This document proposes a new approach to determining hydraulic flow units (HFU) and predicting petrophysical properties through mathematical modeling. It presents equations modeling single-phase laminar flow through cylindrical capillaries, and combines these with the Kozeny-Carmen equation relating permeability and porosity. The modeling results in two proposed approaches to define new terms for a HFU model, including a modified reservoir quality index and flow zone indicator, aimed at characterizing reservoirs and upscaling from core to log data. Future work involves applying available core analysis data to test the proposed models.

1. A New Approach in Hydraulic Flow
Unit Determination and its Application
for Prediction of Petrophysical
Properties
By:
Mohammad Izadi
Supervisor: Dr Ali Ghalambor

2. Contents:
1) Quick Review
2) Mathematical Modeling
3) Future Work

3. Quick Review:
- Reservoir Characterization
- Rock Typing
- Hydraulic Flow Unit

4. Mathematical Modeling
- Poiseuille Equation
Assumptions:
- Steady state flow
- Laminar flow
- One phase flow
- No-slip flow at the wall
- Capillary size tube (micron size
diameter)
1 1p d d
r
x r dr dr
υ
µ
∆
=
∆

5. New Assumption:
R r
2 2 2 2
2 2
water
w
pore
V R L r L R r
S
V R L R
π π
π
− −
= = = 1 wr R S= −
New Radius and Bundle of capillary tube
Mathematical Modeling (cont.)

6. Mathematical Modeling (cont.)
1 1p d d
r
x r dr dr
υ
µ
∆
=
∆
2 21
( ) [ (1 ) ]
4
w
p
r R S r
x
υ
µ
∆
= − −
∆
4 2
1
2 3
0
. . (1 )
[ (1 ) ]
2 8
wR S
w
w
p R Sp
rR S r dr
x x
ππ
µ µ
− ∆ −∆
Φ = − − =
∆ ∆∫
4 2
(1 )
8
wp r S
q
L
π
µ
∆ −
=
∆

7. ckA p
q
Lµ
∆
=
4 2
(1 )
8
wn p r S
q
L
π
µ
∆ −
=
∆
Coupling of Darcy and Poiseuille Equations
4 2
(1 )
8
w
c
n r S
k
A
π −
=
2
p
b c
V n r L
V A L
π
φ = =
2
c
n r
A
π
φ
=
Therefore,
2 2
(1 )
8
wr S
k
φ−
=
Mathematical Modeling (cont.)

8. Kozeny (1927) defined two following parameters:
2
(2 ) 2
( )P
s
V
p
A n rL
S
V n r L r
π
π
= = = (internal surface area per unit of pore volume)
2
(2 ) 2 1
( )
(1 ) (1 )gr
s
V
gr c c
A n rL nr
S
V A L A r
π π
φ φ
= = =
− −
(total area exposed within the pore space per unit of grain volume )
Combining equations ( )
1gr pV VS S
φ
φ
=
−
Substituting
3
2
2 2
1
(1 )
2 (1 )gr
w
V
k S
S
φ
φ
= × × −
−
Mathematical Modeling (cont.)

9. Kozeny and Carmen (1937) obtained the following relationship by
defining the tortuosity and replacing in Darcy equation:
3
2 2 2
1
[ ]
(1 ) 2 gv
k
S
φ
φ τ
=
−
The generalized form of Kozeny-Carmen relationship is given by the equation
3
2 2 2
1
[ ]
(1 ) s gv
k
F S
φ
φ τ
=
−
Fs : Shape Factor, 2 for circular
cylinder
2
sFτ : Kozeny constant
Mathematical Modeling (cont.)

10. With the analogy to previous equations
3
2
2 2 2
1
(1 )
(1 )gr
w
s V
k S
F S
φ
τ φ
= × × −
−
To define the HFU model we need to rearrange the equation to:
We propose two approaches and define new terms.
Mathematical Modeling (cont.)

11. Approach 1
1 1
(1 ) (1 ). gr
w s V
k
S F S
φ
φ φτ
× = ×
− −
Approach 2
(1 )1
(1 ). gr
w
s V
Sk
F S
φ
φ φτ
−
= ×
−
1
(1 )w
k
Sφ
×
− Modified Reservoir Quality Index (MRQI).
1
. grs VF Sτ
Flow Zone Indicator (FZI)
(1 )
φ
φ−
Normalized Porosity
k
φ
Reservoir Quality Index (RQI)
(1 )
(1 )
wSφ
φ
−
−
Modified Normalized Porosity
Mathematical Modeling (cont.)

12. Future Work
1) Completing Routine and Special Core Analysis
2) Applying available data to proposed models and
comparing to the existing model
3) Up scaling with log analysis

13. Thank you!
Questions?