Page 165 - Fundamentals of Reservoir Engineering

DARCY'S LAW AND APPLICATIONS 104

each pore leading to an apparent increased permeability. This phenomenon, which is
3
called the Klinkenberg effect , seldom enters reservoir engineering calculations but is
important in laboratory experiments in which, for convenience, rock permeabilities are
determined by measuring air flow rates through core plugs at pressures close to
4
atmospheric. This necessitates a correction to determine the absolute permeability .

Due to its very low viscosity, the flow velocity of a real gas in a reservoir is much
greater than for oil or water. In a limited region around the wellbore, where the pressure
drawdown is high, the gas velocity can become so large that Darcy's law does not fully
5
describe the flow. This phenomenon, and the manner of its quantification in flow
equations for gas, will be fully described in Chapter 8, sec. 6.

4.3 SIGN CONVENTION

Darcy's empirical law was described in the previous section without regard to sign
convention, it being assumed that all terms in equ. (4.8) were positive. This is adequate
if the law is being used independently to calculate flow rates; however, if equ. (4.8) is
used in conjunction with other mathematical equations then, just as described in
connection with the definition of thermodynamic compressibility in Chapter 1, sec. 4,
attention must be given to the matter of sign convention.

Linear flow

If distance is measured positive in the direction of flow, then the potential gradient dΦ/dl
must be negative in the same direction since fluids move from high to low potential.
Therefore, Darcy's law is

kρ dΦ
u =− (4.9)
µ dl

Radial flow

If production from the reservoir into the well is taken as positive, which is the
convention adopted in this book, then, since the radius is measured as being positive in
the direction opposite to the flow, dΦ/dr is positive and Darcy's law may be stated as

kρ dΦ
u = (4.10)
µ dr

4.4 UNITS: UNITS CONVERSION

In any absolute set of units Darcy's equation for linear flow is

kρ dΦ
u = (4.9)
µ dl

in which the various parameters have the following dimensions

2
3
2
u = L/T; ρ = M/L ; µ = M/LT; I = L and Φ (potential energy/unit mass) = L /T . Therefore,
the following dimensional analysis performed on equ. (4.9):