Multi-scale finite-volume method for elliptic problems in subsurface flow simulation

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Abstract

In this paper we present a multi-scale finite-volume (MSFV) method to solve elliptic problems with many spatial scales arising from flow in porous media. The method efficiently captures the effects of small scales on a coarse grid, is conservative, and treats tensor permeabilities correctly. The underlying idea is to construct transmissibilities that capture the local properties of the differential operator. This leads to a multi-point discretization scheme for the finite-volume solution algorithm. Transmissibilities for the MSFV have to be constructed only once as a preprocessing step and can be computed locally. Therefore this step is perfectly suited for massively parallel computers. Furthermore, a conservative fine-scale velocity field can be constructed from the coarse-scale pressure solution. Two sets of locally computed basis functions are employed. The first set of basis functions captures the small-scale heterogeneity of the underlying permeability field, and it is computed in order to construct the effective coarse-scale transmissibilities. A second set of basis functions is required to construct a conservative fine-scale velocity field. The accuracy and efficiency of our method is demonstrated by various numerical experiments.

Introduction

The level of detail in reservoir description exceeds the computational capability of existing reservoir simulation. This resolution gap is usually tackled by upscaling the fine-scale description to sizes that can be treated by a full-featured simulator. In upscaling, the original model is coarsened using a computationally inexpensive process. In flow-based methods [5], the process is based on single-phase flow. The simulation study is then performed using the coarsened model. These upscaling methods have proved quite successful. However, it is not possible to have a priori estimates of the errors that are present when complex flow processes are investigated using coarse models constructed via simplified settings.

Various fundamentally different multi-scale approaches for flow in porous media have been proposed to accommodate the fine-scale description directly. As opposed to upscaling, the multi-scale approach targets the full problem with the original resolution. The methodology is based on resolving the length and time-scales of interest by maximizing local operations. Arbogast [1] and Arbogast and Bryant [2] presented a mixed finite-element method, where the fine-scale effects are localized by a boundary condition assumption at the coarse element boundaries. Then the small-scale influence is coupled with the coarse-scale effects by numerical Greens functions. Hou and Wu [8] employed the finite-element approach and constructed specific basis functions which capture the small scales. Again, localization is achieved by boundary condition assumptions for the coarse elements. To reduce the effects of these boundary conditions an oversampling technique can be applied. Recently, Chen and Hou [4] applied these ideas in combination with a mixed finite-element approach. Another approach by Beckie et al. [3] is based on large eddy simulation (LES) techniques which are commonly used for turbulence modeling.

Here a new multi-scale finite-volume (MSFV) approach is proposed which employs ideas from the flux-continuous finite-difference (FCFD) scheme developed by Lee et al. [10] for 2D, Lee et al. [11] for 3D, and later implemented in a multi-block simulator by Jenny et al. [9]. We also follow some principal ideas presented by Hou and Wu [8] and Efendiev et al. [6]. Advantages of our method are that it fits nicely into a finite-volume framework, allows for computing effective coarse-scale transmissibilities, treats tensor permeabilities properly, and is conservative at the coarse and fine scales. We will show that the method is computationally efficient and well suited for massively parallel computation. We also discuss how it can be applied to 3D unstructured grids and extended to multi-phase flow.

In Section 2 the flow problem is briefly introduced and in Section 3 the multi-scale finite-volume method is explained. Implementation and efficiency of the method are discussed in 4 Implementation of the MSFV method, 5 Computational efficiency, and numerical results are presented in Section 6. Finally, application to unstructured grids and multi-phase flow are discussed in Section 7. Summary and conclusions are given in Section 8.

Section snippets

Flow problem

We study the following elliptic problem:∇·(λ·∇p)=fonΩ,where p is the pressure and λ is the mobility (permeability, K, divided by the fluid viscosity, μ). The source term f represents wells and in the compressible case time derivatives. The permeability heterogeneity is a dominant factor in dictating the flow behavior in natural porous formations. The heterogeneity of K is usually represented as a complex multi-scale function of space. Moreover, K tends to be a highly discontinuous full tensor.

Multi-scale finite-volume (MSFV) method

We derive a multi-scale approach which results in effective transmissibilities for the coarse-scale problem. Once the transmissibilities are constructed, the method, which employs ideas from the flux-continuous finite-difference (FCFD) scheme developed by Lee et al. [11], does not differ from a finite-volume scheme using multi-point stencils for flux discretization. The approach is conservative and treats tensor permeabilities correctly. It can be easily applied by existing finite-volume codes,

Implementation of the MSFV method

In this section we discuss the necessary steps and implications on the data structure when the MSFV method is implemented. The algorithm consists of six major parts:

  • 1.

    Computation of transmissibilities for coarse-scale fluxes.

  • 2.

    Construction of fine-scale basis functions Φ.

  • 3.

    Computation of the coarse solution at the new time level.

  • 4.

    Reconstruction of the fine-scale velocity field in regions of interest.

  • 5.

    Solution of the transport equations.

  • 6.

    Re-computation of transmissibilities (part 1) and fine-scale basis

Computational efficiency

In order to analyze the computational efficiency of the MSFV method we introduce the following definitions:

nvnumber of volumes of the fine grid
nVnumber of volumes of the coarse grid
nNnumber of nodes of the coarse grid
nttotal number of time steps
aNaverage number of adjacent coarse volumes to a coarse node
aVaverage number of adjacent coarse volumes to a coarse volume
tl(n)CPU time to solve a linear system with n unknowns
tmCPU time for one multiplication
and assume that tl(n)=ctmnα, where c and α

Numerical studies

In order to demonstrate the performance of the MSFV method, we present numerical studies with a broad range of permeability fields. The first test case deals with a homogeneous permeability distribution; the permeability field for the second one is random. Geostatistically generated permeability fields are employed in the remaining examples. Fields with high variability as well as isotropic and anisotropic correlation structures are examined.

Discussion

We have demonstrated that the MSFV method is very powerful for problems of single-phase flow on structured grids. Next, convergence studies show that the MSFV method is consistent with the fine solution. Furthermore, we explain how the MSFV method can be extended for unstructured grids and multi-phase flow problems.

Conclusions

A new multi-scale finite-volume method for elliptic problems describing flow in porous media has been developed, tested and analyzed. The method, which is based on a flux-continuous finite-difference approach, is conservative and treats full tensor permeabilities and nonorthogonal grids correctly. The calculated transmissibilities account for the fine-scale effects. Once they are computed, these transmissibilities can be used by any finite-volume code that can handle multi-point flux

Acknowledgements

This work has been supported by ChevronTexaco Exploration and Production Technology Company and ChevronTexaco/Schlumberger Intersect Alliance Technology.

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