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New General Maximum Entropy Model for Flow Through Porous Media

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Abstract

New experimental and numerical techniques constitute the major recent advancements in the study of flow through porous media. However, a model that duly links the micro- and macroscales of this phenomenon is still lacking. Therefore, the present work describes a new, analytical model suitable for both Darcian and post-Darcian flow. Unlike its predecessors, most of which are based on empirical assessments or on some derivation of the Navier–Stokes equations, the presented model employed the principle of maximum entropy, along with a reduced number of premises. Nevertheless, it is compatible with classic expressions, such as Darcy’s and Forchheimer’s laws. Also, great adherence to previously published experimental results was observed. Moreover, the developed model allows for the delimitation of Darcian and post-Darcian regimes. It enabled the determination of a probabilistic distribution function of flow velocities within the pore space. Further, it bestowed richer interpretations of the physical meanings of principal flow parameters. Finally, through a new quantity called the entropy parameter, the proposed model may serve as a bridge between experimental and numerical findings both at the micro- and macroscales. Therefore, the present research yielded an analytical, entropy-based model for flow through porous media that is sufficiently general and robust to be applied in several fields of knowledge.

Graphic Abstract

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Abbreviations

\(a\) :

Forchheimer’s linear coefficient [L−1 T]

\(a_{\text{P}}\) :

Prony’s equation linear coefficient \(\left[ {\text{T}} \right]\)

\(b\) :

Forchheimer’s quadratic coefficient \(\left[ {{\text{L}}^{ - 2} {\text{T}}^{2} } \right]\)

\(b_{\text{P}}\) :

Prony’s equation quadratic coefficient \(\left[ {{\text{L}}^{ - 1} {\text{T}}^{2} } \right]\)

\(c\) :

Porous medium geometric constant \(\left[ {{\text{M}}^{0} \,{\text{L}}^{0} \,{\text{T}}^{0} \,\Theta ^{0} } \right]\)

\(c_{0}\) :

Porous medium constant obtained by dimensional analysis \(\left[ {{\text{M}}^{0} \,{\text{L}}^{0} \,{\text{T}}^{0} \,\Theta ^{0} } \right]\)

\(c_{\text{V}}\) :

Coefficient of variation of the hydraulic radius \(\left[ {{\text{M}}^{0} \,{\text{L}}^{0} \,{\text{T}}^{0} \,\Theta ^{0} } \right]\)

\(D\) :

Conduit diameter \(\left[ {\text{L}} \right]\)

\(d\) :

Porous medium characteristic length \(\left[ {\text{L}} \right]\)

\(d_{\text{p}}\) :

Particle diameter \(\left[ {\text{L}} \right]\)

\(f_{\sqrt k }\) :

Resistance factor for porous media \(\left[ {{\text{M}}^{0} \,{\text{L}}^{0} \,{\text{T}}^{0} \,\Theta ^{0} } \right]\)

\(g\) :

Gravitational acceleration \(\left[ {{\text{L T}}^{ - 2} } \right]\)

\(H\) :

Shannon’s entropy function \(\left[ {{\text{M}}^{0} \,{\text{L}}^{0} \,{\text{T}}^{0} \,\Theta ^{0} } \right]\)

\(i\) :

Hydraulic gradient \(\left[ {{\text{M}}^{0} \,{\text{L}}^{0} \,{\text{T}}^{0} \,\Theta ^{0} } \right]\)

\(i_{0}\) :

Threshold gradient \(\left[ {{\text{M}}^{0} \,{\text{L}}^{0} \,{\text{T}}^{0} \,\Theta ^{0} } \right]\)

\(K\) :

Darcy’s permeability coefficient \(\left[ {{\text{L T}}^{ - 1} } \right]\)

\(k\) :

Intrinsic permeability coefficient \(\left[ {{\text{L}}^{2} } \right]\)

\(k_{\text{s}}\) :

Pore-shape factor \(\left[ {{\text{M}}^{0} \,{\text{L}}^{0} \,{\text{T}}^{0} \,\Theta ^{0} } \right]\)

\(L\) :

Conduit length \(\left[ {\text{L}} \right]\)

\(M\) :

Entropy parameter \(\left[ {{\text{M}}^{0} \,{\text{L}}^{0} \,{\text{T}}^{0} \,\Theta ^{0} } \right]\)

\(n\) :

Porous medium porosity \(\left[ {{\text{M}}^{0} \,{\text{L}}^{0} \,{\text{T}}^{0} \,\Theta ^{0} } \right]\)

\(n_{\text{p}}\) :

