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Algebraic multiscale method for flow in heterogeneous porous media with embedded discrete fractures (F-AMS)

Matei Tene, Mohammed Saad Al Kobaisi, Hadi Hajibeygi

arXiv:1705.05784v1math.NAcs.CEcs.NAphysics.comp-phphysics.flu-dyn

TL;DR

Flow simulation in heterogeneous fractured formations is challenging because highly conductive fracture networks span multiple length scales. F-AMS builds algebraic multiscale maps for matrix and fracture grids with flexible coarse resolutions and coupling. The reported results indicate robustness to conductivity contrasts and scalability under suitable coarsening, while coupled basis construction can become costly in some cases.

  • Problem

    Heterogeneous matrix rock crossed by highly conductive fractures at multiple length scales poses challenges for flow simulation.

    Source: §1 · p. 2

  • Method

    F-AMS algebraically constructs locally supported basis functions on independently coarsened matrix and fracture grids, with four choices for their coupling.

    Source: §1 · p. 3, §3 · p. 7, §3.1 · p. 12

  • Results

    F-AMS is reported insensitive to fracture-matrix conductivity contrast and scalable with fracture density, domain scale, and heterogeneous properties, but performance degrades with sub-optimal coarsening.

    Source: §5 · pp. 35–36

  • Takeaways & Limitations

    Few fracture coarse nodes can deliver good approximate pressure solutions at fine-scale resolution, and mass-conservative fluxes can be reconstructed after an MSFV stage.

    Source: §3 · p. 11, §5 · p. 36

    With many interconnected or long fractures, Coupled-AMS can expand basis support and densify prolongation, harming computational performance.

    Source: §3.1 · p. 17

Abstract

from arXiv · show

This paper introduces an Algebraic MultiScale method for simulation of flow in heterogeneous porous media with embedded discrete Fractures (F-AMS). First, multiscale coarse grids are independently constructed for both porous matrix and fracture networks. Then, a map between coarse- and fine-scale is obtained by algebraically computing basis functions with local support. In order to extend the localization assumption to the fractured media, four types of basis functions are investigated: (1) Decoupled-AMS, in which the two media are completely decoupled, (2) Frac-AMS and (3) Rock-AMS, which take into account only one-way transmissibilities, and (4) Coupled-AMS, in which the matrix and fracture interpolators are fully coupled. In order to ensure scalability, the F-AMS framework permits full flexibility in terms of the resolution of the fracture coarse grids. Numerical results are presented for two- and three-dimensional heterogeneous test cases. During these experiments, the performance of F-AMS, paired with ILU(0) as second-stage smoother in a convergent iterative procedure, is studied by monitoring CPU times and convergence rates. Finally, in order to investigate the scalability of the method, an extensive benchmark study is conducted, where a commercial algebraic multigrid solver is used as reference. The results show that, given an appropriate coarsening strategy, F-AMS is insensitive to severe fracture and matrix conductivity contrasts, as well as the length of the fracture networks. Its unique feature is that a fine-scale mass conservative flux field can be reconstructed after any iteration, providing efficient approximate solutions in time-dependent simulations.

1 Introduction

Naturally fractured formations combine heterogeneous matrix rock with highly conductive fractures across multiple length scales, making flow simulation challenging. F-AMS addresses the need for flexible multiscale solvers by constructing algebraic coarse-to-fine maps with locally supported basis functions for both media.

1 Introduction

  • Naturally fractured formations combine heterogeneous matrix rock with multiple length scales of highly conductive fractures, challenging flow simulation.These formations arise in applications including hydrocarbon production and geothermal energy exploitation.Source: §1 · p. 2
  • Even after small fractures are homogenized, the degrees of freedom can exceed the scope of classical simulation methods.The paper motivates an efficient multiscale method for heterogeneous fractured porous media.Source: §1 · p. 2
  • Earlier fractured-reservoir multiscale methods did not include 3D heterogeneous cases or commercial-solver benchmarks, and lacked flexible coarse grids for both media.The literature also lacked a multiscale method accommodating fracture networks at different length scales.Source: §1 · p. 3
  • F-AMS algebraically constructs prolongation and restriction operators from locally supported basis functions for matrix and fracture coarse grids.The method uses EDFM and partitions fine-scale cells into primal and dual coarse blocks.Source: §1 · p. 3
  • F-AMS allows arbitrary coarse-grid resolutions in both media and four coupling strategies, from decoupled to fully coupled basis functions.The strategies are Decoupled-AMS, Rock-AMS, Frac-AMS, and Coupled-AMS; full coupling can create a dense coarse system and require truncation.Source: §1 · pp. 3–4
  • The proof-of-concept evaluates 2D and 3D cases, studies algorithm components, and benchmarks CPU times against commercial AMG.The authors report that only a few degrees of freedom per fracture network are needed for a good trade-off between convergence and computational expense.Source: §1 · p. 4

2 Fine-scale discretized system

The fine-scale model represents matrix and fracture pressures on independently adaptable grids, with fracture-matrix exchange and external sources included in the flow system. Large, poorly conditioned systems motivate conservative multiscale solution strategies.

