Permeability 2016-10-25T11:54:45+00:00

Fundamentals of Fluid Flow in Porous Media

 

Chapter 2

Permeability

Permeability is a property of the porous medium that measures the capacity and ability of the formation to transmit fluids. The rock permeability, k, is a very important rock property because it controls the directional movement and the flow rate of the reservoir fluids in the formation. This rock characterization was first defined mathematically by Henry D’ Arcy in 1856. By analogy with electrical conductors, permeability represents reciprocal of residence which porous medium offers to fluid flow.

Poiseuille’s equation for viscous flow in a cylindrical tube is a well-known equation
Poiseuille's Equation for Viscous Flow in Cylindrical Tube
Where:

  • v = fluid velocity, cm/sec
  • d = tube diameter, cm
  • ∆P = pressure loss over length L,
  • μ = fluid viscosity, centipoise
  • L = length over which pressure loss is measured, cm

A more convenient form of Poiseuille’s equation is
Poiseuille's Equation Convenient Form
If assume that the rock is consist of a lot of tube in different group with different radius, total flow rate from this system by using the equation (2‑18)

Example 2-4
Derive Poiseuille’s equation for viscous flow in a horizontal cylindrical tube.

Solution
Consider a horizontal flow in a circular pipe. Assume a disc shape element of the fluid in the middle of the cylinder that is concentric with the tube and with radius equal to rw and length equal to ∆L. The forces on the disc are due to the pressure on the upstream and downstream face of the disc and shear force over the rim of the element Figure 2‑20.

Flow through a Pipe
Figure 2-20: Flow through a Pipe

According to the steady state condition:
Steady State Condition Equation of Permeability

After simplification
Steady State Condition Equation of Permeability Simplified

According to the shear stress definition and using equation (2‑21) and then rearranged and integration on both sides
shear stress definition and Steady State Condition Equation of Permeability and then rearranged and integration on both sides

Maximum velocity at r=0 so
Maximum velocity at r=0 equation

For viscous flow in the cylindrical tube
Viscous Flow in Cylindrical tube equation

From (2‑21), (2‑24) and (2‑25)
Permeability Equation Re-arranged

d = 2rw , so after rearrangement
Poiseuille's equation

A cast of the flow channel in a rock formation is shown in (Figure 2‑21). It is seen that the flow channels are of varying sizes and shapes and are randomly connected. So it is not correct to use the Poiseuille’s equation for flow in the porous media.

Metallic cast of pore space in a consolidated sand
Figure 2-21: Metallic cast of pore space in a consolidated sand

[1]

In 1856, D’ Arcy (commonly known as Darcy) developed a fluid flow equation that has since become one of the standard mathematical tools of the petroleum engineer. He investigated the flow of water through sand filter for water purification. His experimental apparatus showed schematically in (Figure 2‑22)

Schematic Drawing of Darcy Experiment of Flow of Water through Sand
Figure 2-22: Schematic Drawing of Darcy Experiment of Flow of Water through Sand[2]

Darcy interpreted his observation so as to yield result essentially as following equation:
Darcy's Equation

Here Q represents the volume rate of flow of water downward through the cylindrical sand pack of cross sectional area A and height h and K is a proportionality constant. The sand pack was assumed to be fully saturated with water. Later investigator found that Darcy’s law could be extended to other fluid as well as water and that the constant of proportionality K could be written as ( K / μ ) . The final form of the Darcy equation for horizontal linear flow of an incompressible fluid is established through a core sample of length L and a cross-section of area A is
Final form of Darcy Equation for horizontal linear flow of incompressible fluid

Where:

  • K = Proportionality constant or permeability, Darcy’s
  • μ = Viscosity of flowing fluid, cp
  • ( dP / dL ) = Pressure drop per unit length, ( atm / cm )
  • Q = Volumetric flow rate, ( cm3 / sec )

This is a linear law, similar to Newton’s law of viscosity, Ohm’s law of electricity, Fourier ‘s law of heat conduction, and Fick’s law of diffusion.

Substituting the relationship u = Q/A, in equation (2‑28) results in
Darcy's law with v=Q/A

The velocity, u, in Equation 2-29 is not the actual velocity of the flowing fluid but is the apparent velocity determined by dividing the flow rate by the cross-sectional area across which fluid is flowing.

“Darcy” is a practical unit of permeability (in honor of Henry Darcy). A porous material has permeability equal to 1 Darcy if a pressure difference of 1 atm will produce a flow rate of 1 cm3/sec  of a fluid with 1 cP viscosity through a cube having side 1 cm in length. Thus
Dary's Law Equation

One Darcy is a relatively high permeability as the permeabilities of most reservoir rocks are less than one Darcy. In order to avoid the use of fractions in describing permeabilities, the term millidarcy is used. As the term indicates, one millidarcy, i.e., 1 md, is equal to one-thousandth of one Darcy. The negative sign in equation (2‑29) is necessary as the pressure increases in one direction while the length increases in the opposite direction.

Permeability correlations  Radial Version of Darcy’s Law  Permeability and conductivity
ADD TEXT TO SECTION (Section Not Complete)

Example 2-5
Find the Darcy’s equation for isothermal flow of ideal gas.

Solution
Multiply both side of equation (2-28) by density so we have a mass flow equation:

Mass flow equation

But because of isothermal flow, Q x ρ = Qb x ρb which “b” is as a “base condition”. For ideal gas at constant temperature: ρ = ρb ( ρ / ρb). Therefore
Mass flow equation with isothermal flow

Separating variable and integrating
integrating ideal gas constant temp and mass flow equation

Define ̅P as ( P1 + P2 ) / 2  and ̅Q as flow rate at P = ̅P . Then: ̅PQ = PbQb. Substituting in equation (2‑32)
Flow rate of ideal gas could be found from Darcy's law for incompressible fluid when flow rate defined at mean pressure

There for flow rate of ideal gas could be found from the Darcy’s law for the incompressible fluid when the flow rate defined at the algebraic mean pressure.

References

[1] Amyx (1960)

[2] Hubbert (1954)

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