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Mathematical Relations

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This slides shows the proof of new hydraulic flow unit

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Mathematical Relations

  1. 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. 2. Contents: 1) Quick Review 2) Mathematical Modeling 3) Future Work
  3. 3. Quick Review: - Reservoir Characterization - Rock Typing - Hydraulic Flow Unit
  4. 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. 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. 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. 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. 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. 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. 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. 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. 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. 13. Thank you! Questions?

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