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Zhou, Hui (ConocoPhillips Subsurface Technology) | Tchelepi, Hamdi A. (Stanford University)

An efficient two-stage algebraic multiscale solver (TAMS) that converges to the fine-scale solution is described. The first (global) stage is a multiscale solution obtained algebraically for the given fine-scale problem. In the second stage, a local preconditioner, such as the Block ILU (BILU) or the Additive Schwarz (AS) method, is used. Spectral analysis shows that the multiscale solution step captures the low-frequency parts of the error spectrum quite well, while the local preconditioner represents the high-frequency components accurately. Combining the two stages in an iterative scheme results in efficient treatment of all the error components associated with the fine-scale problem. TAMS is shown to converge to the reference fine-scale solution. Moreover, the eigenvalues of the TAMS iteration matrix show significant clustering, which is favorable for Krylov-based methods. Accurate solution of the nonlinear saturation equations (i.e., transport problem) requires having locally conservative velocity fields. TAMS guarantees local mass conservation by concluding the iterations with a multiscale finite-volume step. We demonstrate the performance of TAMS using several test cases with strong permeability heterogeneity and large-grid aspect ratios. Different choices in the TAMS algorithm are investigated, including the Galerkin and finite-volume restriction operators, as well as the BILU and AS preconditioners for the second stage. TAMS for the elliptic flow problem is comparable to state-of-the-art algebraic multigrid methods, which are in wide use. Moreover, the computational time of TAMS grows nearly linearly with problem size.

BILU, boundary condition, coarse grid, equation, grid, iteration, iteration matrix, june 2012, mass conservation, matrix, Modeling & Simulation, msfvm, operator, permeability, permeability field, preconditioner, prolongation operator, reservoir simulation, restriction operator, scaling method, TAM, Tchelepi, Upstream Oil & Gas

Oilfield Places:

- Europe > Norway > North Sea > Tarbert Formation (0.99)
- Europe > Germany > North Sea > Tarbert Formation (0.99)

Zhou, Hui (ConocoPhilips) | Lee, Seong H. (Chevron Energy Technology Company) | Tchelepi, Hamdi A. (Stanford University)

Recent advances in multiscale methods have shown great promise in modeling multiphase flow in highly detailed heterogeneous domains. Existing multiscale methods, however, solve for the flow field (pressure and total velocity) only. Once the fine-scale flow field is reconstructed, the saturation equations are solved on the fine scale. With the efficiency in dealing with the flow equations greatly improved by multiscale formulations, solving the saturation equations on the fine scale becomes the relatively more expensive part. In this paper, we describe an adaptive multiscale finite-volume (MSFV) formulation for nonlinear transport (saturation) equations. A general algebraic multiscale formulation consistent with the operator-based framework proposed by Zhou and Tchelepi (*SPE Journal*, June 2008, pages 267-273) is presented. Thus, the flow and transport equations are solved in a unified multiscale framework. Two types of multiscale operators--namely, restriction and prolongation--are used to construct the multiscale saturation solution. The restriction operator is defined as the sum of the fine-scale transport equations in a coarse gridblock. Three adaptive prolongation operators are defined according to the local saturation history at a particular coarse block. The three operators have different computational complexities, and they are used adaptively in the course of a simulation run. When properly used, they yield excellent computational efficiency while preserving accuracy. This adaptive multiscale formulation has been tested using several challenging problems with strong heterogeneity, large buoyancy effects, and changes in the well operating conditions (e.g., switching injectors and producers during simulation). The results demonstrate that adaptive multiscale transport calculations are in excellent agreement with fine-scale reference solutions, but at a much lower computational cost.

basis function, coarse block, Computation, construction, correction function, equation, formulation, iteration, march 2012, Modeling & Simulation, Multiscale Method, operator, prolongation operator, reservoir simulation, saturation, saturation equation, scaling method, Tchelepi, Timestep, total velocity, transport equation, Upstream Oil & Gas, velocity field

Zhou, Hui (ConocoPhillips) | Tchelepi, Hamdi A. (Stanford University)

Previous research on multiscale methods for subsurface flow aims to obtain an efficient multiscale solution to the fine-scale problems. Such multiscale solution is usually a good approximation to the fine-scale problem. However, it has been reported that the multiscale solution deteriorates for high aspect ratios and channelized structures of permeability. Moreover, the multiscale solution does not converge to the fine-scale solution unless some special techniques are used.

