Modeling and numerical simulations of fractured,vuggy,porus media is a challenging problem which occurs frequently in reservoir engineering.The problem is especially relevant in flow simulations of karst reservoirs wh...Modeling and numerical simulations of fractured,vuggy,porus media is a challenging problem which occurs frequently in reservoir engineering.The problem is especially relevant in flow simulations of karst reservoirs where vugs and caves are embedded in a porous rock and are connected via fracture networks at multiple scales.In this paper we propose a unified approach to this problem by using the StokesBrinkman equations at the fine scale.These equations are capable of representing porous media such as rock as well as free flow regions(fractures,vugs,caves)in a single system of equations.We then consider upscaling these equations to a coarser scale.The cell problems,needed to compute coarse-scale permeability of Representative Element of Volume(REV)are discussed.A mixed finite element method is then used to solve the Stokes-Brinkman equation at the fine scale for a number of flow problems,representative for different types of vuggy reservoirs.Upscaling is also performed by numerical solutions of Stokes-Brinkman cell problems in selected REVs.Both isolated vugs in porous matrix as well as vugs connected by fracture networks are analyzed by comparing fine-scale and coarse-scale flow fields.Several different types of fracture networks,representative of short-and long-range fractures are studied numerically.It is also shown that the Stokes-Brinkman equations can naturally be used to model additional physical effects pertaining to vugular media such as partial fracture with fill-in by some material and/or fluids with suspended solid particles.展开更多
In this paper we use the Generalized Multiscale Finite Element Method(GMsFEM)framework,introduced in[26],in order to solve nonlinear elliptic equations with high-contrast coefficients.The proposed solution method invo...In this paper we use the Generalized Multiscale Finite Element Method(GMsFEM)framework,introduced in[26],in order to solve nonlinear elliptic equations with high-contrast coefficients.The proposed solution method involves linearizing the equation so that coarse-grid quantities of previous solution iterates can be regarded as auxiliary parameters within the problem formulation.With this convention,we systematically construct respective coarse solution spaces that lend themselves to either continuous Galerkin(CG)or discontinuous Galerkin(DG)global formulations.Here,we use Symmetric Interior Penalty Discontinuous Galerkin approach.Both methods yield a predictable error decline that depends on the respective coarse space dimension,and we illustrate the effectiveness of the CG and DG formulations by offering a variety of numerical examples.展开更多
Numerical solution of time-lapse seismic monitoring problems can be challenging due to the presence of finely layered reservoirs.Repetitive wave modeling using fine layered meshes also adds more computational cost.Con...Numerical solution of time-lapse seismic monitoring problems can be challenging due to the presence of finely layered reservoirs.Repetitive wave modeling using fine layered meshes also adds more computational cost.Conventional approaches such as finite difference and finite element methods may be prohibitively expensive if the whole domain is discretized with the cells corresponding to the grid in the reservoir subdomain.A common approach in this case is to use homogenization techniques to upscale properties of subsurface media and assign the background properties to coarser grid;however,inappropriate application of upscaling might result in a distortion of the model,which hinders accurate monitoring of the fluid change in subsurface.In this work,we instead investigate capabilities of a multiscale method that can deal with fine scale heterogeneities of the reservoir layer and more coarsely meshed rock properties in the surrounding domains in the same fashion.To address the 3-D wave problems,we also demonstrate how the multiscale wave modeling technique can detect the changes caused by fluid movement while the hydrocarbon production activity proceeds.展开更多
In this paper we propose a general framework for the uncertainty quantification of quantities of interest for high-contrast single-phase flow problems.It is based on the generalized multiscale finite element method(GM...In this paper we propose a general framework for the uncertainty quantification of quantities of interest for high-contrast single-phase flow problems.It is based on the generalized multiscale finite element method(GMsFEM)and multilevel Monte Carlo(MLMC)methods.The former provides a hierarchy of approximations of different resolution,whereas the latter gives an efficient way to estimate quantities of interest using samples on different levels.The number of basis functions in the online GMsFEM stage can be varied to determine the solution resolution and the computational cost,and to efficiently generate samples at different levels.In particular,it is cheap to generate samples on coarse grids but with low resolution,and it is expensive to generate samples on fine grids with high accuracy.By suitably choosing the number of samples at different levels,one can leverage the expensive computation in larger fine-grid spaces toward smaller coarse-grid spaces,while retaining the accuracy of the final Monte Carlo estimate.Further,we describe a multilevel Markov chain Monte Carlo method,which sequentially screens the proposal with different levels of approximations and reduces the number of evaluations required on fine grids,while combining the samples at different levels to arrive at an accurate estimate.The framework seamlessly integrates the multiscale features of the GMsFEM with the multilevel feature of the MLMC methods following the work in[26],and our numerical experiments illustrate its efficiency and accuracy in comparison with standard Monte Carlo estimates.展开更多
