A new computational strategy for solving two-phase flow in strongly heterogeneous poroelastic media of evolving scales (original) (raw)
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International Journal for Numerical and Analytical Methods in Geomechanics, 2004
The compaction of highly heterogeneous poroelastic reservoirs with the geology characterized by long-range correlations displaying fractal character is investigated within the framework of the stochastic computational modelling. The influence of reservoir heterogeneity upon the magnitude of the stresses induced in the porous matrix during fluid withdrawal and rock consolidation is analysed by performing ensemble averages over realizations of a log-normally distributed stationary random hydraulic conductivity field. Considering the statistical distribution of this parameter characterized by a coefficient of variation governing the magnitude of heterogeneity and a correlation function which decays with a power-law scaling behaviour we show that the combination of these two effects result in an increase in the magnitude of effective stresses of the rock during reservoir depletion. Further, within the framework of a perturbation analysis we show that the randomness in the hydraulic conductivity gives rise to non-linear corrections in the upscaled poroelastic equations. These corrections are illustrated by a self-consistent recursive hierarchy of solutions of the stochastic poroelastic equations parametrized by a scale parameter representing the fluctuating log-conductivity standard deviation. A classical example of land subsidence caused by fluid extraction of a weak reservoir is numerically simulated by performing Monte Carlo simulations in conjunction with finite elements discretizations of the poroelastic equations associated with an ensemble of geologies. Numerical results illustrate the effects of the spatial variability and fractal character of the permeability distribution upon the evolution of the Mohr-Coulomb function of the rock.
International Journal for Numerical Methods in Engineering, 2012
A new multiscale procedure is proposed to compute flow in compressible heterogeneous porous media with geology characterized by power-law covariance structure. At the fine scale, the deformable medium is modeled by the partially coupled formulation of poroelasticity with Young's modulus and permeability treated as stationary random fields represented by their Karhunen-Loève decompositions. The framework underlying the multiscale procedure is based on mapping these random parameters to an auxiliary domain and constructing a family of equivalent stochastic processes at different length scales characterized by the same ensemble mean and covariance function. The poromechanical variables inherit a space-time version of the scaling relations of the random input parameters which allows for constructing a set of multiscale solutions of the same governing equations posed at different space and time scales. A notable feature of the multiscale method proposed herein is the feasibility of solving both the poroelastic model and the Fredholm integral equation for the eigenpairs of the Karhunen-Loève expansion in an auxiliary domain with much lower computational effort and then derive the long term behavior at a coarser scale from a straightforward rescaling of the auxiliary solution. Within the framework of the finite element approximation, in conjunction with the Monte Carlo algorithm, numerical simulations of fluid withdrawal and injection problems in a heterogeneous poroelastic reservoir are performed to illustrate the potential of the method in drastically reducing the computational burden in the computation of the statistical moments of the poromechanical unknowns in large-scale simulations. coupled hydromechanical phenomena in heterogeneous formations may be significantly influenced by the spatial variability of properties that commonly occur in natural multiscale environments (see e.g., ).
A new locally conservative numerical method for two-phase flow in heterogeneous poroelastic media
Computers and Geotechnics, 2013
We construct a new class of locally conservative numerical methods for two-phase immiscible flow in heterogeneous poroelastic media. Within the framework of the so-called iteratively coupled methods and fixed-stress split algorithm we develop mixed finite element methods for the flow and geomechanics subsystems which furnish locally conservative Darcy velocity and transient porosity input fields for the transport problem for the water saturation. Such hyperbolic equation is decomposed within an operator splitting technique based on a predictor-corrector scheme with the predictor step discretized by a higher-order non-oscillatory finite volume central scheme. The proposed scheme adopts an inhomogeneous dual mesh with variable cell size ruled by the local wave speed of propagation to compute numerical fluxes at cell edges. In the limit of small time steps the central scheme gives rise to a semidiscrete formulation for the water saturation capable of incorporating heterogeneous porosity fields and generalized flux functions including the water transport due to the solid phase velocity. Numerical simulations of a water-flooding problem in secondary oil recovery are presented for different realizations of the input random fields (permeability, Young modulus and initial porosity). Comparison between the accuracies of the proposed approach and the traditional one-way coupled hydro-geomechanical formulation are presented. The effects of the cross-correlation between the input random fields and compaction drive mechanism upon finger growth and breakthrough curves are also analyzed. A notable feature of the formulation proposed herein is the accurate prediction of the influence of geomechanical effects upon the unstable movement of the water front, whose evolution is dictated by rock heterogeneity and unfavorable viscosity ratio, without deteriorating the local conservative character of the numerical schemes. (M.A. Murad), mrborges@lncc.br (M. Borges), alexei@lncc.br (J.A. Obregón), maicon@ime.unicamp.br (M. Correa).
