1.3. Fluid flow through porous media 11
organic tissues (bones, cork), and man-made materials like cement and ceramics, which are all porous. Only by assuming that they are porous media, many of their significant features can be explained. Filtration, mechanics, petroleum engineering, bioremediation, and construction engineering are some of the fields that make use of porous media. Other fields include biology and biophysics, geomechanics and geoengineering that also use the concept of porous media. Bothsaturated and unsaturated porous media exist in the world of adsorbents. Saturated and unsaturated porous media both have empty spaces that are partly filled with fluid, while the voids in saturated porous media are completely filled.
Only a small portion of the space is occupied by the liquid, while the rest is occupied by air. Porosity is an essential word that defines how much "empty" space is there in a material. Let us consider a volume VT of a reasonably large porous medium such that
VT =VP +VS,
whereVP is the void volume (pore volume) whileVSis the solid material volume. Porosity, which is defined as the ratio of pore volume to total volume, may be represented and denoted as follows:
= VP
VT = VT −VS
VT . Types of porosity
• Primary porosity (The porosity of a rock or any other natural porous material that happens at the depositing time).
• Secondary porosity (The porosity that grows after deposition of the rock or any other natural porous material).
Further, porosity can be categorized into the following six types depending on the size of the pores:
• Total porosity (It is a ratio of the entire pore space in a rock to its bulk volume).
• Effective porosity (It is a measure of the void space that is filled by recoverable oil and gas),
porosity= Vol. of interconnected pores + Vol. of dead endTotal or bulk vol. of reservoir rock .
• Dual porosity (It is a conceptual idea that there are two overlapping reservoirs which interact. In fractured rock aquifers, the rock mass and fractures are often simulated as being two overlapping but distinct bodies).
• Microporosity (Pore size is smaller than 2 nanometers in diameter).
1.3. Fluid flow through porous media 13
• Mesoporosity (Pore size is greater than 2 nanometers and less than 50 nanometers in diameter).
• Macroporosity (Pore size is greater than 50 nanometers in diameter).
Porosity measuring method
There are several types of methods for finding porosity such as
• Direct method
• Optical method
• Computed tomography method
• Imbibition method
• Gas expansion method.
Motion in three dimensions of small amplitude waves in an undeformable porous medium is studied here with the medium being homogeneous and isotropic. When the seepage fluid velocityU~ and dynamic pressureP are taken into account, the fluid moves according to the continuity equation and the equation of motion such as
∇. ~U = 0, (1.32)
m∂ ~U
∂t + 1
ρ∇P +f ω ~U = 0, (1.33)
where the innertial coefficient is defined as
m= 1 +CM1−
, (1.34)
with CM as the added mass coefficient, f is the friction parameter and is the porosity.
The derivation and physical significance of (1.33) and (1.34) are described in Appendix A.
Now similarly, the irrotational flow inside the porous medium results in a pore velocity potential Φ(x, y, z, t) satisfying
U~ =−∇Φ. (1.35)
Integrating (1.33) gives theBernoulli’s equation for porous medium as follows:
m∂Φ
∂t +f ωΦ + P
ρ = 0. (1.36)
Considering time harmonic solutions in Cartesian coordinates (x, y, z), U , P~ and Φ can be expressed as
U~ =u(x, y, z)e−iωt, P =p(x, y, z)e−iωt, Φ = φ(x, y, z)e−iωt. (1.37) Substituting these values in (1.32) and (1.35), it results in
∇2φ = 0, (1.38)
which is the governing Laplace’s equation.
