AES1310: Rock Fluid Interactions - Part 1 1 Susanne Rudolph [email protected] Darcys law in...

AES1310: Rock Fluid Interactions - Part 1

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Susanne [email protected]

Darcy’s law in heterogeneous medium- Introduction - Averages


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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.


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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 theformation rock due to dissolution of carbonates in

water. Sedimentary porous media (e.g. sandstone) have

layeredstructure. Permeability parallel to layers is mostly

greater thanperpendicular.


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Anisotropy

Stratified formation are defined as anisotropichomogeneous medium when the thickness of the

individuallayer is smaller than the length of interest. In such

cases thepermeability cannot be determined from core samples

becauseit would not display the real permeability. So far, only isotropic media have been considered

wherein thepermeability as constant factor (scalar) in Darcy’s law. Due to anisotropy is the direction of the pressure

gradientvector different than the direction of the Darcy velocity

vectorat a point in the medium.


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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):

1, ,i ix iy iz

P gz P gz P gzu k k k i x y z

x y z


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Anisotropy

Herein kij are the elements of the permeability tensor

withPermeability 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:

xx xy xz

yx yy yz

zx zy zz

k k k

k k k

k k k

k

gz

k

u p


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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 diagonal0 0

0 0

0 0

xx

yy

zz

k

k

k

k


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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.


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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.


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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.

1 2 3 41

1 1...

n

n i ni

A x x x x x x xn n


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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

1 1 2

1 1 11...

n n

i ni

n nH x

x x xx


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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

1/

1/

1 2 31

...nn

n

i ni

G x x x x x


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Relation betweenarithmetic 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)

1/ 1/

1 1

1

1

exp ln

1exp ln

1exp ln

exp ln

n nn n

n i ii i

n

ii

n

ii

n

G x x x

xn

xn

A x


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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.


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Average permeability for heterogeneous media

Two situations are considered: 1) Flow is parallel to layers 2) 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 layersμ: viscosity of fluid; assumed to be equal for all systems

and constantΦ: fluid potentialA: cross-sectional area


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Flow parallel to layers1-D & linear flow


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Driving force described by difference of fluid

potentials (piezometric head) 1 and 2.

1 1 1P g z

2 2 2P g z

ii

Q Q

ii

b b


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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:

2 1i ii i i i i

k kQ u A A A

L L

1 2 1 2

1 1 1

N N Ni i

i i ii i i

k kQ Q A A

L L

i iA w b



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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:

1 2

1

N

ii

Q w TL

1 2 1 2parallelparallel

kQ A w T

L L

ii i

kb T



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Gives:

1 1

1

1

N N

i i i i Ni i

parallel parallel iNi

ii

k b k bk T T

bb

1 2 1 2

1 2 1 2

1 1

parallelparallel

N Ni

i ii i

kQ A w T

L L

kQ A w T

L L



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Rewriting the effective or equivalent permeability for the horizontal flow parallel to the layers gives:

Which is the arithmetic average of the permeability.

1

0

1

N

bi ii

parallel i h

k bk k dx k

b b



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Flow parallel to layers1-D &radial flow

hT


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Flow parallel to layers1-D &radial flow

Driving force described by difference of fluid

potentials (piezometric head) 1 and 2.

1 eP

2 wP

ii

Q Q

ii

h hhT


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Flow rate of each layer expressed by Darcy’s law:

Total flow rate Q: sum of the flow rates through each layer:

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ln ln

i i w ei ii i i

w w

e e

k h p pk h pQ u A

r r

r r

1 1 1

2 2

ln ln

N N Ni i w e e w i

i ii i iw w

e e

k h p p p p kQ Q h

r r

r r

Flow parallel to layers1-D & radial flow


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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:

1

2

ln

Ne w

iiw

e

p pQ T

r

r

2 2

ln ln

parallele w e wT parallel

w w

e e

kp p p pQ h T

r r

r r

ii i

kh T



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Gives:

1

1

N

i i Ni

parallel parallel ii

k hk T T

h

1 1

2 2

ln ln

2 2

ln ln

parallele w e wT parallel

w w

e e

N Ne w e wi

i ii iw w

e e

kp p p pQ h T

r r

r r

p p p pkQ h T

r r

r r



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Flow normal to layers1D & linear flow

Datum level


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Flow normal to layers1D & 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 Δi

Rock Fluid Interactions – Part 1AES1310

i

i

kQ b

L

ii

ii

L L

Datum level


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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:

ii

i

L Qkb

1 1 1

N N Ni i

iii i i i

L Q LQk b kb


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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:

1

Ni

inormal i

LL

k k

normalkQ b

L


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Rewriting the permeability for the horizontal flow normal to the layers gives:

Which is the harmonic average of the permeability.

1 1 1 0

11 1

normal vN N N Li i

ii i ii i i i

L L Lk k

L LL dx

k L k k k


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Flow normal to layers1D & radial flow


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Flow normal to layers1D & 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 Δi. Horizontal flow is

considered, thus change of fluid potential is equal to pressure drop.

1

2

ln

ii

i

i

k h pQ

r

r

i ii i

p p


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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:

1

ln

2

i

ii i

i

rQ

rp

k h

1 1

1 1 1 1

ln ln

2 2

i i

N N N Ni i

i ii i i ii i

r rQ

r rQp p

k h h k

1

1

2

ln i

Ni

i i

h pQ

r

r

k


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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:

1

ln

ln

w

enormal

i

i

i i

r

rk

r

r

k

2

ln

normal

w

e

k h pQ

r

r


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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.

parallel normalk k


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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)

parallel chessboard normalk k k


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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?Bed Thicknes

sH [ft]

Perm[mD]

1 20 100

2 15 200

3 10 300

4 5 400

Bed Length orradiusL or R [ft]

Perm[mD]

1 250 25

2 250 50

3 500 100

4 1000 200

Parallel flowPerpendicular flow

AES1310: Rock Fluid Interactions - Part 1 1 Susanne Rudolph [email protected] Darcys law in...

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Transcript of AES1310: Rock Fluid Interactions - Part 1 1 Susanne Rudolph [email protected] Darcys law in...