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Non-local Multi-continua Upscaling for Flows in Heterogeneous Fractured Media
Eric T. Chung, Yalchin Efendiev, Wing Tat Leung, Yating Wang, Maria Vasilyeva
TL;DR
Flow upscaling in fractured media requires representing multiple continua and their interactions, while fracture-related modes can be missed by existing coarse-space selection. The paper modifies a multiscale framework to construct non-local upscaled equations from localized problems and continuum-specific basis functions. Numerical examples report good accuracy with small localization, including a 0.008% error for H = 1/20 with four oversampling layers.
Problem
Fracture-associated basis functions can correspond to very small eigenvalues and be omitted from the coarse space, while fractured-flow upscaling requires multiple continua.
Source: §1.1 · p. 2, §1.2 · p. 3Method
The method modifies GMsFEM to build continuum-specific basis functions and local oversampled problems for computing non-local transfer and effective properties.
Source: §3.1.2 · p. 7, §6 · p. 17Results
0.008% error ∥uT −¯u∥L2 was reported for H = 1/20 with four oversampling layers, alongside good accuracy with small localization.
Source: §1.3 · p. 4, §5.1 · p. 13Takeaways & Limitations
The upscaled variables are average pressures in each fracture and matrix continuum, giving them physical meaning for practical simulations.
Source: §1.3 · p. 4, §6 · p. 17Localization requires identifying and separating each fracture network, and more realistic and complicated fracture systems are left for future work.
Source: §1.3 · p. 3, §5.1 · p. 11
Abstract
from arXiv · showhide
In this paper, we propose a rigorous and accurate non-local (in the oversampled region) upscaling framework based on some recently developed multiscale methods [10]. Our proposed method consists of identifying multi-continua parameters via local basis functions and constructing non-local (in the oversampled region) transfer and effective properties. To achieve this, we significantly modify our recent work proposed within Generalized Multiscale Finite Element Method (GMsFEM) in [10] and derive appropriate local problems in oversampled regions once we identify important modes representing each continua. We use piecewise constant functions in each fracture network and in the matrix to write an upscaled equation. Thus, the resulting upscaled equation is of minimal size and the unknowns are average pressures in the fractures and the matrix. We note that the use of non-local upscaled model for porous media flows is not new, e.g., in [14], the authors derive non-local approach. Our main contribution is identifying appropriate local problems together with local spectral modes to represent each continua. The model problem for fractures assumes that one can identify fracture networks. The resulting non-local equation (restricted to the oversampling region, which is several times larger compared to the target coarse block) has the same form as \cite{Hamdi_Nonlocal} with much smaller local regions. We present numerical results, which show that the proposed approach can provide good accuracy.
1 Introduction
The paper adapts multiscale methods for flow-based upscaling in fractured media, where multiple continua require effective properties and transfer terms. It constructs localized, non-local upscaled equations using separate fracture networks and matrix continua.
1.1 Flow-based upscaling methods
- Multiple-continua upscaling computes effective properties and transfer coefficients between continua, while conventional flow-based upscaling computes effective properties for coarse blocks.These computations are performed locally.Source: §1.1 · p. 2
1.2 Multiscale methods and their relation
- Multiscale methods compute basis functions rather than effective properties, and GMsFEM uses local spectral decomposition to select multiple basis functions representing continua.GMsFEM can automatically identify basis functions for each continuum.Source: §1.2 · p. 2
- Fracture-associated basis functions correspond to very small eigenvalues, and the corresponding eigenfunction may be excluded from the coarse space, motivating a modified GMsFEM framework.The proposed modification is recast as non-local upscaling suited to flow-based upscaling.Source: §1.2 · p. 3
1.3 This paper
- Localization requires identifying and separating each fracture network because fracture effects are global and cannot otherwise be localized.The method assumes the separate networks within each coarse block are known.Source: §1.3 · p. 3
- The method uses local oversampled problems to compute non-local transmissibilities and effective properties for multiple continua.Each local problem is constrained to represent one continuum while being orthogonal to the others; the resulting equation is defined over a neighborhood of the target coarse block.Source: §1.3 · p. 3
- Numerical comparisons with fine-grid models report good accuracy with small localization, while the resulting variables represent average pressures in each continuum.The paper notes that these physically interpretable variables avoid the need to extract physical parameters from the variables, as in its previous GMsFEM framework.Source: §1.3 · p. 4
2 Preliminaries
The preliminaries set up a heterogeneous single-phase flow problem and introduce the fine and coarse grids used for its numerical approximation. They also define the matrix and fracture regions and their permeabilities.
2 Preliminaries
- The model is a single-phase flow equation with pressure u, source g, and heterogeneous high-contrast permeability κ(x).The numerical examples use zero Neumann boundary conditions.Source: §2 · p. 4
- The fractured domain is divided into matrix and fracture regions, whose permeabilities can differ by orders of magnitude.The matrix permeability is κm, and the i-th fracture permeability is κi; di denotes fracture aperture.Source: §2 · pp. 4–5
- A conforming refinement defines a fine mesh with size h from a coarse mesh with size H, and oversampled regions extend around coarse blocks.The fine mesh is assumed to resolve small-scale information, while the coarse mesh contains many fine-scale features.Source: §2 · p. 5
3 The non-local multi-continua upscaling
The method represents each continuum with local basis functions, then uses them to construct non-local transfer and effective properties over oversampled regions. For fractured media with known networks, simplified constant bases yield a minimal coarse system whose variables represent average pressures.
