Page 314 - Fundamentals of Reservoir Engineering

REAL GAS FLOW: GAS WELL TESTING 249

Note that in reaching this stage it has not been necessary to make any restrictive
assumptions about the viscosity being independent of pressure or that the pressure
gradients are small and hence squared pressure gradient terms are negligible, as was
implicit in the approach of Russell and Goodrich.

Therefore, the problem has already been partially solved but it should be noted that the
term φµc/k in equ. (8.11) is not a constant, as it was in the case of liquid flow, since for
a real gas both µ and c are highly pressure dependent. Equation (8.11) is therefore, a
non-linear form of the diffusivity equation.
Continuing with the argument; in order to derive an inflow equation under semi-steady
state flow conditions, then applying the simple material balance for a well draining a
bounded part of the reservoir at a constant rate

∂ p ∂ V
cV =− = − q (5.8)
t ∂ t ∂

and for the drainage of a radial volume element

∂ p q
=− (5.10)
2
φ
t ∂ π rh c
e
Also, using equ. (8.9)

()
∂ mp 2p ∂ p 2p q
= ⋅ = − ⋅ (8.12)
2
t ∂ µΖ t ∂ µΖ π rh c
φ
e
and substituting equ. (8.12) in (8.11) gives
()
1 ∂ r ∂ mp =− φµ c ⋅ 2p ⋅ q
2
φ
r ∂ r r ∂ k µΖ π r h c
e
or

()
1 ∂ r ∂ mp =− 2 pq (8.13)
2
r ∂ r r ∂ π r kh Ζ res
e
Furthermore, using the real gas equation of state,

pq T
= p sc q sc
Ζ res T sc

equ. (8.13) can be expressed as

()
1 ∂ r ∂ mp =− 2p q sc ⋅ T (8.14)
sc
2
r ∂ r r ∂ π rkh T sc
e