Number of experimental points \(\left[ {{\text{M}}^{0} \,{\text{L}}^{0} \,{\text{T}}^{0} \,\Theta ^{0} } \right]\)

\(P\) :

Cumulative density function (CDF)

\(p\) :

Probability density function (PDF)

\(q\) :

Flow average bulk velocity \(\left[ {{\text{L T}}^{ - 1} } \right]\)

\(\text{Re}_{\sqrt k }\) :

Reynolds number (with \(\sqrt k\) as characteristic length) for porous media \(\left[ {{\text{M}}^{0} \,{\text{L}}^{0} \,{\text{T}}^{0} \,\Theta ^{0} } \right]\)

\(\bar{R}_{\text{h}}\) :

Porous medium average hydraulic radius \(\left[ {\text{L}} \right]\)

\(t\) :

Hydraulic tortuosity of the porous medium \(\left[ {{\text{M}}^{0} \,{\text{L}}^{0} \,{\text{T}}^{0} \,\Theta ^{0} } \right]\)

\({\mathcal{U}}\) :

Representative elementary volume (REV) \(\left[ {{\text{L}}^{3} } \right]\)

\(\varvec{u}\) :

Velocity vector at a given point \(\varvec{x}\) in the fluid phase \(\left[ {{\text{L T}}^{ - 1} } \right]\)

\(u\) :

Magnitude of \(\varvec{u}\)\(\left( {u = \left\| \varvec{u} \right\| = \sqrt {u_{x}^{2} + u_{y}^{2} + u_{z}^{2} } } \right)\)\(\left[ {{\text{L T}}^{ - 1} } \right]\)

\(u_{{\rm max}}\) :

Maximum velocity occurring in the pore space \(\left[ {{\text{L T}}^{ - 1} } \right]\)

\(\bar{u}\) :

Average (Dupuit’s) velocity in the pore space \(\left[ {{\text{L T}}^{ - 1} } \right]\)

\(\varvec{x}\) :

Point in a 3-D Cartesian space flow domain \(\left[ {\text{L}} \right]\)

\(\Delta h\) :

Head loss \(\left[ {\text{L}} \right]\)

\(\varepsilon_{0}\) :

Momentum-transfer coefficient at the wall \(\left[ {{\text{L}}^{2} {\text{T}}^{ - 1} } \right]\)

\(\lambda\) :

Lagrange multiplier \(\left[ {{\text{M}}^{0} \,{\text{L}}^{0} \,{\text{T}}^{0} \,\Theta ^{0} } \right]\)

\(\mu\) :

Dynamic viscosity of fluid \(\left[ {{\text{M}}\, {\text{L}}^{ - 1} \, {\text{T}}^{ - 1} } \right]\)

\(\nu\) :

Kinematic viscosity of fluid \(\left[ {{\text{L}}^{2} {\text{T}}^{ - 1} } \right]\)

\(\nu_{t}\) :

Eddy viscosity \(\left[ {{\text{L}}^{2} {\text{T}}^{ - 1} } \right]\)

\(\xi \left( u \right)\) :

Projection of isovel in plane perpendicular to principal flow \(\left[ {\text{L}} \right];\)

\(\xi_{0}\) :

Minimum value of ξ\(\left[ {\text{L}} \right]\)

\(\xi_{{\rm max} }\) :

Maximum value of ξ\(\left[ {\text{L}} \right]\)

\(\rho\) :

Specific mass of fluid \(\left[ {{\text{M L}}^{ - 3} } \right]\)

\(\sigma_{\text{s}}\) :

Surface tension \(\left[ {{\text{M}} {\text{T}}^{ - 2} } \right]\)

\(\bar{\tau }_{0}\) :

Average shear stress at the solid surfaces \(\left[ {{\text{M}}\, {\text{L}}^{ - 1} \,{\text{T}}^{ - 2} } \right]\)

\(\phi\) :

Functional relationship

\(\psi\) :

General quantity/function

\(\psi_{{\rm max} }\) :

Maximum value assumed by \(\psi\)

\(\psi_{{\rm min} }\) :

Minimum value assumed by \(\psi\)

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Acknowledgements

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (Capes)—Finance Code 001.

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Lofrano, F.C., Morita, D.M., Kurokawa, F.A. et al. New General Maximum Entropy Model for Flow Through Porous Media. Transp Porous Med 131, 681–703 (2020). https://doi.org/10.1007/s11242-019-01362-3

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Keywords

  • Porous media
  • Fluid flow
  • Principle of maximum entropy
  • Analytical model
  • Velocity distribution within the pore space