

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.

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.

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

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.

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.