Accuracy and convergence properties of the fixed‐stress iterative solution of two‐way coupled poromechanics

Castelletto, Nicola; White, Joshua A.; Tchelepi, Hamdi A.

doi:10.1002/nag.2400

Cited by 130 publications

(91 citation statements)

References 53 publications

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“…Then, the fully discrete system becomes

([], \begin{matrix} center \\ array K & array - {bold-italicL}^{T} \\ array L & array F \end{matrix}) {([], \begin{matrix} center \\ array δ u \\ array δ p \end{matrix})}^{n} \to ([], \begin{matrix} center \\ array K & array - {bold-italicL}^{T} \\ array 0 & array F + S \end{matrix}) {([], \begin{matrix} center \\ array δ u \\ array δ p \end{matrix})}^{n} - ([], \begin{matrix} center \\ array 0 & array 0 \\ array - L & array S \end{matrix}) {([], \begin{matrix} center \\ array δ u \\ array δ p \end{matrix})}^{n - 1},

where S is calculated by fixing the rate of mean stress and by introducing the term b 2 / K d r locally in each element. () According to Kim et al() and Mikelic and Wheeler, the fixed‐stress split provides unconditional stability and convergence in time, whereas the fixed‐strain and drained splits are neither unconditionally stable nor convergent. Similar to the fixed‐stress method, the fixed‐strain split method solves flow first, fixing the strain field locally, and solves geomechanics at the next step.…”

Section: Time Discretization and Solution Strategymentioning

confidence: 99%

“…Remark The fixed‐stress split can be applied to a preconditioner of the monolithic method. () Then, Equation can be modified as

([], \begin{matrix} center \\ array K & array - {bold-italicL}^{T} \\ array L & array F \end{matrix}) {([], \begin{matrix} center \\ array δ u \\ array δ p \end{matrix})}^{n, false(k false)} \to ([], \begin{matrix} center \\ array K & array - {bold-italicL}^{T} \\ array 0 & array F + S \end{matrix}) {([], \begin{matrix} center \\ array δ u \\ array δ p \end{matrix})}^{n, false(k false)} - ([], \begin{matrix} center \\ array 0 & array 0 \\ array - L & array S \end{matrix}) {([], \begin{matrix} center \\ array δ u \\ array δ p \end{matrix})}^{n, false(k - 1 false)},

where ( k ) denotes the iteration level.…”

Section: Time Discretization and Solution Strategymentioning

confidence: 99%

“…where S is calculated by fixing the rate of mean stress and by introducing the term b 2 ∕K dr locally in each element. 31,36 According to Kim et al 31,32 and Mikelic and Wheeler, 35 the fixed-stress split provides unconditional stability and convergence in time, whereas the fixed-strain and drained splits are neither unconditionally stable nor convergent. Similar to the fixed-stress method, the fixed-strain split method solves flow first, fixing the strain field locally, and solves geomechanics at the next step.…”

Section: Sequential Scheme: Fixed-stress Splitmentioning

confidence: 99%

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Spatial stability for the monolithic and sequential methods with various space discretizations in poroelasticity

Yoon

Kim

2018

Numerical Meth Engineering

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Summary We investigate spatial stability with various numerical discretizations in displacement and pressure fields for poroelasticity. We study 2 sources of the early time instability: discontinuity of pressure and violation of the inf‐sup condition. We consider both compressible and incompressible fluids by employing the monolithic, stabilized monolithic, and fixed‐stress sequential methods. Four different spatial discretization schemes are used: Q1Q1, Q2Q1, Q1P0, and Q2P0. From mathematic analysis and numerical tests, the piecewise constant finite volume method for flow provides stability at the early time for the case of the pressure discontinuity. On the other hand, a piecewise continuous (or higher‐order) interpolation of pressure shows spatial oscillation, having lower limits of time step size, although lower approximations of pressure than displacement can alleviate the oscillation. For an incompressible fluid, Q2Q1 can be better than Q1P0, because Q1P0 might not satisfy the inf‐sup condition. However, regardless of fluid compressibility and the pressure discontinuity, the fixed‐stress method can effectively stabilize the oscillation without an artificial stabilizer. Even when Q1P0 and Q1Q1 with the monolithic method cannot satisfy the inf‐sup condition, the fixed‐stress method can yield the full‐rank linear system, providing stability. Thus, the fixed‐stress method with Q1P0 can effectively circumvent the aforementioned 2 types of instability.

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“…Then, the fully discrete system becomes

([], \begin{matrix} center \\ array K & array - {bold-italicL}^{T} \\ array L & array F \end{matrix}) {([], \begin{matrix} center \\ array δ u \\ array δ p \end{matrix})}^{n} \to ([], \begin{matrix} center \\ array K & array - {bold-italicL}^{T} \\ array 0 & array F + S \end{matrix}) {([], \begin{matrix} center \\ array δ u \\ array δ p \end{matrix})}^{n} - ([], \begin{matrix} center \\ array 0 & array 0 \\ array - L & array S \end{matrix}) {([], \begin{matrix} center \\ array δ u \\ array δ p \end{matrix})}^{n - 1},

Section: Time Discretization and Solution Strategymentioning

confidence: 99%

“…Remark The fixed‐stress split can be applied to a preconditioner of the monolithic method. () Then, Equation can be modified as