In this paper,we present the Multiscale Finite Element Method(MsFEM)for problems on rough heterogeneous surfaces.We consider the diffusion equation on oscillatory surfaces.Our objective is to represent small-scale fea...In this paper,we present the Multiscale Finite Element Method(MsFEM)for problems on rough heterogeneous surfaces.We consider the diffusion equation on oscillatory surfaces.Our objective is to represent small-scale features of the solution via multiscale basis functions described on a coarse grid.This problem arises in many applications where processes occur on surfaces or thin layers.We present a unified multiscale finite element framework that entails the use of transformations that map the reference surface to the deformed surface.The main ingredients of MsFEM are(1)the construction of multiscale basis functions and(2)a global coupling of these basis functions.For the construction of multiscale basis functions,our approach uses the transformation of the reference surface to a deformed surface.On the deformed surface,multiscale basis functions are defined where reduced(1D)problems are solved along the edges of coarse-grid blocks to calculate nodalmultiscale basis functions.Furthermore,these basis functions are transformed back to the reference configuration.We discuss the use of appropriate transformation operators that improve the accuracy of the method.The method has an optimal convergence if the transformed surface is smooth and the image of the coarse partition in the reference configuration forms a quasiuniform partition.In this paper,we consider such transformations based on harmonic coordinates(following H.Owhadi and L.Zhang[Comm.Pure and Applied Math.,LX(2007),pp.675-723])and discuss gridding issues in the reference configuration.Numerical results are presented where we compare the MsFEM when two types of deformations are used formultiscale basis construction.The first deformation employs local information and the second deformation employs a global information.Our numerical results showthat one can improve the accuracy of the simulations when a global information is used.展开更多
基金The authors would like to thank the China Petroleum&Chemical Corporation(SINOPEC),for supporting this work.
文摘Modeling and numerical simulations of fractured,vuggy,porus media is a challenging problem which occurs frequently in reservoir engineering.The problem is especially relevant in flow simulations of karst reservoirs where vugs and caves are embedded in a porous rock and are connected via fracture networks at multiple scales.In this paper we propose a unified approach to this problem by using the StokesBrinkman equations at the fine scale.These equations are capable of representing porous media such as rock as well as free flow regions(fractures,vugs,caves)in a single system of equations.We then consider upscaling these equations to a coarser scale.The cell problems,needed to compute coarse-scale permeability of Representative Element of Volume(REV)are discussed.A mixed finite element method is then used to solve the Stokes-Brinkman equation at the fine scale for a number of flow problems,representative for different types of vuggy reservoirs.Upscaling is also performed by numerical solutions of Stokes-Brinkman cell problems in selected REVs.Both isolated vugs in porous matrix as well as vugs connected by fracture networks are analyzed by comparing fine-scale and coarse-scale flow fields.Several different types of fracture networks,representative of short-and long-range fractures are studied numerically.It is also shown that the Stokes-Brinkman equations can naturally be used to model additional physical effects pertaining to vugular media such as partial fracture with fill-in by some material and/or fluids with suspended solid particles.
基金supported by the DOE and NSF(DMS 0934837 and DMS 0811180)supported by Award No.KUS-C1-016-04made by King Abdullah University of Science and Technology(KAUST).
文摘In this paper we use the Generalized Multiscale Finite Element Method(GMsFEM)framework,introduced in[26],in order to solve nonlinear elliptic equations with high-contrast coefficients.The proposed solution method involves linearizing the equation so that coarse-grid quantities of previous solution iterates can be regarded as auxiliary parameters within the problem formulation.With this convention,we systematically construct respective coarse solution spaces that lend themselves to either continuous Galerkin(CG)or discontinuous Galerkin(DG)global formulations.Here,we use Symmetric Interior Penalty Discontinuous Galerkin approach.Both methods yield a predictable error decline that depends on the respective coarse space dimension,and we illustrate the effectiveness of the CG and DG formulations by offering a variety of numerical examples.
