1. Darcy’s law in heterogeneous medium - Introduction Averages
Susanne Rudolph

2. Heterogeneity of porous media
Homogenous: A medium is homogenous with respect to a property if the property is independent of position within the medium.
Isotropic: A medium is isotropic with respect to a property if the property is independent of the direction within the medium.
Anisotropic: If at one point in the medium a property such as permeability varies with the direction.

3. Anisotropy Reservoirs are commonly anisotropic with respect to the
permeability.
Anisotropy of permeability is due to evolution of formations;
e.g. in carbonate rocks formation of channels within the
formation rock due to dissolution of carbonates in water.
Sedimentary porous media (e.g. sandstone) have layered
structure. Permeability parallel to layers is mostly greater than
perpendicular.

4. Anisotropy Stratified formation are defined as anisotropic
homogeneous medium when the thickness of the individual
layer is smaller than the length of interest. In such cases the
permeability cannot be determined from core samples because
it would not display the real permeability.
So far, only isotropic media have been considered wherein the
permeability as constant factor (scalar) in Darcy’s law.
Due to anisotropy is the direction of the pressure gradient
vector different than the direction of the Darcy velocity vector
at a point in the medium.

5. Anisotropy Assume a porous medium with an arbitrary orientation with
respect to the coordinate system and the pressure gradient
points in the x-direction.
Due to anisotropy the flow rates in the different directions
are not the same.
Darcy’s law for anisotropic media is (Ferrandon 1948):

6. Anisotropy Herein kij are the elements of the permeability tensor with
Permeability values depend on the orientation of the medium with respect to the coordinate system:
With this, Darcy’s law can be written in vector notation as:

7. Anisotropy
If it is assumed that the anisotropic medium is ‘orthotropic’
(they have 3 mutually orthogonal principal axes) and if the
coordinate axes are aligned with the principal axes of the
medium  permeability tensor is diagonal
An orthotropic material has two or three mutually orthogonal two-fold axes of rotational symmetry so that its mechanical properties are, in general, different along the directions of each of the axes. Orthotropic materials are thus anisotropic; their properties depend on the direction in which they are measured. An isotropic material, in contrast, has the same properties in every direction.
One common example of an orthotropic material with two axes of symmetry would be a polymer reinforced by parallel glass or graphite fibers. The strength and stiffness of such a composite material will usually be greater in a direction parallel to the fibers than in the transverse direction. Another example would be a biological membrane, in which the properties in the plane of the membrane will be different from those in the perpendicular direction. Such materials are sometimes called transverse isotropic.

8. Heterogeneous Media
Reservoirs are commonly heterogeneous. A reservoir consists of ‘patches’ with different properties.
Often reservoir simulations are performed applying a cylindrical homogeneous structure.
Main interest of reservoir engineers is the flow through porous medium and its understanding.
Only in recent years heterogeneity is taken into account in order to analyze reservoir behavior.
Geological or geostatistical models provide are detailed description of the heterogeneity of the reservoir.

9. Heterogeneous Media
Details can often not be incorporated in reservoir simulation models.
Permeability values have to be averaged.
Averaging procedure has to be conducted with care to avoid erroneous averaged values.
Mathematics provides computation of means such as arithmetic, harmonic and geometric mean.

10. Arithmetic mean
- Sum of all the values of which the arithmetic mean has to be determined divided by the number of summed values.
- For a set of data X = (x1, x2, x3,…,xn) the arithmetic mean is:
Note:
- If the arithmetic mean is determined of values varying strongly in value/order of magnitude, it might give an erroneous high average value.
- Arithmetic mean can only be taken from values with the same reference.

11. Harmonic mean
- The number of values divided by the sum of the reciprocal values of the property.
For a set of data X = (x1, x2, x3,…, xn) the harmonic mean is:
Derived from electrotechnique computing the avarage resistance of a electrical circuit with two resistors in parallel

12. Geometric mean
Indicates the central tendency or a typical value of a set of parameters.
For a set of data X = (x1, x2, x3,…, xn) the geometric mean is:
Can only applied to possitive values.
Is smaller or equal to the arithmetic mean of the same data set
Is closely related to arithmetic mean

13. Relation between arithmetic and geometric mean
For a set of data X = (x1, x2, x3,…,xn) the geometric mean can be written as an arithmetic mean by taking the natural logarithm:
For positive values of xi: Hn(x) ≤ Gn(x) ≤ An(x)

14. Average permeability for heterogeneous media
After recalling the meaning of the different ways to take mean values, the averaging of the permeability is considered for heterogeneous reservoirs
Use of average permeability only for simple flow cases.
Rock system composed of distinct layers with different properties.
Only flow of a homgeneous fluid (,  = constant); therefore use of hydraulic conductivity analogous to use of permeability.

