1. INSTRUCTOR © 2017, John R. Fanchi
All rights reserved. No part of this manual may be reproduced in any form without the express written permission of the author.
© 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

2. To the Instructor
The set of files here are designed to help you prepare lectures for your own course using the text Introduction to Petroleum Engineering, J.R. Fanchi and R.L. Christiansen (Wiley, 2017)
File format is kept simple so that you can customize the files with relative ease using your own style. You will need to supplement the files to complete the presentation topics.

3. PERMEABILITY © 2017, John R. Fanchi
All rights reserved. No part of this manual may be reproduced in any form without the express written permission of the author.
© 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

4. Outline Darcy’s Law and Permeability Permeability Averaging
Transmissibility
Measures of Permeability Heterogeneity
Darcy’s Law with Directional Permeability

5. DARCY’S LAW AND PERMEABILITY
© 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

6. Henri Darcy (1856) Empirical fit of water flow through packed sand
Observation
Flow rate proportional to pressure gradient
Assumptions:
1. Single phase water
2. Homogeneous sand
3. Vertical flow
4. Non-reactive fluid
5. Single geometry
© 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

7. Darcy’s Law – HorizontalQ A P1 P2
q = Flow rate
k = Proportionality constant
A = Cross-sectional area
μ = Fluid viscosity
p = Pressure
L = Length or distance of flow
Integrate assuming constant q, k, A, μ:
© 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

8. Permeability Defined Darcy’s Law q Fluid flow rate of 1.0 cc/sec 
Fluid viscosity of 1.0 cp A
Sample cross-sectional area of 1.0 cm2 L
Sample length of 1.0 cm Δp
= p1 - p2
= Pressure gradient of 1.0 atm k
Permeability of 1.0 darcy
Darcy’s Law

9. Permeability Dimensional Analysis
Permeability unit is “length squared” or “area”
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10. Permeability Interpretation
Permeability describes flow in a given sample for given fluid and set of experimental conditions
Permeability may be viewed as mathematical convenience
Describes statistical behavior of given flow experiment
Permeability has meaning as statistical representation of a large number of pores

11. Tortuosity
Area
Grain H
Pore
Flow
Path B
Flow
Path A

12. Actual vs. Apparent Velocity
Apparent velocity = velocity of fluid flowing through linear conduit
Actual Velocity = velocity of fluid flowing through tortuous path
Note: Vact > Vapp
© 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

13. Poiseuille’s Equation
Poiseuille’s Equation: calculate flow q through n cylindrical capillary tubes of radius r and length L
Where Atube = π r2
In Darcy’s Law, flow in a core can be approximated by:
Equating Poiseuille’s eq and Darcy’s law gives  - k relationship:
© 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

14. ϕ-k from Poiseuille’s Equation
 - k relationship from Poiseuille’s eq and Darcy’s law:
Permeability = k, ft2
Number of tubes = n
Radius of tube = r, ft
Cross-sectional area of core = A, ft2
Perm Conversion factors:
1 md = × m2 = 1.06 × ft2
© 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

15. Darcy’s Law for a Core Darcy’s law Volumetric flow rate = q, bbl/day
Permeability = k, md
Cross-sectional area of core = A, ft2
Viscosity = μ, cp
Pressure change across core = Δp, psia
Length of core = L, ft
© 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

16. Examples of Permeability
Porous Medium
Permeability
Coal
0.1 to 200 md
Shale
<0.005 md
Loose sand (well sorted)
1 to 500 md
Partially consolidated sandstone
0.2 to 2 d
Consolidated sandstone
Tight gas sandstone
< 0.01 md
Limestone
Diatomite
1 to 10 md
© 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

17. Qualitative Permeability Ranges
Range (md)
Description
≈ 10-6
Shale (nanodarcy)
≈ 10-3
Tight (microdarcy)
1 – 10
Low (millidarcy)
10 – 250
Moderate
> 250
High
© 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

