Page 173 - Fundamentals of Reservoir Engineering

DARCY'S LAW AND APPLICATIONS 111

and substituting this in equ. (4.6) gives the real gas potential as

RT p Zdp
Φ= + gz (4.19)
M p b p

But, since

RT Z dp
dΦ= dp gdz = + gdz (4.20)
+
Mp ρ

then the gradient of the gas potential in the flow direction is simply

dΦ = 1 dp + g dz (4.21)
dl ρ dl dl

and Darcy's equation for linear flow is again

ρ
kdΦ k dp dz
u =− =− + ρ g (4.12)
µ dl µ dl dl
The above merely illustrates that real gas flow can be described using precisely the
same form of equations as for an incompressible liquid.

4.6 DATUM PRESSURES

An alternative way of expressing the potential of any fluid is

ψ = ρΦ = p + ρgz

where ψ is the psi-potential and has the units-potential per unit volume. Using this
function, Darcy's law becomes

ρ
kA dΦ kA dψ
q =− =− (4.22)
µ dl µ dl
The ψ potential is also frequently referred to as the "datum pressure", since the
function represents the pressure at any point in the reservoir referred to the datum
plane, as illustrated in fig. 4.3.