Stochastic computational modeling of reservoir compaction due to fluid withdrawal
2002
The compaction of highly heterogeneous poroelastic reservoirs with the geology characterized by long-range correlations displaying fractal character is investigated within the framework of the stochastic computational modelling. The influence of reservoir heterogeneity upon the magnitude of the stresses induced in the porous matrix during fluid withdrawal and rock consolidation is analysed by performing ensemble averages over realizations of a log-normally distributed stationary random hydraulic conductivity field. Considering the statistical distribution of this parameter characterized by a coefficient of variation governing the magnitude of heterogeneity and a correlation function which decays with a power-law scaling behaviour we show that the combination of these two effects result in an increase in the magnitude of effective stresses of the rock during reservoir depletion. Further, within the framework of a perturbation analysis we show that the randomness in the hydraulic conductivity gives rise to non-linear corrections in the upscaled poroelastic equations. These corrections are illustrated by a self-consistent recursive hierarchy of solutions of the stochastic poroelastic equations parametrized by a scale parameter representing the fluctuating log-conductivity standard deviation. A classical example of land subsidence caused by fluid extraction of a weak reservoir is numerically simulated by performing Monte Carlo simulations in conjunction with finite elements discretizations of the poroelastic equations associated with an ensemble of geologies. Numerical results illustrate the effects of the spatial variability and fractal character of the permeability distribution upon the evolution of the Mohr-Coulomb function of the rock.
Advances in Water Resources, 2009
In this work we develop a new multiscale procedure to compute numerically the statistical moments of the stochastic variables which govern single phase flow in heterogeneous porous media. The technique explores the properties of the log-normally distributed hydraulic conductivity, characterized by power-law or exponential covariances, which shows invariance in its statistical structure upon a simultaneous change of the scale of observation and strength of heterogeneity. We construct a family of equivalent stochastic hydrodynamic variables satisfying the same flow equations at different scales and strengths of heterogeneity or correlation lengths. Within the new procedure the governing equations are solved in a scaled geology and the numerical results are mapped onto the original medium at coarser scales by a straightforward rescaling. The new procedure is implemented numerically within the Monte Carlo algorithm and also in conjunction with the discretization of the low-order effective equations derived from perturbation analysis. Numerical results obtained by the finite element method show the accuracy of the new procedure to approximated the two first moments of the pressure and velocity along with its potential in reducing drastically the computational cost involved in the numerical modeling of both power-law and exponential covariance functions.