基金support of Mega-grant of the Russian Federation Government(N 14.Y26.31.0013)。
文摘Numerical solution of time-lapse seismic monitoring problems can be challenging due to the presence of finely layered reservoirs.Repetitive wave modeling using fine layered meshes also adds more computational cost.Conventional approaches such as finite difference and finite element methods may be prohibitively expensive if the whole domain is discretized with the cells corresponding to the grid in the reservoir subdomain.A common approach in this case is to use homogenization techniques to upscale properties of subsurface media and assign the background properties to coarser grid;however,inappropriate application of upscaling might result in a distortion of the model,which hinders accurate monitoring of the fluid change in subsurface.In this work,we instead investigate capabilities of a multiscale method that can deal with fine scale heterogeneities of the reservoir layer and more coarsely meshed rock properties in the surrounding domains in the same fashion.To address the 3-D wave problems,we also demonstrate how the multiscale wave modeling technique can detect the changes caused by fluid movement while the hydrocarbon production activity proceeds.
基金Y.Efendiev’s work is partially supported by the U.S.Department of Energy Office of Science,Office of Advanced Scientific Computing Research,Applied Mathematics program under Award Number DE-FG02-13ER26165 and the DoD Army ARO ProjectThe research of B.Jin is partly supported by NSF Grant DMS-1319052.
文摘In this paper we propose a general framework for the uncertainty quantification of quantities of interest for high-contrast single-phase flow problems.It is based on the generalized multiscale finite element method(GMsFEM)and multilevel Monte Carlo(MLMC)methods.The former provides a hierarchy of approximations of different resolution,whereas the latter gives an efficient way to estimate quantities of interest using samples on different levels.The number of basis functions in the online GMsFEM stage can be varied to determine the solution resolution and the computational cost,and to efficiently generate samples at different levels.In particular,it is cheap to generate samples on coarse grids but with low resolution,and it is expensive to generate samples on fine grids with high accuracy.By suitably choosing the number of samples at different levels,one can leverage the expensive computation in larger fine-grid spaces toward smaller coarse-grid spaces,while retaining the accuracy of the final Monte Carlo estimate.Further,we describe a multilevel Markov chain Monte Carlo method,which sequentially screens the proposal with different levels of approximations and reduces the number of evaluations required on fine grids,while combining the samples at different levels to arrive at an accurate estimate.The framework seamlessly integrates the multiscale features of the GMsFEM with the multilevel feature of the MLMC methods following the work in[26],and our numerical experiments illustrate its efficiency and accuracy in comparison with standard Monte Carlo estimates.
基金supported by the US Army 62151-MA,DOE and NSF(DMS 0934837,DMS 0724704,and DMS 0811180)supported by Award No.KUS-C1-016-04,made by King Abdullah University of Science and Technology(KAUST).
文摘In this paper,we present the Multiscale Finite Element Method(MsFEM)for problems on rough heterogeneous surfaces.We consider the diffusion equation on oscillatory surfaces.Our objective is to represent small-scale features of the solution via multiscale basis functions described on a coarse grid.This problem arises in many applications where processes occur on surfaces or thin layers.We present a unified multiscale finite element framework that entails the use of transformations that map the reference surface to the deformed surface.The main ingredients of MsFEM are(1)the construction of multiscale basis functions and(2)a global coupling of these basis functions.For the construction of multiscale basis functions,our approach uses the transformation of the reference surface to a deformed surface.On the deformed surface,multiscale basis functions are defined where reduced(1D)problems are solved along the edges of coarse-grid blocks to calculate nodalmultiscale basis functions.Furthermore,these basis functions are transformed back to the reference configuration.We discuss the use of appropriate transformation operators that improve the accuracy of the method.The method has an optimal convergence if the transformed surface is smooth and the image of the coarse partition in the reference configuration forms a quasiuniform partition.In this paper,we consider such transformations based on harmonic coordinates(following H.Owhadi and L.Zhang[Comm.Pure and Applied Math.,LX(2007),pp.675-723])and discuss gridding issues in the reference configuration.Numerical results are presented where we compare the MsFEM when two types of deformations are used formultiscale basis construction.The first deformation employs local information and the second deformation employs a global information.Our numerical results showthat one can improve the accuracy of the simulations when a global information is used.
