Validity of Darcy's law under transient conditions

Mongan, Charles E.

doi:10.3133/pp1331

Cited by 5 publications

(10 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The discussions and the results presented in the previous sections neglected transient flow behavior within the porous domain. However, unsteady flow characteristics are indispensable in a wide variety of applications such as the ones observed in aquifers and oil-bearing strata [Mongan, 1985], and composite manufacturing applications based on resin transfer molding [Nakshatrala et al, 2006;Pacquaut et al, 2012] where two different fibers are usually used, providing two different pathways for the fluid. In this section, the proposed mixed formulation is extended to the transient case.…”

Section: An Extension To Transient Analysismentioning

confidence: 99%

Modeling flow in porous media with double porosity/permeability: A stabilized mixed formulation, error analysis, and numerical solutions

Joodat

Nakshatrala

Ballarini

2018

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

The flow of incompressible fluids through porous media plays a crucial role in many technological applications such as enhanced oil recovery and geological carbon-dioxide sequestration. The flow within numerous natural and synthetic porous materials that contain multiple scales of pores cannot be adequately described by the classical Darcy equations. It is for this reason that mathematical models for fluid flow in media with multiple scales of pores have been proposed in the literature. However, these models are analytically intractable for realistic problems. In this paper, a stabilized mixed four-field finite element formulation is presented to study the flow of an incompressible fluid in porous media exhibiting double porosity/permeability. The stabilization terms and the stabilization parameters are derived in a mathematically consistent manner, and the computationally convenient equal-order interpolation of all the field variables is shown to be stable. A systematic error analysis is performed on the resulting stabilized weak formulation. Representative problems, patch tests and numerical convergence analyses are performed to illustrate the performance and convergence behavior of the proposed mixed formulation in the discrete setting. The accuracy of numerical solutions is assessed using the mathematical properties satisfied by the solutions of this double porosity/permeability model. Moreover, it is shown that the proposed framework can perform well under transient conditions and that it can capture well-known instabilities such as viscous fingering.The mathematical models pertaining to the flow in porous media with multiple pore-networks are complex and involve numerous field variables. It is not always possible to derive analytical solutions to these mathematical models, and one has to resort to numerical solutions for realistic problems. Different approaches are available for developing formulations for multi-field mathematical models. Mixed finite element formulations, which offer the flexibility of using different approximations for different field variables, are particularly attractive for multi-field problems. Accurate numerical solutions have been obtained using mixed finite element for various porous media models; for example, see [Badia and Codina, 2010;Choo and Borja, 2015;Masud and Hughes, 2002;Nakshatrala and Rajagopal, 2011;Nakshatrala et al., 2006]. Moreover, many of the mathematical models pertaining to the multiple pore-networks, and in particular, the mathematical model considered in this paper, cannot be written in terms of a single-field variable. Although mixed methods are considered a powerful tool, especially for modeling flow problems in porous media, they suffer from some restrictions. To obtain stable and convergent solutions, a mixed formulation should satisfy the Ladyzhenskaya-Babuška-Brezzi (LBB) stability condition [Babuška, 1973;Brezzi and Fortin, 1991]. Numerical instability of the solution and probable spurious oscillations in the profile of unknown variables are the main co...

show abstract

Section: An Extension To Transient Analysismentioning

confidence: 99%

Modeling flow in porous media with double porosity/permeability: A stabilized mixed formulation, error analysis, and numerical solutions

Joodat

Nakshatrala

Ballarini

2018

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

show abstract

“…The permeability κ of the medium is taken as 1.0 cm/s, and the storativity S for the given porous region using 7000 capillary tubes is expected to be 0.6. The transient effect is studied for 10 time constants with time interval of approximately 1.7 msec [35]. The initial velocity at t = 0 is assumed to be zero, that is, the flow starts from rest.…”

Section: Transient Studymentioning

confidence: 99%

“…6 for quadrilateral and triangular meshes. In the transient plots, the validation of the transient velocity is achieved using Mongan's model [35]. The empirical formula for transient velocity is given as where a is the radius of the cylindrical tubes, G 1 is the pressure gradient step and T c1 is the time constant defined as follows:…”

Section: Transient Studymentioning

confidence: 99%

“…where ρ is the density and µ is the dynamic viscosity of the water. The parameter α 1 is the first root of the Bessel function that is equal to 2.405 and the value of a is set to 0.01 cm [35]. For the validation process, 7000 capillary tubes are used in a region of 2 cm 2 , i.e., a porosity of 0.6.…”

Section: Transient Studymentioning

confidence: 99%

“…The emerging system of linear equations is well-posed and is subsequently solved by the CG method for unknown pressure and velocity of the fluid. The numerical results of the transient model is validated by an analytical model proposed by Mongan [35]. The partial results of the stabilized mixed Galerkin transient method of sequential and parallel implementations have been published in [36,37].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stabilized Mixed Finite Element Method for Transient Darcy Flow

Ansari

Hussain

Ahmad

et al. 2017

Transactions of the Canadian Society for Mechanical Engineering

View full text Add to dashboard Cite

Darcy flow is a steady-state model for laminar flow of a fluid through a porous medium. The present work proposes an extended model of laminar Darcy flow by introducing dynamic pressure and velocity to the classical formulation. The solution of the proposed time-space model is attained by discretizing the problem with a stabilized mixed Galerkin method in space and a forward Euler method in time. The resulting matrix equation is well-posed and is solved using the conjugate gradient (CG) method. The error analysis of the numerical solutions confirms convergence to the actual model.

show abstract