In this paper, we propose an efficient two-stage algebraic multiscale (TAMS) method that converges to the fine-scale solution. The TAMS method consists of two stages, namely global and local stages. In the global stage, a multiscale solution is obtained purely algebraically from the fine-scale matrix. In the local stage, a local solution is constructed from a local preconditioner such as Block ILU(0) (BILU) or Additive Schwarz (AS) method. Spectral analysis shows that the multiscale solution step captures the low-frequency spectra in the original matrix very well and when combined with a local preconditioner that represents the high-frequency spectra, the full spectra can be well approximated. Thus the TAMS is guaranteed to converge to the fine-scale solution. Moreover, the spectra of the TAMS method tend to cluster together, which is a favorable property for Krylov subspace methods to converge fast. We also show that local mass conservation can be preserved if a multiscale solution step with the finite-volume restriction operator is applied before the iterative procedure exits. This allows for the TAMS method to be used to build efficient approximate solutions for multiphase flow problems that need to solve transport equations.

We test the numerical performance of the TAMS method using several challenging large-scale problems (fine-scale grid in the magnitude of one million) with complex heterogeneous structures and high aspect ratios. Different choices in the TAMS algorithm are employed, including the Galerkin or finite-volume type of restriction operator, BILU or AS preconditioner for the second stage, and the size of blocks for BILU and AS. The performance of TAMS is comparable or superior to the state-of-the-art algebraic multigrid (AMG) preconditioner when some optimal choice in the TAMS method is adopted. Moreover, the convergence of the TAMS method is insensitive to problem sizes, and the CPU time is almost linear to problem sizes. These indicate the TAMS method is efficient and

robust for large-scale problems.

boundary condition, coarse grid, convergence, equation, grid, iteration, local preconditioner, mass conservation, matrix, Modeling & Simulation, multiscale finite-volume method, operator, permeability, preconditioner, programming language, prolongation operator, reservoir simulation, restriction operator, scaling method, spe 141473, spectra, TAM, Upstream Oil & Gas

Oilfield Places:

- Europe > Norway > North Sea > Tarbert Formation (0.99)
- Europe > Germany > North Sea > Tarbert Formation (0.99)

Technology:

Tchelepi, Hamdi A. (Stanford University) | Lee, Seong Hee (Chevron ETC) | Zhou, Hui

Recent advances in multiscale methods have shown great promise in modeling multiphase flow in highly detailed heterogeneous domains. Existing multiscale methods, however, solve for the flow field (pressure and total-velocity) only. Once the fine-scale flow field is reconstructed, the saturation equations are solved on the fine scale. With the efficiency in dealing with the flow equations greatly improved by multiscale formulations, solving the saturation equations on the fine scale becomes the relatively more expensive part. In this paper, we describe an adaptive multiscale finite-volume (MSFV) formulation for the nonlinear transport (saturation) equations. A general algebraic multiscale formulation consistent with the operator based framework proposed by Zhou and Tchelepi (SPEJ 13:267-173) is presented. Thus, the flow and transport equations are solved in a unified multiscale framework. Two types of multiscale operators, namely restriction and prolongation, are used to construct the multiscale saturation solution. The restriction operator is defined according to the local sum of the fine-scale transport equations in a coarse gridblock. Three adaptive prolongation operators are defined according to the local saturation history at a particular coarse block. The three operators have different computational complexity, and they are used adaptively in the course of a simulation run. When properly used, they yield excellent computational efficiency while preserving accuracy. This adaptive multiscale formulation has been tested using several challenging problems with strong heterogeneity, large buoyancy effects, and changes in the well operating conditions (e.g., switching injectors and producers during simulation). The results demonstrate that adaptive multiscale transport calculations are in excellent agreement with fine-scale reference solutions, but with a much lower computational cost.

basis function, coarse block, Computation, construction, equation, fine-scale velocity field, formulation, Jenny, Modeling & Simulation, Multiscale Method, operator, prolongation operator, reservoir simulation, restriction operator, saturation, saturation equation, scaling method, spe 119183, Tchelepi, time step, transport equation, Upstream Oil & Gas, velocity field

Lee, Seong Hee (Chevron ETC) | Wang, Xiaochen (Stanford University) | Zhou, Hui (Stanford University) | Tchelepi, Hamdi A.