2 Fine-scale discretized system

  • The flow equations include matrix and fracture source terms, including injection and production wells.For nonlinear flows, the system is linearized and iteratively solved in a Newton-Raphson loop.Source: §2 · pp. 5–7
  • Fracture-matrix exchange terms are formulated so total flux between an intersecting fracture element and matrix volume is equal.The coupling uses effective interface mobility and a connectivity index based on fracture overlap and average element separation.Source: §2 · pp. 5–6
  • Matrix and fracture pressures are solved on separate domains, with independently adaptable grids and fractures represented in a lower-dimensional space.The independent grids can be adapted to each medium’s physics, including fractures generated or closed during simulation.Source: §2 · p. 6
  • Realistic systems can contain millions of unknowns, while heterogeneous matrix properties and conductive fractures worsen conditioning.Classical upscaling is unsuitable for the highly resolved fractures; conservative multiscale methods preserve fine-scale data and allow flux reconstruction during iterative error reduction.Source: §2 · p. 7

3 Algebraic Multiscale Method for Heterogeneous Porous Media with Embedded Discrete Fractures (F-AMS)

F-AMS builds algebraic coarse-to-fine maps for matrix and fracture domains using locally supported basis functions and flexible coarse grids. It offers four coupling strategies and alternative restriction operators, balancing coupling, sparsity, and mass-conservative flux reconstruction.

3 Algebraic Multiscale Method for Heterogeneous Porous Media with Embedded Discrete Fractures (F-AMS)

  • F-AMS superimposes primal and dual coarse grids on matrix and fracture domains, then uses local basis functions to map between coarse and fine scales.The primal grid partitions the domain into non-overlapping blocks; connecting coarse nodes defines the overlapping dual grid.Source: §3 · p. 7, §3.1 · pp. 11–12
  • Fracture pressure is explicitly represented and coarsened like matrix pressure, while each well is assigned one fine-scale degree of freedom in the experiments.F-AMS permits arbitrary coarse-grid resolutions in both matrix and fractures.Source: §3 · p. 9

3.1 Basis function formulations

  • The four coupling strategies range from zero cross-medium interpolation in Decoupled-AMS to nonzero interpolation in both directions in Coupled-AMS.Frac-AMS and Rock-AMS each retain one-way coupling; Decoupled-AMS yields the sparsest prolongation, while Coupled-AMS can increase accuracy and operator density.Source: §3 · p. 10

3.2 The F-AMS solution algorithm

  • F-AMS supports MSFV, MSFE, or hybrid MSMIX restriction; MSFV enables conservative flux reconstruction, whereas MSFE gives an SPD coarse system when the fine-scale system is SPD.For multiphase flow with pressure iterations stopped before machine accuracy, an MSFV stage and conservative flux reconstruction are used before transport.Source: §3 · p. 11, §3.2 · p. 18

3.1 Basis function formulations

  • Independent matrix and fracture grids allow cross-medium connections to blocks of different wirebasket ranks, motivating basis functions with different degrees of coupling.Coupled-AMS merges adjacent dual blocks when cells across them have nonzero transmissibility; this can enlarge basis support and densify the prolongation, so merging limits or truncation with row rescaling may be used.Source: §3.1 · pp. 12, 15, 17

3.2 The F-AMS solution algorithm

  • For highly heterogeneous reservoirs, F-AMS is paired with a fine-scale smoother; the described implementation uses ILU(0), while additional media-specific smoothing remains future work.The paper notes that severe matrix-fracture transmissibility contrast can produce a high condition number, and leaves the impact of another smoothing stage beyond its scope.Source: §3.2 · p. 18

4 Numerical Results

The experiments evaluate F-AMS on 2D and 3D fractured reservoirs, varying fracture and matrix properties, coarsening strategies, and domain sizes. They measure setup and solution CPU times and compare them with SAMG.

