Fluid flow through porous media

The movement of a fluid through porous media is described by the combination of Darcy's law with the principle of conservation of mass in order to express the capillary force or fluid velocity as a function of various other parameters including the effective pore radius, liquid viscosity or permeability.[https://hal.archives-ouvertes.fr/hal-02310722/document#page=6 Tailoring porous media for controllable capillary flow] Journal of colloid and interface science (2019), 539, 379-387 However, the use of Darcy's law alone does not produce accurate results for heterogeneous media like shale, and tight sandstones, where there is a huge proportion of nanopores. This necessitates the use of a flow model that considers the weighted proportion of various flow regimes like Darcy flow, transition flow, slip flow, and free molecular flow.{{cite journal | last1=Saurabh | first1=Suman | title=Flow regime evolution and stress-dependent permeability in nanoporous rocks | date=2022-02-15 | url=https://www.sciencedirect.com/science/article/abs/pii/S0016236121022857| journal=Fuel| volume=310 | page=122413 |publisher=Elsevier|doi=10.1016/j.fuel.2021.122413 | bibcode=2022Fuel..31022413S | s2cid=243474407 }}

The basic law governing the flow of fluids through porous media is Darcy's Law, which was formulated by the French civil engineer Henry Darcy in 1856 on the basis of his experiments on vertical water filtration through sand beds.{{cite journal | doi=10.1007/BF01036523 | volume=1 | title=Flow in porous media I: A theoretical derivation of Darcy's law | journal=Transport in Porous Media | pages=3–25|year = 1986|last1 = Whitaker|first1 = Stephen| issue=1 | bibcode=1986TPMed...1....3W | s2cid=121904058 }}

According to Darcy's law, the fluid's viscosity, effective fluid permeability, and fluid pressure gradient determine the flow rate at any given location in the reservoir.{{cite journal |last1=Johns |first1=Russell T. |title=Oil Recovery |journal=Encyclopedia of Energy |date=1 January 2004 |pages=701–713 |doi=10.1016/b0-12-176480-x/00255-2 |url=https://doi.org/10.1016/B0-12-176480-X/00255-2 |publisher=Elsevier |isbn=9780121764807 |language=en}}

{{Equation box 1

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|equation=:$Q=\backslash frac\{-k\; A\}\{\backslash mu\}\; \backslash frac\{(p\_b\; -\; p\_a)\}\{L\}$

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For transient processes in which the flux varies from point to-point, the following differential form of Darcy’s law is used.

{{Equation box 1

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|equation=:$Q=\; \backslash frac\; \{-k\; A\}\{\backslash mu\}\; \backslash left(\; \backslash frac\{d\; p\}\{d\; x\}\; \backslash right)$

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Darcy's law is valid for situation where the porous material is already saturated with the fluid. For the calculation of capillary imbibition speed of a liquid to an initially dry medium, Washburn's or Bosanquet's equations are used.

Mass conservation of fluid across the porous medium involves the basic principle that mass flux in minus mass flux out equals the increase in amount stored by a medium.{{cite book|last1=Bear|first1=Jacob|title=Dynamics of fluids in Porous Media|url=https://books.google.com/books?id=fBMeVSZ_3u8C&dq=Mass+conservation+in+porous+media&pg=PP1|isbn=9780486131801|date=2013-02-26|publisher=Courier Corporation }} This means that total mass of the fluid is always conserved. In mathematical form, considering a time period from $t$ to $\backslash Delta\; t$, length of porous medium from $x$ to $\backslash Delta\; x$ and $m$ being the mass stored by the medium, we have

:$[A\backslash rho\; (x)q(x)\; -\; A\backslash rho\; (x\; +\; \backslash Delta\; x)q(x\; +\; \backslash Delta\; x\; )]\backslash Delta\; t\; =\; m(t\; +\; \backslash Delta\; t)\; -\; m(t)\; .$

Furthermore, we have that $m\; =\; \backslash rho\; V\_p$, where $V\_p$ is the pore volume of the medium between $x$ and $x+\backslash Delta\; x$ and $\backslash rho$ is the density. So $m\; =\; \backslash rho\; V\_p\; =\; \backslash rho\; \backslash phi\; V\; =\; \backslash rho\; \backslash phi\; A\backslash Delta\; x,$ where $\backslash phi$ is the porosity. Dividing both sides by $A\backslash Delta\; x$, while $\backslash Delta\; x\; \backslash rightarrow\; 0$, we have for 1 dimensional linear flow in a porous medium the relation

:$\backslash frac\{-d(\backslash rho\; q)\}\{d\; x\}\; =\; \backslash frac\{d(\backslash rho\; \backslash phi)\}\{d\; t\}\; ~~~~\; (i)$

In three dimensions, the equation can be written as

:$\backslash frac\{d(\backslash rho\; q)\}\{d\; x\}\; +\; \backslash frac\{d(\backslash rho\; q)\}\{d\; y\}\; +\; \backslash frac\{d(\backslash rho\; q)\}\{d\; z\}\; =\; \backslash frac\{-d(\backslash rho\; \backslash phi)\}\{d\; t\}$

The mathematical operation on the left-hand side of this equation is known as the divergence of $\backslash rho\; q$, and represents the rate at which fluid diverges from a given region, per unit volume.