We propose an upscaling method that is based on dynamic simulation of a given model in which the accuracy of the upscaled model is continuously monitored via indirect error-measures. If the indirect measures are bigger than a specified tolerance, the upscaled model is dynamically updated with approximate fine scale information that is reconstructed by a multi-scale finite volume method (Jenny et al., JCP 217; 627-641, 2006). Upscaling of multi-phase flow entails a detailed flow information in the underlying fine scale. We apply adaptive prolongation and restriction operators for flow and transport equations in constructing an approximate fine scale solution. This new method eliminates inaccuracy associated with the traditional upscaling method which relies on prescribed inaccurate boundary conditions in computing upscaled variables. The new upscaling algorithm is validated for two-phase, incompressible flow in two dimensional porous media with heterogeneous permeabilities. It is demonstrated that the dynamically upscaled model achieves high numerical efficiency than the fine-scale models and also provides an excellent agreement with the reference solution computed from fine-scale simulation.

adaptive reconstruction, basis function, boundary condition, coarse grid, equation, fine scale, grid, Jenny, Lee, Modeling & Simulation, multiphase flow, Oil Recovery, operator, porous media, reconstruction, reference solution, reservoir simulation, saturation, scaling method, transmissibility, upscaled model, Upstream Oil & Gas

Zhou, Hui (Stanford University) | Tchelepi, Hamdi A. (Stanford University)

Multiscale methods have been developed for accurate and efficient numerical solution of flow problems in large-scale heterogeneous reservoirs. A scalable and extendible Operator-Based Multiscale Method (OBMM) is described here. OBMM is cast as a general algebraic framework. It is natural and convenient to incorporate more physics in OBMM for multiscale computation. In OBMM, two operators are constructed: prolongation and restriction. The prolongation operator is constructed by assembling the multiscale basis functions. The specific form of the restriction operator depends on the coarse-scale discretization formulation (e.g., finitevolume or finite-element). The coarse-scale pressure equation is obtained algebraically by applying the prolongation and restriction operators to the fine-scale flow equations. Solving the coarse-scale equation results in a high-quality coarse-scale pressure. The finescale pressure can be reconstructed by applying the prolongation operator to the coarse-scale pressure. A conservative fine-scale velocity field is then reconstructed to solve the transport (saturation) equation. We describe the OBMM approach for multiscale modeling of compressible multiphase flow. We show that extension from incompressible to compressible flows is straightforward. No special treatment for compressibility is required. The efficiency of multiscale formulations over standard fine-scale methods is retained by OBMM. The accuracy of OBMM is demonstrated using several numerical examples including a challenging depletion problem in a strongly heterogeneous permeability field (SPE 10).

The accuracy of simulating subsurface flow relies strongly on the detailed geologic description of the porous formation. Formation properties such as porosity and permeability typically vary over many scales. As a result, it is not unusual for a detailed geologic description to require 10^{7}-10^{8} grid cells. However, this level of resolution is far beyond the computational capability of state-of-the-art reservoir simulators (10^{6} grid cells). Moreover, in many applications, large numbers of reservoir simulations are performed (e.g., history matching, sensitivity analysis and stochastic simulation). Thus, it is necessary to have an efficient and accurate computational method to study these highly detailed models.

Multiscale formulations are very promising due to their ability to resolve fine-scale information accurately without direct solution of the global fine-scale equations. Recently, there has been increasing interest in multiscale methods. Hou and Wu (1997) proposed a multiscale finite-element method (MsFEM) that captures the fine-scale information by constructing special basis functions within each element. However, the reconstructed fine-scale velocity is not conservative. Later, Chen and Hou (2003) proposed a conservative mixed finite-element multiscale method. Another multiscale mixed finite element method was presented by Arbogast (2002) and Arbogast and Bryant (2002). Numerical Green functions were used to resolve the fine-scale information, which are then coupled with coarse-scale operators to obtain the global solution. Aarnes (2004) proposed a modified mixed finite-element method, which constructs special basis functions sensitive to the nature of the elliptic problem. Chen et al. (2003) developed a local-global upscaling method by extracting local boundary conditions from a global solution, and then constructing coarse-scale system from local solutions. All these methods considered incompressible flow in heterogeneous porous media where the pressure equation is elliptic.

A multiscale finite-volume method (MsFVM) was proposed by Jenny et al. (2003, 2004, 2006) for heterogeneous elliptic problems. They employed two sets of basis functions--dual and primal. The dual basis functions are identical to those of Hou and Wu (1997), while the primal basis functions are obtained by solving local elliptic problems with Neumann boundary conditions calculated from the dual basis functions.