Further, substituting the values into (1.33) gives
∇p
ρ −iωγu= 0, (1.39)
whereγ =m+if. Following small amplitudeη(x, y, t)of the motion along the free surface and p=ρgη, the linearised dynamic boundary condition (1.36) at z = 0 becomes
η= iωγ
g Φ. (1.40)
The linearised kinematic condition is
∂Φ
∂z = ∂η
∂t at z = 0. (1.41)
Combining (1.40) and (1.41) gives us the mean free surface condition at z = 0 as
∂φ
∂z −γKφ= 0. (1.42)
For f = 0and m = 1, i.e., γ = 1, the porous medium gets converted into a water region and hence, the corresponding free surface condition (1.42) reduces to
∂φ
∂z −Kφ= 0. (1.43)
For a constant bottom, we can define the impermeable bottom condition as
∂φ
∂z = 0 at z =−h. (1.44)
Similar theory can be applied for the flow in a two-layer porous medium with free surface at z = 0, impermeable bottom at z = −h and linear interface at z = −hI. Hence, the
1.3. Fluid flow through porous media 15
potentials satisfy the governing equations as follows:
∇2φ1 = 0, for −hI ≤z ≤0, (1.45)
∇2φ2 = 0, for −h≤z≤ −hI. (1.46) The mean free surface condition at the upper layer gives
∂φ1
∂z −γKφ1 = 0. (1.47)
The impermeable bottom condition at the lower layer gives
∂φ2
∂z = 0. (1.48)
Further, the continuity of momentum from linearised Bernoulli’s equation ofporous medium gives another pair of boundary conditions as follows:
∂φ1
∂z = ∂φ2
∂z , (1.49)
ρ1 ∂φ1
∂z −γKφ1
=ρ2 ∂φ2
∂z −γKφ2
. (1.50)
For a two-layer porous medium flow, we consider a harmonic motion in they-direction as well as in time. Hence, the velocity potential Φj(x, y, z, t) for j = 1,2 in each fluid layer can be written asΦj(x, y, z, t) =Re
φj(x, z)ei(kyy−ωt)
where ky = k1sinθ with k1 being the incident wavenumber.
Similarly as in Section 1.2.1, substituting Φj in equations (1.45) and (1.46), the Laplace’s equation gets transformed into a Helmholtz equation and subsequently, by ap- plying equations (1.47) to (1.50), the spatial potentials can be written as
φ1(x, z) =e±iqxsinhk(h−hI)(Kγsinhkz+kcoshkz))
KγcoshkhI−ksinhkhI , (1.51) φ2(x, z) =e±iqxcoshk(z+h), (1.52) where q = p
k2−ky2 and (±) sign stands for the direction of the waves. Wavenumber k can be found from the dispersion relation, by using equations (1.51) and (1.52) in equation (1.50), as
K2γ2(ρ+ cothkhIcothk(h−hI))−kKγ(cothkhI+ cothk(h−hI)) +k2(1−ρ) = 0.
(1.53) For real γ, this equation has two pairs of symmetric real roots denoting the progressive modes and an infinite number of symmetric purely imaginary roots denoting evanescent
modes. However, due to complexγ, all the roots are complex of the formkn=±(an+ ibn) with an and bn are real and the existence of the roots can be found in Appendix B. Two sorts of solution of potential are produced depending on k.
The potential function for such an n-th mode can be written as φ1(x, z) =e±iqnxsinhkn(h−hI)(Kγsinhknz+kcoshknz))
KγcoshknhI−ksinhknhI , (1.54) φ2(x, z) =e±iqnxcoshkn(z+h). (1.55)
Darcy’s law for flow in porous media
By Darcy’s law, the movement of liquids and gases in porous media may be described. It was laboratory tests that first presented Darcy’s law as an empirical connection in 1856.
Incompressible flow through a homogeneous porous medium may be represented by using Darcy’s law in terms of laminar stationary flow as
U~ =−K µ∇P,
where,U~ andP represent the velocity and pressure of the fluid passing through the pores, respectively, K denotes the permeability of the medium and µ is the dynamic viscosity.
The relation between particle diameter (dp) and porosity () is defined by the Carman- Kozeny equation as follows:
K = d2pγ2 180(1−γ)2.
Permeability has meter2 as the dimension of measurement. The Darcy number is the ratio of permeability and the total length of the medium. The bigger the Darcy number, the larger the particle diameter and the smaller the pores are. The Darcy number 0 denotes the absence of any fluid in the medium.
Darcy’s law applies to fluid flow through porous medium when the structure is thin.
The pressure differential across the porous barrier determines the flow velocity. It is now possible to calculate the hydrodynamic pressure P(x, y, z, t) = Re
p(x, y, z)e−iωt from the linearized Bernoulli’s equation as p= iρωφ, where ρ is the density of the fluid.
Therefore, the condition at the interface can be written as W(x, y, z) = ibρω
µ (φ+−φ−) =G(φ+−φ−), (1.56) where±denotes the velocity potentials at the boundary of the porous wall. Dimensionless porous-effect parameter G = ibρω
µ is the term used by Chwang in [20], where k is the wavenumber. It may be expressed asG=GR+iGI whereGRsignifies the real component
1.4. Wave propagation over dissipative sea-beds 17