3.1.1 General spectral setup
- Harmonic snapshots extend fine-grid boundary data into local regions, providing a snapshot space for spectral dimension reduction.The local region may be a coarse block, neighborhood, or oversampled region; randomized boundary conditions can reduce snapshot cost.Source: §3.1.1 · p. 6
- Small-eigenvalue modes form the offline basis, with all channel-representing modes retained in the auxiliary space.The general spectral setup can be used when separate fracture networks cannot be identified.Source: §3.1.1 · pp. 6–7
3.1.2 Simplified basis functions representing continua
- For fractured media, simplified constant bases represent each known fracture network and the matrix, minimizing degrees of freedom and yielding average-pressure variables.The construction assumes the separate fracture networks are known.Source: §3.1.2 · pp. 8–9
- Constraint-energy-minimizing bases are constructed in oversampled regions to provide spatial decay and distinguish continua.The simplified basis uses constants within separate fracture networks and the matrix.Source: §3.1.2 · p. 8
3.2.1 General spectral basis
- The general spectral basis functions define local transmissibilities by pairing bases across coarse blocks.The indices distinguish basis functions within blocks from the coarse-block indices.Source: §3.2.1 · p. 9
3.2.2 Simplified basis
- For known fracture networks, the simplified basis construction produces non-local transfer and effective properties between continua over oversampled regions.Here, m and n denote continua, while i and j denote coarse blocks.Source: §3.2.2 · p. 10
3.3 Approximation using local multiscale basis
- The approximation assembles transmissibilities into a coarse system whose row sums are zero, ensuring mass conservation.The system size reflects one matrix continuum plus the discrete fractures in each coarse block.Source: §3.3 · p. 10
4 Time-dependent problem
The paper also treats time-dependent single-phase flow using the spatial upscaling developed earlier. It contrasts a fine-scale backward-Euler finite-element solve with a coarse solve using non-local operators.
4 Time-dependent problem
- The time-dependent single-phase fine-scale problem is discretized with standard finite elements and backward Euler in time.The fine-scale mass and stiffness matrices and right-hand side form its matrix system.Source: §4 · pp. 10–11
- The coarse time-dependent solve uses non-local stiffness and approximate mass matrices defined for each continuum.The coarse stiffness operator is the previously defined AT.Source: §4 · p. 11
5 Numerical results
The steady-state examples report accurate upscaled solutions with small oversampling regions, alongside rapidly decaying transmissibility and localized local solutions. A time-dependent example also reports good accuracy against the averaged fine-scale solution.
5.1 Steady state case
- The steady-state tests use coarse meshes H = 1/10 and H = 1/20, with 282 and 927 coarse-scale degrees of freedom, respectively.Source: §5.1 · p. 11
- Local distributions decay, and with 6 oversampling layers the local solution almost vanishes outside the oversampling region, indicating that effects can be localized.Source: §5.1 · p. 13, Figure 6 · p. 13
- For ∥uT −¯u∥L2, error is 0.008% at H = 1/20 with 4 layers and 1.46% at H = 1/10 with 2 layers.The authors report good convergence against the averaged fine-scale solution.Source: Table 1 · p. 13, §5.1 · p. 13, Table 2 · p. 15
- Transmissibility influence is almost within 4 oversampling layers; along the slab it decays fast, with large errors at 1 layer but quite accurate values at 4.Source: §5.1 · pp. 13, 15, Figure 8 · p. 15, Figure 9 · p. 16
5.2 Time-dependent case
- At t = 0.1, 0.5, and 1.0, the time-dependent coarse-scale solution shows good accuracy against the averaged fine-scale solution.Source: §5.2 · p. 15, Table 3 · p. 17
6 Conclusions
The proposed non-local upscaling approach provides good accuracy in numerical examples, comparing average pressures and examining decay of local upscaled quantities. It constructs continua-specific effective properties and transfer terms using local problems in oversampled regions.
6 Conclusions
- The numerical examples show good accuracy, with average pressures compared and decay of local upscaled quantities examined.The examples are described as simplistic.Source: §6 · p. 17
- Piecewise constant functions for each fracture network and the matrix define continua whose unknowns are average pressures, yielding a small upscaled system.Local problems for effective properties are formulated for each continua to be orthogonal to the other continua.Source: §6 · p. 17
- Constraints from local solutions are used to compute non-local effective transmissibilities for each continua within oversampled regions several times larger than the target coarse block.The resulting non-local equation is defined in a small neighborhood of the coarse block.Source: §6 · p. 17
- The examples are simplistic, and the authors plan to consider more general fracture systems and analytical modeling of non-local upscaled quantities in future work.The paper notes that non-local upscaling for porous-media flows has been studied previously, including in.Source: §6 · p. 17