Page 130 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

110 Thomas Russell et al.

3.4.3 Calculation of impedance
The next step is to calculate the impedance of the well. The impedance is
defined as the normalized inverse of the well index.

II 0ðÞ
JTðÞ 5 ; (3.85)
II TðÞ
where the well index is the flow rate per unit pressure drop:
q
II TðÞ 5 : (3.86)
ΔP

At T 5 0 there are no suspended particles. Therefore, the initial rate
follows from:

2πk @p
q 0 5 r : (3.87)
μ @r

The initial pressure drop is given by:
q 0 μ
Δp 0 52 ln X w Þ: (3.88)
ð
4πk
The dimensionless initial pressure drop is therefore:
1
ΔP 0 52 ln X w Þ: (3.89)
ð
2
It will from this point be assumed that the production rate from the
well is constant. From Eq. (3.85), it then follows that the impedance in
the damaged zone is:

ð 1
ΔPTðÞ βφS ai S s
J D 5 5 1 2 dX: (3.90)
ð
ΔP 0 ln X w Þ X
X w
The total well index is the sum of that in damaged zone and that in
undamaged zone:
ΔPTðÞ 1 P e 2 P i
J 5 J D 1 J UD 5 : (3.91)
ΔP 0
An example calculation of impedance is shown in Fig. 3.14 with field
data taken from an oil-producing well. The associated critical retention
function is given in Fig. 3.15. The fitted impedance curve using the
above model (black curve) shows good agreement with the data.