Fluid flow through porous media

In fluid mechanics, fluid flow through porous media is the manner in which fluids behave when flowing through a porous medium, for example sponge or wood, or when filtering water using sand or another porous material. As commonly observed, some fluid flows through the media while some mass of the fluid is stored in the pores present in the media.

Governing law

Symbol	Description
$Q$	Volumetric flow rate [m^3/s]
$\kappa$	Permeability of porous medium [m^2]. The permeability is a function of material type, and also varies with stress, temperature, etc.
$\mu$	Fluid viscosity [Pa.s]
$A$	Cross-sectional area of Porous medium [m^2]
${\displaystyle (p_{b}-p_{a})}$	Pressure drop across medium [Pa]
$L$	Length of sample [m]

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.^[1]

$Q=\frac{-\kappa A}{\mu} \frac{(p_b - p_a)}{L}$

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

${\displaystyle Q={\frac {-\kappa A}{\mu }}\left({\frac {dp}{dx}}\right)}$

Mass conservation

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.^[2] This means that total mass of the fluid is always conserved. In mathematical form, considering a time period from $t$ to $\Delta t$, length of porous medium from $x$ to $\Delta x$ and $m$ being the mass stored by the medium, we have

${\displaystyle [A(x)\rho (x)q(x)-A(x+\Delta x)\rho (x+\Delta x)q(x+\Delta x)]\Delta t=m(t+\Delta t)-m(t).}$

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

${\displaystyle {\frac {-d(\rho q)}{dx}}={\frac {d(\rho \phi )}{dt}}~~~~(i)}$

In three dimensions, the equation can be written as

${\displaystyle {\frac {d(\rho q)}{dx}}+{\frac {d(\rho q)}{dy}}+{\frac {d(\rho q)}{dz}}={\frac {-d(\rho \phi )}{dt}}}$

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

Diffusion Equation

Material Type	Compressibility (m^2N^−1 or Pa^−1)[3]
Clay	10^−6 - 10^−8
Sand	10^−7 - 10^−9
Gravel	10^−8 - 10^−10
Jointed rock	10^−8 - 10^−10
Sound Rock	10^−9 - 10^−11
Water (beta)	4.4 x 10^−10

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

${\displaystyle {\frac {d(\rho \phi )}{dt}}=\rho {\frac {d\phi }{dt}}+\phi {\frac {d\rho }{dt}}=\rho {\frac {d\phi }{dP}}{\frac {dP}{dt}}+\phi {\frac {d\rho }{dP}}{\frac {dP}{dt}}=\rho \phi \left[{\frac {1}{\phi }}{\frac {d\phi }{dP}}+{\frac {1}{\rho }}{\frac {d\rho }{dP}}\right]\quad {\frac {dP}{dt}}=\rho \phi [c_{\phi }+c_{f}]{\frac {dP}{dt}}~~~~(ii)}$

Here, $c_{f}$ = compressibility of the fluid and ${\displaystyle c_{\phi }}$ = compressibility of porous medium.^[4] Now considering the left hand side of the mass conservation equation, which is given by Darcy's Law as

${\displaystyle {\frac {-d(\rho q)}{dx}}={\frac {-d}{dx}}\left[{\frac {-\rho k}{\mu }}{\frac {dP}{dx}}\right]\quad ={\frac {k}{\mu }}\left[\rho {\frac {d^{2}P}{dx^{2}}}+{\frac {d\rho }{dP}}{\frac {dP}{dx}}{\frac {dP}{dx}}\right]\quad ={\frac {\rho k}{\mu }}\left[{\frac {d^{2}P}{dx^{2}}}+\left({\frac {1}{\rho }}{\frac {d\rho }{dP}}\right)\left({\frac {dP}{dx}}\right)^{2}\right]\quad ={\frac {\rho k}{\mu }}\left[{\frac {d^{2}P}{dx^{2}}}+c_{f}\left({\frac {dP}{dx}}\right)^{2}\right]\quad ~~~~(iii)}$

Equating the results obtained in ${\displaystyle (ii)}$ & ${\displaystyle (iii)}$, we get

${\displaystyle {\frac {d^{2}P}{dx^{2}}}+c_{f}\left({\frac {dP}{dx}}\right)^{2}={\frac {\phi \mu (c_{f}+c_{\phi })}{k}}{\frac {dP}{dt}}}$

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

${\displaystyle {\frac {dP}{dt}}={\frac {k}{\phi \mu c_{t}}}{\frac {d^{2}P}{dx^{2}}}}$

where ${\displaystyle c_{t}=c_{f}+c_{\phi }}$.^[5]

References

1. "Flow in porous media I: A theoretical derivation of Darcys law". Transport in Porous Media. 1: 3–25. doi:10.1007/BF01036523.
2. Bear, Jacob. Dynamics of fluids in Porous Media.
3. https://eng.ucmerced.edu/people/jfisher/.../EnveEss110_20081110.pdf
4. Ohirhian, Peter. "Steady State Compressible Fluid Flow in Porous Media".
5. Zimmerman, Dr. R.W. "Flow in porous media". imperial.ac.uk.

External links

• Fundamentals of Fluid Flow in Porous Media