4 Numerical Results

  • The study varies coarsening strategies, fracture–matrix conductivity ratios, fracture densities, domain sizes, and heterogeneous fracture conductivities across 2D and 3D test cases.The 2D case uses a heterogeneous matrix and complex fracture network; its network is extruded for the 3D simulations.Source: §4 · p. 19
  • F-AMS setup and solution CPU times are measured and compared with the industrial-grade SAMG solver.Some experiments also break down time across initialization, operator construction, fine-system construction, coarse-system construction, solution, and smoothing stages.Source: §4 · pp. 19–20
  • The transmissibility ratio Tratio denotes the ratio of average fracture transmissibility to average matrix transmissibility.All test cases use a finite-element restriction operator, and direct LU solvers for coarse systems and basis functions.Source: §4 · p. 19

4.1 Distance-based fracture coarsening

F-AMS automatically coarsens fracture networks using a distance-based graph traversal. The distance parameter controls spacing between coarse nodes and can reduce the network to a single coarse node.

4.1 Distance-based fracture coarsening

  • A breadth-first graph traversal selects fracture coarse nodes so that each pair is separated by at least dmin cells.The fracture network is represented with fracture lines as arcs and intersections as nodes; dmin acts as the coarsening factor.Source: §4.1 · p. 20
  • Setting dmin = ∞ produces a single coarse node, while dmin = 20 cells yields multiple nodes on the illustrated network.The coarse-node illustration is for a network with 575 fine-scale cells; its aperture is magnified for clarity.Source: §4.1 · p. 20, Figure 8 · p. 20
  • Non-vertex fine cells form edges on the dual-coarse grid; primal[i] indexes a cell’s primal block and level[i] gives its distance to the nearest vertex.The algorithm builds the fracture coarse-grid structure required by F-AMS.Source: §4.1 · p. 21

4.2 F-AMS convergence

In the 2D test, Rock-AMS converges well across fracture coarsening factors, while the other strategies reach similar behavior after adding a few fracture coarse degrees of freedom. Initial interpolation quality also depends on the coupling strategy and fracture-grid resolution.

4.2 F-AMS convergence

  • The 2D convergence test embeds a 575-cell fracture network in a heterogeneous 64×64 matrix, with pressure-constrained wells on the west and east boundaries.The test-case illustration shows the pressure solution for Tratio = 100.Source: §4.2 · p. 21, Figure 9 · p. 22
  • With one fracture coarse degree of freedom, Rock-AMS and Coupled-AMS give more accurate initial pressure interpolations than Decoupled-AMS and Frac-AMS in this test.The matrix rock heterogeneity is the main source of approximation error; adding a few fracture coarse degrees of freedom improves Decoupled-AMS, Frac-AMS, and Coupled-AMS, while Rock-AMS is insensitive to that change.Source: §4.2 · pp. 21–22, Figure 10 · p. 23
  • Rock-AMS converges well regardless of fracture coarsening factor, while the other strategies reach similar convergence behavior with only a few additional fracture coarse degrees of freedom.Iterations continue until the residual 2-norm reaches 10^-6.Source: §4.2 · p. 22

4.3 Basis function truncation

In the 3D test, truncating small basis-function values reduces computational effort while largely preserving convergence for thresholds from 10^-2. The largest reported benefit is for Coupled-AMS, with locality maintained after truncation.

4.3 Basis function truncation

  • Fracture coarse nodes are selected on the network’s X-Y projection, then the grids are extruded along Z with the specified vertical coarsening ratio.The test uses the fracture network previously embedded in the 2D case.Source: §4.3 · pp. 23–24
  • Coupled-AMS can have especially dense prolongation and wider basis-function support in the high-density 3D fracture case, increasing coarse-system construction and solution effort.Truncation removes basis-function values below α and rescales affected rows to preserve partition of unity.Source: §4.3 · pp. 24–25
  • A truncation threshold of α = 10^-2 gives Coupled-AMS a speedup factor of 2 versus the un-truncated case, while preserving basis-function locality.The very restrictive α = 10^-1 increases iteration counts; from α = 10^-2, convergence is no longer much affected and reduced FLOPS improve efficiency.Source: §4.3 · p. 25
  • The 3D test uses a 64×64×64 matrix grid and a 575×64 fracture grid, with matrix and fracture coarsening ratios of 8×8×8 and 8×8, respectively.The shown pressure solution uses Tratio = 100.Source: Figure 13 · p. 26

4.4 Sensitivity to the coarsening factor

F-AMS performance depends on matrix and fracture coarsening choices. Coupled-AMS adapts across coarsening ratios, while similar matrix and fracture ratios give the best results in each tested row.