Using product rule(and chain rule) on right hand side of the above mass conservation equation (i),

:$\backslash frac\{d(\backslash rho\; \backslash phi)\}\{d\; t\}\; =\; \backslash rho\; \backslash frac\{d\; \backslash phi\}\{d\; t\}\; +\; \backslash phi\; \backslash frac\{d\; \backslash rho\}\{d\; t\}\; =\; \backslash rho\; \backslash frac\{d\; \backslash phi\}\{d\; P\}\; \backslash frac\{d\; P\}\{d\; t\}\; +\; \backslash phi\; \backslash frac\{d\; \backslash rho\}\{d\; P\}\; \backslash frac\{d\; P\}\{d\; t\}\; =\; \backslash rho\; \backslash phi\; \backslash left\; [\backslash frac\{1\}\{\backslash phi\}\; \backslash frac\{d\; \backslash phi\}\{d\; P\}\; +\; \backslash frac\{1\}\{\backslash rho\}\; \backslash frac\{d\; \backslash rho\}\{d\; P\}\; \backslash right]\; \backslash quad\; \backslash frac\{d\; P\}\{d\; t\}\; =\; \backslash rho\; \backslash phi\; [c\_\backslash phi\; +\; c\_f]\backslash frac\{d\; P\}\{d\; t\}\; ~~~~\; (ii)$

Here, $c\_f$ = compressibility of the fluid and $c\_\backslash phi$ = compressibility of porous medium.{{cite journal|last1=Ohirhian|first1=Peter|title=Steady State Compressible Fluid Flow in Porous Media}} Now considering the left hand side of the mass conservation equation, which is given by Darcy's Law as

:$\backslash frac\{-d(\backslash rho\; q)\}\{d\; x\}\; =\; \backslash frac\{-d\}\{dx\}\; \backslash left\; [\backslash frac\{-\; \backslash rho\; k\}\{\backslash mu\}\; \backslash frac\{d\; P\}\{d\; x\}\; \backslash right]\; \backslash quad\; =\; \backslash frac\{k\}\{\backslash mu\}\; \backslash left\; [\backslash rho\; \backslash frac\{d^2\; P\}\{d\; x^2\}\; +\; \backslash frac\{d\; \backslash rho\}\{d\; P\}\; \backslash frac\{d\; P\}\{d\; x\}\; \backslash frac\{d\; P\}\{d\; x\}\; \backslash right]\; \backslash quad\; =\; \backslash frac\{\backslash rho\; k\}\{\backslash mu\}\; \backslash left[\backslash frac\{d^2\; P\}\{d\; x^2\}\; +\; \backslash left\; (\backslash frac\{1\}\{\backslash rho\}\; \backslash frac\{d\; \backslash rho\}\{d\; P\}\backslash right)\backslash left(\backslash frac\{d\; P\}\{d\; x\}\; \backslash right)^2\; \backslash right\; ]\; \backslash quad\; =\; \backslash frac\{\backslash rho\; k\}\{\backslash mu\}\; \backslash left\; [\backslash frac\{d^2\; P\}\{d\; x^2\}\; +\; c\_f\; \backslash left(\backslash frac\{d\; P\}\{d\; x\}\backslash right)^2\; \backslash right]\; \backslash quad\; ~~~~\; (iii)$

Equating the results obtained in $(ii)$ & $(iii)$, we get

:$\backslash frac\{d^2\; P\}\{d\; x^2\}\; +\; c\_f\backslash left(\backslash frac\{d\; P\}\{d\; x\}\backslash right)^2\; =\; \backslash frac\{\backslash phi\; \backslash mu\; (c\_f\; +\; c\_\backslash phi)\}\{k\}\; \backslash frac\{d\; P\}\{d\; t\}$

The second term on the left side is usually negligible, and we obtain the diffusion equation in 1 dimension as

:$\backslash frac\{d\; P\}\{d\; t\}\; =\; \backslash frac\{k\}\{\backslash phi\; \backslash mu\; c\_t\}\; \backslash frac\{d^2\; P\}\{d\; x^2\}$

where $c\_t\; =\; c\_f\; +\; c\_\backslash phi$.{{cite web|last1=Zimmerman|first1=Dr. R.W.|title=Flow in porous media|url=http://imperial.ac.uk}}

{{Reflist}}

• {{cite book|last=Acheson|first=D. J.|title=Elementary Fluid Dynamics|publisher=Clarendon Press|year=1990|isbn=0-19-859679-0}}
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• {{cite book|last=Chanson|first=H.|author-link=Hubert Chanson|title=Applied Hydrodynamics: An Introduction to Ideal and Real Fluid Flows|publisher=CRC Press, Taylor & Francis Group, Leiden, The Netherlands, 478 pages|year=2009|isbn=978-0-415-49271-3}}
• {{cite book|last=Clancy|first=L. J.|title=Aerodynamics|publisher=Pitman Publishing Limited|location=London|year=1975|isbn=0-273-01120-0}}
• {{citation | last1=Nazarenko | first1=Sergey | year=2014 | title=Fluid Dynamics via Examples and Solutions | publisher=CRC Press (Taylor & Francis group) | isbn=978-1-43-988882-7 }}

• [http://perminc.com/resources/fundamentals-of-fluid-flow-in-porous-media/ Fundamentals of Fluid Flow in Porous Media]
• [https://geology.buzz/threads/porosity.104/ Geology Buzz: Porosity]
• [https://careers.slb.com/inside-schlumberger/transformative-technology/defining-permeability Defining Permeability ]
• [https://imechanica.org/files/JCIS-2019-Tailoring%20porous%20media%20for%20controllable%20capillary%20flow_0.pdf Tailoring porous media to control permeability]
• [http://perminc.com/resources/fundamentals-of-fluid-flow-in-porous-media/chapter-2-the-porous-medium/permeability/ Permeability of Porous Media]
• [http://techalive.mtu.edu/meec/module06/Permeability.htm Graphical depiction of different flow rates through materials of differing permeability]

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