Fluid flow in porous media and Carter’s equation¶

Incompressible flow¶

The incompressible fluid refers to the fluid in which the density of a fluid parcel (infinitesimal control volume) is independent of the pressure under the constant temperature. The equation of state can be written as:

where: \(V, p\) and \(\rho\) are the bulk volume, net pressure and density of the flow, respectively.

The incompressible fluid does not exist in the real world and it is only used to simplify some mathematical models.

Weakly compressible fluid¶

The weakly compressible fluid is the fluid such that the volume change is proportional to the pressure change under the constant temperature with a constant proportional coefficient known as the isothermal compressibility \(c_1\). The equation of state can be written as:

where: the constant \(\rho_0\) and \(p_0\) are the density and net pressure at a reference state.

The second equation of Eqn (3) can be rewritten as Eqn (4) by ignoring higher-order terms of its Taylor expansions:

Usually, liquid can be treated as weakly compressible fluid because the compressibilities are of order of \(10^{-4}\) . The fluid discussed in this document refers to the weakly compressible fluid, unless specified.

Compressible fluid¶

Compressible fluid refers to the fluid whose density changes significantly with pressure or temperature change. Gas usually behaves as compressible fluid that can be described by the real gas law:

where:

• \(p\) is the net fluid pressure, with unit [\(MPa\)]
• \(Z\) is the real gas compressibility factor, dimensionless
• \(V\) is the gas volume, with unit [\(m^3\)]
• \(V\) is the gas volume, with unit [\(kmol\)]
• \(R\) is the real gas constant, with unit [\(\frac{MPa \cdot m^3}{kmol \cdot K}\)]
• \(T\) is the gas temperature, with unit [\(K\)]

All the parameters in this document are with S.I. unit, unless specified.

And the isothermal gas compressibility is defined as:

The subscript \(T\) represents that the partial derivative is taken at constant temperature.

When \(Z=1\), the gas becomes ideal and Eqn (6) is reduced to the Boyle-Mariotte law:

Preliminaries¶

Before going to the Darcy law, some preliminary concepts need clarifying by presenting two concepts:

• The velocity of fluid flow through porous media \(v\): It is the volumetric rate of the fluid flow \(Q\) passing through a unit cross-section \(A\) of the porous media:

  (8)\[v = \frac{Q}{A}\]

• The average velocity(seepage) of fluid particle \(U\): It is defined as the volumetric rate of the fluid flow \(Q\) passing through a unit cross-section \(A_{effective}\) of the porous network:

  (9)\[U = \frac{Q}{A_\mathrm{effect}}=\frac{Q}{A\phi}\]

The two concepts are illustrated in Figure 3 in which the dash arrows indicate the flow does not pass through the skeleton but he whole area is used for calculation.

Figure 3: Velocity and seepage

It is necessary to define the seepage of the fluid particle \(U\) because the pore structure of porous media is so complicated that the velocity of one fluid particle inside is of little significance to characterize the whole fluid flow. The physical meaning of \(U\) is the average of the velocity field of a porous media control volume. The relation of \(v\) and \(U\) is:

Eqn (10) is also known as the Dupuit-Forchheimer relation.

Darcy law¶

The Darcy law was first deduced by Darcy as an empirical equation (Eqn (11)) to describe the one-dimensional (\(1D\)) flow for a series experiment in which he varied the amount of water flowing through a cross-sectional area \(A\) of sand filling tube as shown in Figure 4 .

Figure 4: A schematic view of Darcy experiment [3]

The result of the Darcy experiment indicate that the volumetric rate \(Q\) of the fluid flow passing through the cross-section is proportional to the cross-section \(A\), inversely proportional to the tube length \(L\) and proportional to the piezometric head difference of the two sides of the tube (\(\triangle h = h_1-h_2\)) in a quasistatic process, which means the system passes continuously through a sequence of equilibrium states [4]

\(K\) is the hydraulic conductivity defined as:

where:

• \(k\) is the permeability of the porous media, with S.I. unit [\(m^2\)], which is a measure of the ability of porous media to allow fluids to pass through it
• \(\mu\) is the fluid viscosity, with S.I. unit [\(Pa \bullet sec\)]

For the fluid per unit mass in a quasistatic process, the energy conservation yields:

The three terms at the right-hand side of Eqn (13) represent the pressure potential, position potential energy and kinetic energy of this unit-mass fluid, respectively. And \(H\) is the Hubbert’s potential energy per unit mass of fluid. Notice that \(z\) is the elevation of the point at which the piezometric head is being considered above the datum level and has the relation with the piezometric head \(h\) :

Because the seepage is small, the kinetic term can be ignored then \(h=H\) . Then we can write \(h\) for point \(1\) and \(2\) in Figure 4 , respectively:

Since \(z_1-z_2=L\sin\theta\):

where: \(\theta\) is the inclination of the tube with respect to the horizontal datum.

Substitution of Eqn (16) and Eqn (12) into Eqn (11) :

Eqn (17) is the one-dimensional Darcy law.

The ” \(-\) “sign is due to the opposition of the velocity direction and pressure increase direction.