4.4 Sensitivity to the coarsening factor

  • Coarsening sensitivity was tested by varying matrix and fracture factors, with isotropic factors motivated by the point-wise ILU(0) smoother.The factor is the average fine-cell count per primal coarse block along each axis; fracture lengthwise coarsening uses dmin.Source: §4.4 · p. 26
  • Figure 16 varies matrix coarsening across columns and fracture coarsening across rows; bar-top labels give iterations to a 10^-6 residual 2-norm.Figure 15 illustrates Coupled-AMS basis support before and after truncation for a coarse node in a low-permeability region and a fracture network with one coarse DOF.Source: Figure 15 · p. 27, Figure 16 · p. 28
  • Similar matrix and fracture coarsening ratios give the optimum results in each tested row.Coupled-AMS adapts to the coarsening ratio, but its convergence is surpassed when the rock and fracture ratios differ substantially.Source: §4.4 · p. 29
  • With α = 10^-2, Coupled-AMS basis construction and coarse-system solution are not significantly more costly than the alternative strategies.Its wider basis support adds computational cost; truncation limits the extra cost.Source: §4.4 · p. 29
  • The study adopts coarsening ratio 8 in each direction for both media; compared with ratio 4, this yields more efficient coarse systems.Using F-AMS as a GMRES preconditioner can also significantly reduce linear iterations.Source: §4.4 · p. 29

4.5 Sensitivity to the transmissibility ratio

The coupling strategies respond differently as matrix-fracture transmissibility contrast changes. At Tratio ≥ 10^6, neither F-AMS nor SAMG reaches the specified tolerance in a Richardson loop, but both converge as GMRES preconditioners.

4.5 Sensitivity to the transmissibility ratio

  • Coupled-AMS remains relatively insensitive to Tratio, while Rock-AMS degrades and Frac-AMS improves as the fracture network becomes more conductive.Decoupled-AMS needs the most iterations when matrix and fracture transmissibilities are close, where two-way coupling is most pronounced.Source: §4.5 · p. 29
  • At Tratio ≥ 10^6, neither F-AMS nor SAMG reaches a 10^-6 residual norm in a Richardson loop; both converge when used as GMRES preconditioners.The experiments use a coarsening factor of 8 for both media.Source: §4.5 · p. 29
  • Figure 17 compares F-AMS performance across matrix-fracture transmissibility ratios; bar-top labels give iterations to a 10^-6 residual 2-norm.The convergence-history comparison in Figure 18 shows the Richardson-loop and GMRES outcomes at Tratio = 10^6.Source: Figure 17 · p. 30, Figure 18 · p. 30

4.6 CPU benchmark study

In GMRES-preconditioned tests, F-AMS and SAMG show comparable performance and scalability across transmissibility contrasts, fracture densities, domain sizes, and heterogeneous fracture properties. F-AMS results depend strongly on choosing an appropriate coarsening strategy.

4.6 CPU benchmark study

  • All F-AMS coupling strategies perform similarly when used as GMRES preconditioners, although Coupled-AMS has a more complex implementation.The benchmark uses converged GMRES-preconditioned runs; SAMG is a commercial black-box package, limiting direct CPU-breakdown comparisons.Source: §4.6 · p. 31
  • F-AMS can reconstruct a fine-scale mass-conservative flux field after any iteration stage once the coarse system with RFV restriction is solved.This benchmark is intended to demonstrate scalability.Source: §4.6 · p. 31

4.6.1 Transmissibility contrast

  • F-AMS iteration counts and CPU time are insensitive to fracture-matrix transmissibility contrast in the benchmark tests.The comparison uses Decoupled-AMS and SAMG as GMRES preconditioners; similar performance was observed for other F-AMS coupling strategies.Source: §4.6.1 · p. 31, Figure 19 · p. 32

4.6.2 Fracture density

  • With fracture coarsening fixed at 8 × 8, F-AMS maintains virtually the same convergence rate across fracture-count cases when preconditioning GMRES to a 10−6 residual 2-norm.The slight CPU-time increase is mainly due to computing extra fracture basis functions and constructing and solving a slightly larger coarse-scale linear system. Multiple coarse-scale DOFs per fracture network allow F-AMS to scale with fracture length and density, unlike the single-basis setup in.Source: §4.6.2 · p. 32, Figure 21 · p. 33

4.6.3 Domain scale

  • F-AMS and SAMG show similar CPU-time scalability as domain size grows, with both maintaining convergence rates in the tested cases.The tested domains range from 32^3 matrix and 320 × 32 fracture cells to 256^3 matrix and 2117 × 256 fracture cells; the study used Tratio = 10^2 and selected coarsening factors.Source: §4.6.3 · p. 33, Figure 23 · p. 34