2009
In this study, a stochastic model is proposed to solve poroelastic problems in heterogeneous porous media. The model is constructed using the first-order second-moment method to investigate the dynamic behaviors of statistical mean and covariance of the change in pore water pressure and displacement. Although several variables can be simultaneously treated as random in the model, the Darcy conductivity is selected as the only random variable for this preliminary investigation due to its large variation compared to other mechanical and hydrogeological properties in natural environments. The constructed model is general in multiple dimensions; however, the one-dimensional case is taken as an example to demonstrate the use of the stochastic model. This model is validated using analytical and numerical solutions from the literature. Numerical experiments are then performed to investigate the boundary effects on the coupled fluid pressure and mechanical deformation in elastic porous media. The results show that the dynamic behavior of a coupled flow-stress system is more complex than a system that does not consider the deformation of porous media. Loading effects on deformation and pore pressure are instantaneous while the effect of discharge takes time to propagate from the boundary through the whole domain. Both loading and discharge boundary conditions can significantly affect the uncertainty of the system response. In the scenario combining loading at the top boundary and discharge at the bottom boundary, the mean total settlement and the average flux satisfy the relationship of superposition to be the sum of the separated effects of loading at top boundary and discharge at bottom boundary, but the variances do not. The proposed stochastic poroelastic method can be applied to hydrological issues that concern the interaction of flow and geomechanics.
Early stochastic studies focused on steady-state, gravity-dominated unsaturated flow in unbounded domains In this study, on the basis of the van Genuchten-Mualem constitu- (e.g., Yeh et al., 1985a,b; Russo, 1993; Yang et tive relationship, we develop a general nonstationary stochastic model for transient, variably saturated flow in randomly heterogeneous me- . dia with the method of moment equations. We first derive partial Under these conditions, the unsaturated flow field is differential equations governing the statistical moments of the flow stationary, and hence analytical or semianalytical soluquantities by perturbation expansions and then implement these equations are possible. Recently, some researchers investitions under general conditions with the method of finite differences. gated the effects of boundary conditions on steady-state The nonstationary stochastic model developed is applicable to the flow and the consequent effects of flow nonstationarity entire domain of bounded, multidimensional vadose zones or intein one-dimensional semibounded domains (Andersson grated unsaturated-saturated systems in the presence of random or and Shapiro, 1983; Indelman et al., 1993) or two-dimendeterministic recharge and sink-source and in the presence of sional bounded domains (Zhang and Winter, 1998). It multiscale, nonstationary medium features. We demonstrate the has been found that the simpler, gravity-dominated flow model with some two-dimensional examples of unsaturated and integrated unsaturated-saturated flows. The validity of the developed Zhiming Lu and Dongxiao Zhang, Hydrology, Geochemistry, and Geology Group (EES-6), MS T003, Los Alamos National Laboratory, mathematical complexity, the van Genuchten-Mualem Los Alamos, NM 87545. Received 31 Jan. 2002. *Corresponding author (donzhang@lanl.gov).
A fully coupled multiphase flow and geomechanics solver for highly heterogeneous porous media
Journal of Computational and Applied Mathematics, 2013
This paper introduces a fully coupled multiphase flow and geomechanics solver that can be applied to modeling highly heterogeneous porous media. Multiphase flow in deformable porous media is a multiphysics problem that considers the flow physics and rock physics simultaneously. To model this problem, the multiphase flow equations and geomechanical equilibrium equation must be tightly coupled. Conventional finite element modeling of coupled flow and geomechanics does not conserve mass locally since it uses continuous basis functions. Mixed finite element discretization that satisfies local mass conservation of the flow equation can be a good solution for this problem. In addition, the stabilized finite element method for discretizing the saturation equation minimizes numerical diffusion and provides better resolution of saturation solution.
In this paper, we solve a problem describing the mechanical de- formations of a porous medium in the presence of a monophasic linear flow or a two phase nonlinear flow with the purpose of modelizing subsidence of hydrocarbon reservoirs. An essential characteristics of this problem is that the mechanical deformation and the flow have different time scales. In petroleum industry, one uses different very efficient simulators for theflow problem and the mechanical deformations, which enables to handle complex models. Therefore it is necessary to be able to combine as efficiently as possiblethe exploitation of these simulators. We propose two alternative splitting approaches. The first one is the staggered algorithm used by engineers, which amounts to a Gauss- Seidel method in the one phase linear case. The second approach is based upon the preconditioned conjugate gradient method. We use a numerical multi-scale method in both of these algorithms. We compare these two approaches and we s...