The differential equation of Eqn (17) is:

When the gravity is ignored, the permeability \(k\) becomes a second-rank tensor for a three-dimensional Darcy law in tensor form:

where:

• the subscripts \(i\) and \(j\) represent the coordinate components \(x\) , \(y\) and \(z\);
• the subscript comma “,” represent the partial derivative, i.e. \(p,_j=\frac{\partial p}{\partial j}\)
• it follows the Einstein summation convention that when an index variable appears twice in a single term, it implies summation of that term over all the values of the index: \(k_{ij}p,_j=k_{ix}\frac{\partial p}{\partial x}+k_{iy}\frac{\partial p}{\partial y}+k_{iz}\frac{\partial p}{\partial z}\)

The S.I. unit of the permeability is \(m^2\) while another practical unit Darcy [D] is also used in honor of the developer of the Darcy law such that \(1(D)=1(\mu m)^2\).

Constant permeability and porosity¶

The governing equation of one-dimensional fluid flow in a homogeneous and isotropic porous medium is derived from the combination of:

• The fluid equation of state (weakly compressible assumed)

  (22)\[\begin{split}\left\{ \begin{align} & c_1 = -\frac{1}{V}\frac{dV}{dp} = \frac{1}{\rho}\frac{d\rho}{dp} \\ & \rho = \rho_0 [1+c_1(p-p_0)] \\ \end{align} \right.\end{split}\]

• The Darcy law without body force:

  (23)\[v_x = -\frac{k}{\mu} \frac{\partial p}{\partial x}\]

• The continuity equation:

  (24)\[\frac{\partial (\phi \rho)}{\partial t} = -\frac{\partial (\rho v_x)}{\partial x}\]

Substitute Eqn (23) into the right-hand side of Eqn (24):

From Eqn (22):

Substitute Eqn (26) and Eqn (22) into Eqn (25):

Substitute Eqn (27) into the left-hand side of Eqn (24):

Combine Eqn (27) and Eqn (28):

With the definition of the piezometric conductivity:

Eqn (29) can be rewritten as:

Eqn (31) is the governing equation of the 1D Newtonian laminar flow in a homogeneous isotropic porous medium by assuming the fluid is weakly compressible and the permeability and porosity are both constant.

Elastic porous medium¶

If the porous medium is elastic, i.e. the permeability and porosity are both functions of pressure in the similar form of the density:

where: \(k_0\) and \(\phi_0\) are constant permeability and porosity at the reference state

Then the left side of Eqn (24) becomes:

Because all the compressibilities are of order \(10^{-4}\) , Eqn (33) can be written as:

The right side of Eqn (24) becomes:

Because the term \(\frac{\partial^2 p}{\partial x^2}\) and \((\frac{\partial p}{\partial x})^2\) are of the same order, we have:

And the compressibilities are of order of \(10^{-4}\): \((c_1+c_k)(p-p_0) \ll 1\) , thus Eqn (35) can be rewritten as:

Combine Eqn (34) and Eqn (36) with introducing the total compressibility \(c_t=c_1+c_\phi\) :

And: the conductivity \(K=\frac{k_0} {\mu \rho_0 c_t}\)

Eqn (37) is the governing equation of the 1D Newtonian laminar flow in a homogeneous isotropic elastic porous medium.

To sum up, Eqn (37) and Eqn (31) can be unified by defining on the conductivity \(K\) based on the total compressibility \(c_t\).

Analytical solution to the 1D fluid flow leakoff¶

An element of one pressurized fracture with leakoff can be illustrated in Figure 6 . An infinite homogeneous and isotropic porous medium is located at the positive part of \(x\) axis while the interface between the medium and fracture is located at \(x=0\) (\(y\) axis), where the fluid pressure keeps constant \(p_c\) since time \(t=0\) . The fluid is leaking with velocity \(v\) through the element to the porous medium and reaches the infinity with pressure dropping to the constant pore pressure \(p_0\). At the initial time \(t=0\) , the fluid pressure within the medium is \(p_0\) everywhere.

Figure 6: Schematic view of one pressurized crack with leakoff

Then, the fluid flow in the porous medium can be treated as a 1D seepage model based on the governing equation Eqn (38)

with boundary conditions and initial condition as:

The boundary conditions and initial condition are “self-similar” (see here for details), and the analytical solution is:

Derivation of the Carter’s equation¶

Once we get the analytical solution as Eqn (40), the leakoff velocity at the fracture/medium interface can be obtained with the following procedure:

Substitution of Eqn (41) into Darcy law Eqn (19) for 1D fluid in homogeneous and isotropic porous medium:

At the fracture/medium interface \(x=0\) , the leakoff velocity may be written with defining the pressure drop from the interface to remote \(\triangle p = p_c-p_0\) :

\(v_L\) is called leakoff velocity in hydraulic fracturing to measure how fast the fluid inside a fracture leaks to the surrounding porous medium. If it takes time \(t_0\) for the element to get exposed to the fluid, the \(t\) inside Eqn (43) should be replaced by \(t-t_0\) because this time-square-root term in the denominator represents how long the element has been exposed (\(t-t_0\)) instead of the actual time \(t\) :

Then the well-known Carter’s leakoff coefficient comes to us by [6]:

Then Eqn (44) can be rewritten as:

Eqn (46) is the Carter’s equation [6] to model the leakoff in hydraulic fracturing, which is based on the solution to 1D fluid flow in a porous medium .