4.6.4 Heterogeneous fractures

  • For strongly heterogeneous fracture permeability spanning six orders of magnitude, F-AMS and SAMG perform comparably with an appropriate coarsening ratio.The reported ratio is 6 × 6 × 8 in the matrix and 4 × 8 in the fractures; adapting coarse grids to conductivity or enriching prolongation are future directions.Source: §4.6.4 · pp. 33–34, Figure 24 · p. 35

4.6 CPU benchmark study

  • Across the benchmark study, F-AMS performance and scalability are comparable to SAMG; the authors describe this as a promising result for naturally fractured media.For realistic multiphase cases, a few F-AMS iterations followed by flux reconstruction can provide efficient approximate solutions.Source: §4.6.3 · p. 33, §4.6.4 · p. 34

5 Conclusions

F-AMS provides a flexible multiscale framework for fractured porous media, with independently chosen fracture coarse grids and four matrix-fracture coupling strategies. The results report comparable scalability to SAMG under appropriate coarsening, while performance can degrade sharply with suboptimal choices.

5 Conclusions

  • F-AMS constructs local basis functions for matrix and fracture coarse nodes and supports arbitrary fracture coarse-grid resolutions and four coupling strategies.The choices let users adjust the trade-off between setup cost and convergence rate.Source: §5 · p. 35
  • For all tested cases, truncating prolongation entries at α = 10^-2 was found optimum, supporting efficiency especially for Coupled-AMS.The paper also introduces distance-based automatic fracture coarsening for independently gridded fracture and matrix domains.Source: §5 · p. 35
  • F-AMS is reported to match SAMG's insensitivity to conductivity contrast and scalability across fracture density, domain scale, and heterogeneous properties, given appropriate coarsening.Performance can degrade dramatically with a suboptimal coarsening strategy; enhanced coarsening, different smoothers, and enriched prolongation are proposed directions.Source: §5 · pp. 35–36
  • All coupling strategies perform similarly as GMRES preconditioners, making Decoupled-AMS attractive for commercial simulation because of its convenient implementation.The conclusion also identifies stress-dependent fracture networks, capillarity, and gravity in multiphase flow as future extensions.Source: §5 · p. 36

A Algebraic Formulation of the F-AMS prolongation operators

The prolongation operators are derived from a matrix–fracture fine-scale system reordered by wirebasket rank. Four strategies encode different levels of matrix–fracture coupling under localization assumptions.

A Algebraic Formulation of the F-AMS prolongation operators

  • The fine-scale system is wirebasket-reordered for matrix and fracture unknowns; full matrix–fracture off-diagonal blocks motivate basis functions with different coupling levels.Fractures have Face, Edge, and Vertex cells, while the matrix also has Internal cells.Source: A Algebraic Formulation of the F-AMS… · pp. 36–37

A.1 Decoupled-AMS

  • Decoupled-AMS removes matrix–fracture coupling and applies localization by dropping connections to lower-ranked neighboring cells and associated transmissibilities.The resulting approximate system is used to construct the prolongation operator.Source: §A.1 · pp. 37–38
  • After solving the coarse system, Decoupled-AMS obtains prolongation algebraically; higher-rank basis values serve as boundary conditions for lower-rank cells.The approximate system is invertible algebraically because it is upper triangular, and the wirebasket ordering is then transformed back to natural ordering.Source: §A.1 · pp. 38–39

A.2 Frac-AMS

  • Frac-AMS includes matrix-to-fracture transmissibilities when computing basis functions, while applying localization to other connections.Its prolongation includes basis sub-blocks for matrix and fracture cells, including fracture values used in matrix-edge construction.Source: §A.2 · pp. 39–40

A.3 Rock-AMS

  • Rock-AMS sets matrix-to-fracture transmissibilities to zero and applies localization when constructing its approximate system and prolongation.Its prolongation includes sub-blocks spanning matrix and fracture cells.Source: §A.3 · pp. 40–41

A.4 Coupled-AMS

  • Coupled-AMS merges adjacent matrix and fracture Face and Edge blocks, then constructs the prolongation from the localized, reordered system.The merged sets are F = F^m ∪ F^f and E = E^m ∪ E^f, with coarse nodes treated as one set regardless of location.Source: §A.4 · p. 41

B Coarsening ratios used during scale sensitivity test

The scale-sensitivity experiments compare alternative primal-grid coarsening ratios and show that F-AMS performance depends strongly on the chosen ratios.

B Coarsening ratios used during scale sensitivity test

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