Generalized Newtonian fluid flow through a porous medium

Numerical methods are proposed for the nonlinear Stokes-Biot system modeling interaction of a free fluid with a poroelastic structure. We discuss time discretization and decoupling schemes that allow the fluid and the poroelastic structure computed independently using a common stress force along the interface. The coupled system of nonlinear Stokes and Biot is formulated as a least-squares problem with constraints, where the objective functional measures violation of some interface conditions. The local constraints, the Stokes and Biot models, are discretized in time using second-order schemes. Computational algorithms for the least-squares problems are discussed and numerical results are provided to compare the accuracy and efficiency of the algorithms. K E Y W O R D Sdomain decomposition, fluid structure interaction, poroelasticity INTRODUCTIONThe interaction of a fluid with a poroelastic material is of great importance in many engineering and biological applications. In such applications, a free fluid interacts with a deformable porous medium, where the structure is poroelastic, accounting for both the effects of porosity and elasticity. These interactions can be found in many applications such as reservoir engineering, groundwater flow, blood flow through vessels, and more. 1-4 Specifically for a blood flow, the interaction between blood and the arteries, as well as the effect of a blood clot, is a fluid-poroelastic system. 2 Other related applications are fluid flows through aquifers, bodies of permeable rock that can contain or transmit groundwater, where fluids such as water or oil flow through materials such as clay, sand, or sandstone. 3,5 There are a wide variety of solution algorithms proposed for fluid-structure interaction (FSI) problems of the Navier-Stokes coupled with the linear elasticity, but each is limited by computational complexity and stability. [6][7][8] While studies on the system of fluid and elastic structure have been active for the last 20 years, interaction of fluid flows with a poroelastic structure has received less attention and there are relatively a few numerical results reported for such problems. FSI problems involving an incompressible fluid and a poroelastic structure are often represented by the coupled Navier-Stokes and Biot (or Stokes-Biot) system with conditions on the interface between fluid and structure subdomains. A monolithic approach is computationally complex as it requires solving a large linear system; therefore, one needs the development of efficient and appropriate preconditioners for the discretized linear system. Ambartsumyan et al 9 studied a monolithic scheme for the Stokes equations coupled with the quasi-static Biot system, based on the Lagrange multiplier method. The proposed monolithic scheme was analyzed for stability and error estimation. Recently, a nonlinear Stokes-Biot system was considered for the analysis of well-posedness in Reference 10, Int J Numer Meth Fluids. 2020;92:687-702.wileyonlinelibrary.com/journal/fld Consider the domain of a fluid-...

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“…Remark We choose the same function space ∑ for the structure velocity,

\dot{η}

, as η , which requires higher regularity of

\dot{η}

than what it appears as in and . In order to define the trace of

\dot{η}

for the interface conditions on Γ I without the regularity assumption, regularization of

\dot{η}

should be considered by, for example, Laplace smoothing or local averaging, for lifting of an L 2 function to H 1 , see Reference .…”

Section: Model Equations and Semidiscrete Systemmentioning

confidence: 99%

Second‐order time discretization for a coupled quasi‐Newtonian fluid‐poroelastic system

Kunwar

Lee

Seelman

2019

Numerical Methods in Fluids

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“…Here, we extend the model in [1,38] to unsteady non-Darcy flow generalized Forchheimer's law. The work can be extended straightforwardly to viscosity models for generalized Newtonian fluids, including the Power law, the Cross model and the Carreau model [24,25].…”

Section: Goal and Positioning Of The Papermentioning

confidence: 99%

“…It is easy to verify that ξ satisfies the assumption (A1) . For more details see [24,25] and the references therein.…”

Section: Extension To Other Flow Models: the Cross Modelmentioning

confidence: 99%

Robust linear domain decomposition schemes for reduced non-linear fracture flow models

