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![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
∆ −
=
∆](https://image.slidesharecdn.com/2dg-124123488846-phpapp01/85/Mathematical-Relations-6-320.jpg)
![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.)](https://image.slidesharecdn.com/2dg-124123488846-phpapp01/85/Mathematical-Relations-9-320.jpg)
Mathematical Relations
- 1. A New Approachin 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: - ReservoirCharacterization - Rock Typing - Hydraulic Flow Unit
- 4. Mathematical Modeling - PoiseuilleEquation 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 22 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.) 11p 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) definedtwo 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 analogyto 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) CompletingRoutine and Special Core Analysis 2) Applying available data to proposed models and comparing to the existing model 3) Up scaling with log analysis
- 13.