15. Average permeability for heterogeneous media
Two situations are considered:
Flow is parallel to layers
Flow is normal/perpenticular to layers
ki: permeability;
bi: thickness of layer i;
Qi: flow rate through layer I;
W: width of the layers; same for all layers
μ: viscosity of fluid; assumed to be equal for all systems and constant
Φ: fluid potential
A: cross-sectional area

16. Flow parallel to layers 1-D & linear flow

17. Flow parallel to layers 1-D & linear flow
Driving force described by difference of fluid potentials (piezometric head) 1 and 2.

18. Flow parallel to layers 1-D & linear flow
Flow rate of each layer expressed by Darcy’s law:
Total flow rate Q: sum of the flow rates through each layer:
With the cross-sectional area:

19. Flow parallel to layers 1-D & linear flow
Transmissitivity: product of the thickness and the permeability over the visocsity
This gives then:
Combining this result with the same flow rate Q through a porous medium of the thickness b described in terms of the equivalent permeability kparallel or transmittivity Tparallel:

20. Flow parallel to layers 1-D & linear flow
Gives:

21. Flow parallel to layers 1-D & linear flow
Rewriting the effective or equivalent permeability for the horizontal flow parallel to the layers gives:
Which is the arithmetic average of the permeability.

22. Flow parallel to layers 1-D &radial flowhT

23. Flow parallel to layers 1-D &radial flow
Driving force described by difference of fluid potentials (piezometric head) 1 and 2. hT

24. Flow parallel to layers 1-D & radial flow
Flow rate of each layer expressed by Darcy’s law:
Total flow rate Q: sum of the flow rates through each layer:

25. Flow parallel to layers 1-D & radial flow
Transmissitivity: product of the thickness and the permeability over the visocsity
This gives then:
Combining this result with the same flow rate Q through a porous medium of the thickness hT described in terms of the equivalent permeability kparallel or transmittivity Tparallel:

26. Flow parallel to layers 1-D & radial flow
Gives:

27. Flow normal to layers 1D & linear flow
Datum
level

28. Flow normal to layers 1D & linear flow
Horizontal flow normal or perpendicular to layers:
Total flow rate per unit width is unchanged. The total drop of the head Δ is now the sum of the drop of heads for each layer Δi
Datum
level
Rock Fluid Interactions – Part 1 AES1310

29. Flow normal to layers 1D & linear flow
The drop of the piezometric head of each section of the layer is described by Darcy’s law:
The total piezometric head is then:

30. Flow normal to layers 1D & linear flow
Combining the equation again with the result obtained regarding the porous medium described by an equivalent or effective permeability knormal for the flow with the same flow rate Q through a porous medium of the length L:
Gives:

31. Flow normal to layers 1D & linear flow
Rewriting the permeability for the horizontal flow normal to the layers gives:
Which is the harmonic average of the permeability.

32. Flow normal to layers 1D & radial flow

33. Flow normal to layers 1D & radial flow
Now we consider the horizontal flow normal or perpendicular to layers occurs. For this case the total flow rate per unit width is unchanged. The total drop of the head Δ is now the sum of the drop of heads for each layer Δi. Horizontal flow is considered, thus change of fluid potential is equal to pressure drop.

34. Flow normal to layers 1D & radial flow
The drop of the piezometric head of each section of the layer is described by Darcy’s law:
The total piezometric head is then:
And the flow rate:

35. Flow normal to layers 1D & radial flow
Combining the equation again with the result obtained regarding the porous medium described by an equivalent or effective permeability knormal for the flow with the same flow rate Q through a radial porous medium with an outer radius re and a inner radius rw:
Gives:

36. Averaging of permeability
From the equations to describe the flow parallel and normal to layers follows that the equivalent permeabilities of parallel flow are larger than of the normal flow:
This can be prooved by considering frist two layers and then increasing the number of layers while computing the knormal and kparallel.

37. Averaging of permeability
Note: The geometric mean is commonly used for the description of the average permeability in a chessboard reservoir (= area is subdivided in blocks of equal size)
Cardwell and Parsons showed for chessboard arrangement that the equivalent permeability lays between the one of parallel flow and the one of normal flow.
This is in agreement what we saw before:
Hn(x) ≤ Gn(x) ≤ An(x)

38. Averaging of permeability
Determine the average permeability of the situation described in the tables for linear flow and radial flow.
What are the ratios of the separate flows in these beds?
What are the ratios of the separate piezometric heads in these beds?
Perpendicular flow
Parallel flow
Bed
Length or
radius
L or R [ft]
Perm
[mD] 1
250 25 2 50 3
500
100 4
1000
200
Bed
Thickness
H [ft]
Perm
[mD] 1 20
100 2 15
200 3 10
300 4 5
400