Existing multiscale methods (Aarnes 2004; Arbogast 2002; Chen and Hou 2003; Hou and Wu 1997; Jenny et al. 2003) deal with the incompressible flow problem only. However, compressibility will be significant if a gas phase is present. Gas has a large compressibility, which is a strong function of pressure. Therefore, there can be significant spatial compressibility variations in the reservoir, and this is a challenge for multiscale modeling. Very recently, Lunati and Jenny (2006) considered compressible multiphase flow in the framework of MsFVM. They proposed three models to account for the effects of compressibility. Using those models, compressibility effects were represented in the coarse-scale equations and the reconstructed fine-scale fluxes according to the magnitude of compressibility.

Motivated to construct a flexible algebraic multiscale framework that can deal with compressible multiphase flow in highly detailed heterogeneous models, we developed an operator-based multiscale method (OBMM). The OBMM algorithm is composed of four steps: (1) constructing the prolongation and restriction operators, (2) assembling and solving the coarse-scale pressure equations, (3) reconstructing the fine-scale pressure and velocity fields, and (4) solving the fine-scale transport equations.

OBMM is a general algebraic multiscale framework for compressible multiphase flow. This algebraic framework can also be extended naturally from structured to unstructured grid. Moreover, the OBMM approach may be used to employ multiscale solution strategies in existing simulators with a relatively small investment.

Artificial Intelligence, basis function, boundary condition, coarse block, coarse-scale operator, coarse-scale pressure, compressibility, compressible flow, equation, fine-scale pressure, formulation, Jenny, matrix, Multiscale Method, OBMM, operator, operator-based multiscale method, primal coarse block, prolongation operator, reservoir simulation, restriction operator, saturation, scaling method, Upstream Oil & Gas

Zhou, Hui (Stanford University) | Tchelepi, Hamdi A.

**Abstract **Recently, multiscale methods have been developed for accurate and efficient numerical solution of large-scale heterogeneous reservoir problems. A scalable and extendible Operator Based Multiscale Method (OBMM) is described here. OBMM is cast as a general algebraic framework of the multiscale method. It is very natural and convenient to incorporate more physics in OBMM for multiscale computation.

In OBMM, two multiscale operators are constructed: prolongation and restriction. The prolongation operator can be constructed by assembling basis functions, and the specific form of the restriction operator depends on the coarse-scale discretization formulation (e.g., finite-volume or finite-element). The coarse-scale pressure equation is obtained algebraically by applying the prolongation and restriction operators on the finescale flow equations. Solving the coarse-scale equation results in a high quality coarse-scale pressure. The fine scale pressure can be reconstructed by applying the prolongation operator to the coarse-scale pressure. A conservative fine-scale velocity field is then reconstructed to solve the transport equation.

As an application example, we study multiscale modeling of compressible flow. We show that the extension of modeling from incompressible to compressible flow is really straightforward for OBMM . No special treatment for compressibility is required. The efficiency of multiscale methods over standard fine-scale methods is retained by OBMM. The accuracy of OBMM is demonstrate by several challenging cases including highly compressible multiphase flow in a strongly heterogeneous permeability field (SPE 10).

**Introduction **The accuracy of simulating subsurface flow relies strongly on the detailed geologic description of the porous formation. Formation properties such as porosity and permeability typically vary over many scales. As a result, it is not unusual for a detailed geologic description to require

The multiscale method is very promising due to its ability to resolve fine-scale information accurately without direct solution of the global fine-scale equations. Recently, there has been increasing interest in multiscale methods. Hou and Wu4 proposed a multiscale finite-element method (MsFEM) that captures the fine-scale information by constructing special finite element basis functions within each element. However, the reconstructed fine-scale velocity is not conservative. Later, Chen and Hou proposed a conservative mixed finite-element multiscale method. Another multiscale mixed finite-element method has been presented by Arbogast1 and Arbogast and Bryant2. Numerical Green functions were used to resolve the fine-scale information, which are then coupled with coarse-scale operators to obtain the global solution. These methods considered incompressible flow in heterogeneous porous media where the

flow equation is elliptic.

Artificial Intelligence, basis function, coarse block, coarse-scale operator, coarse-scale pressure, compressibility, compressible flow, Denote, equation, fine-scale method, fine-scale pressure, fine-scale velocity, incompressible flow, multiphase flow, Multiscale Method, OBMM, operator, primal coarse block, prolongation operator, reservoir simulation, restriction operator, scaling method, scientific computing, transport equation, Upstream Oil & Gas

Technology:

Thank you!