Ahmed¹,

Fumagalli²,

Budiša³

et al. 2019

Preprint

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In this work, we consider compressible single-phase flow problems in a porous media containing a fracture. In the latter, a non-linear pressure-velocity relation is prescribed. Using a non-overlapping domain decomposition procedure, we reformulate the global problem into a non-linear interface problem. We then introduce two new algorithms that are able to efficiently handle the non-linearity and the coupling between the fracture and the matrix, both based on linearization by the so-called L-scheme. The first algorithm, named MoLDD, uses the L-scheme to resolve the non-linearity, requiring at each iteration to solve the dimensional coupling via a domain decomposition approach. The second algorithm, called ItLDD, uses a sequential approach in which the dimensional coupling is part of the linearization iterations. For both algorithms, the computations are reduced only to the fracture by pre-computing, in an offline phase, a multiscale flux basis (the linear Robin-to-Neumann co-dimensional map), that represent the flux exchange between the fracture and the matrix. We present extensive theoretical findings and in particular, the stability and the convergence of both schemes are obtained, where usergiven parameters are optimized to minimise the number of iterations. Examples on two important fracture models are computed with the library PorePy and agree with the developed theory.

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“…Two smoothers which satisfy Aη s 1 and Aη s 2 are discussed in [9]. One is a local averaging operator and the other a differential smoothing operator.…”

Section: Assumptions On η Smentioning

confidence: 99%

“…volume averaging [20], homogenization [1], mixture theory [17]. Recently in [9] we considered the case of (steady-state) generalized Newtonian fluid flow through a porous medium, modeled by equations (1.1), (1.2), with µ ef f κ(η) −→ β(|u|). With the general assumptions that β(·) was a positive, bounded, Lipschitz continuous function, bounded away from zero, and with β(|u|) replaced with β(|u s |), existence of a solution was established.…”

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confidence: 99%

Nonlinear Darcy fluid flow with deposition

Ervin

Lee

Ruiz-Ramírez

2017

Journal of Computational and Applied Mathematics

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In this article we consider a model of a filtration process. The process is modeled using the nonlinear Darcy fluid flow equations with a varying permeability, coupled with a deposition equation. Existence and uniqueness of the solution to the modeling equations is established. A numerical approximation scheme is proposed and analyzed, with an a priori error estimate derived. Numerical experiments are presented which support the obtained theoretical results.The lack of regularity of the fluid velocity, u ∈ H div (Ω), leads to an open question of the existence of a solution to (1.1)-(1.3). In order to obtain a modeling system of equations for which a solution can be shown to exist, we replace η in (1.1) by an smoothed porosity, η s . The approach of regularizing the model with the introduction of η s is, in part, motivated by the Darcy fluid flow equations, which can be derived by averaging, e.g. volume averaging [20], homogenization [1], mixture theory [17]. Recently in [9] we considered the case of (steady-state) generalized Newtonian fluid flow through a porous medium, modeled by equations (1.1), (1.2), with µ ef f κ(η) −→ β(|u|). With the general assumptions that β(·) was a positive, bounded, Lipschitz continuous function, bounded away from zero, and with β(|u|) replaced with β(|u s |), existence of a solution was established. Two smoothing operators for u were presented. One was a local averaging operator, whereby u s (x) is obtained by averaging u in a neighborhood of x. The second smoothing operator, which is nonlocal, computes u s (x) using a differential filter applied to u. That is, u s is given by the solution to an elliptic differential equation whose right hand side is u. For establishing the existence of a solution the key property that the smoothing operator needs to satisfy is that it transform a weakly convergent sequence in L 2 (Ω) into a sequence which converges strongly in L ∞ (Ω).

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Generalized Newtonian fluid flow through a porous medium

Cited by 10 publications

References 16 publications

Second‐order time discretization for a coupled quasi‐Newtonian fluid‐poroelastic system

Second‐order time discretization for a coupled quasi‐Newtonian fluid‐poroelastic system

Robust linear domain decomposition schemes for reduced non-linear fracture flow models

Nonlinear Darcy fluid flow with deposition

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