Page 190 - Fundamentals of Reservoir Engineering

RADIAL DIFFERENTIAL EQUATION FOR FLUID FLOW 128

dr

h qρ | r qρ | r + dr

r e r w
r

Fig. 5.1 Radial flow of a single phase fluid in the vicinity of a producing well.

Consider the flow through a volume element of thickness dr situated at a distance r
from the centre of the radial cell. Then applying the principle of mass conservation
Mass flow rate − Mass flow rate = Rate of change of mass in
IN OUT the volume element

qρ rdr − qρ r = 2rh dr ∂ ρ
πφ
+
t ∂
where 2πrhφdr is the volume of the small element of thickness dr. The left hand side of
this equation can be expanded as

∂ (q ) ∂ ρ
ρ
π φ
qρ r + dr − qρ r = 2 rh dr
r ∂ t ∂
which simplifies to

ρ
∂ (q ) ∂ ρ
= 2rh (5.2)
πφ
r ∂ t ∂
By applying Darcy's Law for radial, horizontal flow it is possible to substitute for the flow
rate q in equ. (5.2) since

π
2 khr ∂ p
q =
µ r ∂
giving

∂ 2khr ∂ p ∂ ρ
π
π
ρ = 2rhφ
r ∂ µ r ∂ t ∂
or

1 ∂ kρ ∂ p ∂ ρ
ρ = φ (5.3)
rr ∂ µ r ∂ t ∂

The time derivative of the density appearing on the right hand side of equ. (5.3) can be
expressed in terms of a time derivative of the pressure by using the basic
thermodynamic definition of isothermal compressibility