([], \begin{matrix} center \\ array K & array - {bold-italicL}^{T} \\ array L & array F \end{matrix}) {([], \begin{matrix} center \\ array δ u \\ array δ p \end{matrix})}^{n, false(k false)} \to ([], \begin{matrix} center \\ array K & array - {bold-italicL}^{T} \\ array 0 & array F + S \end{matrix}) {([], \begin{matrix} center \\ array δ u \\ array δ p \end{matrix})}^{n, false(k false)} - ([], \begin{matrix} center \\ array 0 & array 0 \\ array - L & array S \end{matrix}) {([], \begin{matrix} center \\ array δ u \\ array δ p \end{matrix})}^{n, false(k - 1 false)},

where ( k ) denotes the iteration level.…”

Section: Time Discretization and Solution Strategymentioning

confidence: 99%

Section: Sequential Scheme: Fixed-stress Splitmentioning

confidence: 99%

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Spatial stability for the monolithic and sequential methods with various space discretizations in poroelasticity

Yoon

Kim

2018

Numerical Meth Engineering

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“…For a more general Biot system, numerical solutions are also available. These solutions are obtained using various space discretization methods, including, for example, the mixed finite element method (e.g., Ferronato et al, 2010;Korsawe et al, 2006), the finite volume method (e.g., Nordbotten, 2014), and their combinations (e.g., Castelletto et al, 2015b), and they can be advanced in time in either a fully coupled manner or a proven stable sequentially coupled manner (e.g., Kim et al, 2011;White et al, 2016). example, within the framework of the finite element method, Juanes et al (2002) proposed a hypersurface element technique for global-to-local mapping of the integration over lower dimensional fractures.…”

Section: Introductionmentioning

confidence: 99%

Fully Coupled Nonlinear Fluid Flow and Poroelasticity in Arbitrarily Fractured Porous Media: A Hybrid‐Dimensional Computational Model

Jin

Zoback

2017

JGR Solid Earth

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We formulate the problem of fully coupled transient fluid flow and quasi‐static poroelasticity in arbitrarily fractured, deformable porous media saturated with a single‐phase compressible fluid. The fractures we consider are hydraulically highly conductive, allowing discontinuous fluid flux across them; mechanically, they act as finite‐thickness shear deformation zones prior to failure (i.e., nonslipping and nonpropagating), leading to “apparent discontinuity” in strain and stress across them. Local nonlinearity arising from pressure‐dependent permeability of fractures is also included. Taking advantage of typically high aspect ratio of a fracture, we do not resolve transversal variations and instead assume uniform flow velocity and simple shear strain within each fracture, rendering the coupled problem numerically more tractable. Fractures are discretized as lower dimensional zero‐thickness elements tangentially conforming to unstructured matrix elements. A hybrid‐dimensional, equal‐low‐order, two‐field mixed finite element method is developed, which is free from stability issues for a drained coupled system. The fully implicit backward Euler scheme is employed for advancing the fully coupled solution in time, and the Newton‐Raphson scheme is implemented for linearization. We show that the fully discretized system retains a canonical form of a fracture‐free poromechanical problem; the effect of fractures is translated to the modification of some existing terms as well as the addition of several terms to the capacity, conductivity, and stiffness matrices therefore allowing the development of independent subroutines for treating fractures within a standard computational framework. Our computational model provides more realistic inputs for some fracture‐dominated poromechanical problems like fluid‐induced seismicity.

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“…In [11], the convergence of this method is proven in the fully discrete case when space-time finite element methods are used. In [12], the authors present a very 35 interesting approach which consists to re-interpret the fixed-stress split scheme as a preconditioned-Richardson iteration with a particular block-triangular preconditioning operator. In fact, the four commonly used sequential splitting methods, i.e.…”

Section: Introductionmentioning

confidence: 99%

On the fixed-stress split scheme as smoother in multigrid methods for coupling flow and geomechanics

Gaspar

Rodrigo

2017

Computer Methods in Applied Mechanics and Engineering

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Please cite this article as: F.J. Gaspar, C. Rodrigo, On the fixed-stress split scheme as smoother in multigrid methods for coupling flow and geomechanics, Comput. Methods Appl. Mech. Engrg. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain . 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 On the fixed-stress split scheme as smoother in multigrid methods for coupling flow and geomechanics This solver combines the advantages of being a fully coupled method, with the benefit of decoupling the flow and the mechanics part in the smoothing algorithm. Moreover, the fixed-stress split smoother is based on the physics of the problem, and therefore all parameters involved in the relaxation are based on the physical properties of the medium and are given a priori. A local Fourier analysis is applied to study the convergence of the multigrid method and to support the good convergence results obtained. The proposed geometric multigrid algorithm is used to solve several tests in semi-structured triangular grids, in order to show the good behaviour of the method and its practical utility.

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Accuracy and convergence properties of the fixed‐stress iterative solution of two‐way coupled poromechanics

Cited by 130 publications

References 53 publications

Spatial stability for the monolithic and sequential methods with various space discretizations in poroelasticity

Spatial stability for the monolithic and sequential methods with various space discretizations in poroelasticity

Fully Coupled Nonlinear Fluid Flow and Poroelasticity in Arbitrarily Fractured Porous Media: A Hybrid‐Dimensional Computational Model

On the fixed-stress split scheme as smoother in multigrid methods for coupling flow and geomechanics

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