Fundamentals of Fluid Flow in Porous Media
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
- 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
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)
Derive Poiseuille’s equation for viscous flow in a horizontal cylindrical tube.
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.
Figure 2-20: Flow through a Pipe
According to the steady state condition:
According to the shear stress definition and using equation (2‑21) and then rearranged and integration on both sides
Maximum velocity at r=0 so
For viscous flow in the cylindrical tube
From (2‑21), (2‑24) and (2‑25)
d = 2rw , so after rearrangement
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.