A New Approach in Hydraulic Flow Unit Determination and its Application

for Prediction of Petrophysical Properties

By: Mohammad Izadi

Supervisor: Dr Ali Ghalambor

Contents:

1) Quick Review

2) Mathematical Modeling

3) Future Work

Quick Review:

- Reservoir Characterization

- Rock Typing

- Hydraulic Flow Unit

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 dr

x r dr dr

υµ

∆ =∆

New Assumption:

R r

2 2 2 2

2 2water

wpore

V R L r L R rS

V R L R

π ππ

− −= = = 1 wr R S= −

New Radius and Bundle of capillary tube

Mathematical Modeling (cont.)

Mathematical Modeling (cont.)

1 1p d dr

x r dr dr

υµ

∆ =∆

2 21( ) [ (1 ) ]

4 w

pr R S r

xυ

µ∆= − −∆

4 21 2 3

0

. . (1 )[ (1 ) ]

2 8

wR Sw

w

p R SprR S r dr

x x

ππµ µ

− ∆ −∆Φ = − − =∆ ∆∫

4 2(1 )

8wp r S

qL

πµ

∆ −=∆

ckA pq

Lµ∆=

4 2(1 )

8wn p r S

qL

πµ

∆ −=∆

Coupling of Darcy and Poiseuille Equations

4 2(1 )

8w

c

n r Sk

A

π −=

2p

b c

V n r L

V A L

πφ = =2

c

n rA

πφ

=

Therefore, 2 2(1 )

8wr S

kφ−=

Mathematical Modeling (cont.)

Kozeny (1927) defined two following parameters:

2

(2 ) 2

( )P

sV

p

A n rLS

V n r L r

ππ

= = = (internal surface area per unit of pore volume)

2(2 ) 2 1( )

(1 ) (1 )gr

sV

gr c c

A n rL nrS

V A L A r

π πφ φ

= = =− −

(total area exposed within the pore space per unit of grain volume )

Combining equations ( )1gr pV VS S

φφ

=−

Substituting

32

2 2

1(1 )

2 (1 )gr

wV

k SS

φφ

= × × −−

Mathematical Modeling (cont.)

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

kS

φφ τ

=−

The generalized form of Kozeny-Carmen relationship is given by the equation

3

2 2 2

1[ ]

(1 ) s gv

kF S

φφ τ

=−

Fs : Shape Factor, 2 for circular cylinder

2sFτ : Kozeny constant

Mathematical Modeling (cont.)

With the analogy to previous equations

32

2 2 2

1(1 )

(1 )gr

ws V

k SF S

φτ φ

= × × −−

To define the HFU model we need to rearrange the equation to:

We propose two approaches and define new terms.

Mathematical Modeling (cont.)

Approach 1 1 1

(1 ) (1 ).grw s V

k

S F S

φφ φτ

× = ×− −

Approach 2 (1 )1

(1 ).gr

w

s V

Sk

F S

φφ φτ

−= ×−

1

(1 )w

k

Sφ×

− Modified Reservoir Quality Index (MRQI).

1

.grs VF Sτ

Flow Zone Indicator (FZI)

(1 )

φφ−

Normalized Porosity

k

φReservoir Quality Index (RQI)

(1 )

(1 )wSφ

φ−− Modified Normalized Porosity

Mathematical Modeling (cont.)

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

Thank you!

Questions?