18. Factors Affecting Permeability
1. Packing of grains
2. Rock texture
3. Grain size distribution
4. Angularity
5. Cementation
6. Rock compaction
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19. Methods of Estimating K
1. Well Test Analysis
2. Well Logs
3. Core Experiments
4. Statistical Correlations
5. Simulation Methods
Note the scale of measurement and the “directness” of each approach.
© 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

20. Direct Measurement vs. Correlation
Well test analysis assuming Darcy flow.
Measurement scale is effective drainage area.
Well Logs assume various correlations.
Measurement scale is inches to feet.
Core analysis is the most “direct”, although correlation takes place.
Measurement scale is inches.
Statistical Methods: neural networks and geostatistics blend information.
Inexact science.
Simulation: Attribute an effective perm at a coarse scale.
© 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

21. PERMEABILITY AVERAGING
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22. Flow Through Beds in Parallel
Arithmetic average
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23. Flow Through Beds in Series
Harmonic average
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24. TRANSMISSIBILITY
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25. Series application of Darcy’s Law for neighboring blocks
Flow Between Blocks
Series application of Darcy’s Law for neighboring blocks
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26. Transmissibility Estimate flow rate between blocks
with Darcy phase transmissibility between blocks
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27. MEASURES OF PERMEABILITY HETEROGENEITY
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28. Heterogeneity and Anisotropy
Homogeneous
(no special place)
Isotropic
(no special direction)
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29. HETEROGENEITY MEASURES
Heterogeneity: Spatial variability of permeability
DYKSTRA-PARSONS COEFFICIENT
(SPE 20156, Lake and Jensen, 1989)
LORENZ COEFFICIENT
(SPE 20156, Lake and Jensen, 1989)

30. DYKSTRA-PARSONS COEFFICIENT (SPE 20156, Lake and Jensen, 1989)
Estimate for log normal permeability distribution
Where kA = arithmetic average
And kH = harmonic average
Range: 0  VDP  1
Homogeneous Reservoir: VDP = 0 (in this case, KA = KH)
Increasing heterogeneity increases VDP
Typical Reservoir Values: 0.4  VDP  0.9

31. LORENZ COEFFICIENT (SPE 20156, Lake and Jensen, 1989)
Plot cum flow capacity Fm vs cum thickness Hm where
for n = number of reservoir layers.
Arrange layers in order of decreasing permeability
Thus I=1 has thickness h1 and the largest perm k1
While I=n has thickness hn and the smallest perm kn
By Definition, 0  Fm  1 and 0  Hm  1 for 0  m  n

32. THE LORENZ CURVE
Lorenz curve
1.0 C B Fm A
0.0
0.0 Hm
1.0
The Lorenz coefficient LC is 2*Area between the Lorenz curve ABC in the figure and the diagonal AC.

33. PROPERTIES OF THE LORENZ COEFFICIENT
Range: 0  LC  1
Homogeneous Reservoir: LC = 0
Increasing heterogeneity increases LC
Typical Reservoir Values: 0.2  LC  0.6
© 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

34. ESTIMATING THE LORENZ COEFFICIENT
Assume all perms have equal probability and use trapezoidal rule to estimate area, thus
Ordering of perm is not necessary with this estimate.
Note: if ki = kj (homogeneous case), then LC = 0.

35. DARCY’S LAW WITH DIRECTIONAL PERMEABILITY
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36. Directional Permeability
Permeability may depend on direction
Vertical perm (kz, perpendicular to bedding plane) often about 1/10 horizontal perm (kx, ky, parallel to bedding plane)
Reservoir simulators typically use diagonalized tensor
Permeability may be heterogeneous kz kx ky
© 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

37. Anisotropy and Flow A. Isotropic (Kx = Ky) B. Anisotropic (Kx ≠ Ky)
© 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

38. DARCY’S LAW AND PHASE POTENTIAL
Darcy’s Law for single phase flow in 1-D:
Phase Potential of phase i:

39. 3-D EXTENSION OF DARCY’S LAW
Permeability Tensor

40. PRINCIPAL AXES Coordinate system {x’, y’, z’}
Typically assume diagonalized tensor aligned with principal axes θ x y x' y'
Coordinate
Rotation

41. QUESTIONS?
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42. SUPPLEMENT
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