PETE 311 (07C) — Darcy's Law — Parallel and Series Flow Equations Slide — 1

Petroleum Engineering 311Reservoir Petrophysics

Thomas A. Blasingame, Ph.D., P.E.Department of Petroleum Engineering — Texas A&M University

College Station, TX 77843-3116 (USA)+1.979.845.2292

t-blasingame@tamu.edu

Darcy's Law — Parallel and SeriesLIQUID Flow Equations

01 November 2007

PETE 311 (07C) — Darcy's Law — Parallel and Series Flow Equations Slide — 2

Horizontal Linear Flow: Parallel Flow (1/2)Flow Balance:

321 qqqqt ++=

Pressure Equilibrium:

321 ppppt Δ=Δ=Δ=Δ

Distance (L)

Layer Rates:

tjjj pLB

whkq Δ= μ

Total Rates:

tt pLB

whkq Δ= μ

PETE 311 (07C) — Darcy's Law — Parallel and Series Flow Equations Slide — 3

Horizontal Linear Flow: Parallel Flow (2/2)Flow Balance:

nt qqqq +++= ... 21Substituting Layer Rate Relations:

tnnttt pLB

whkpLB

whkpLB

whkpLB

whk Δ++Δ+Δ=Δ ... 2211 μμμμCancelling Like Terms:

jj

n

jnn hkhkhkhkhk

12211 ...

=∑=+++=

Defining the Average Permeability:

jj

n

jhk

hk

1 1

=∑=

This result is valid for BOTH the horizontal radial and linear flow-in-parallel cases.

PETE 311 (07C) — Darcy's Law — Parallel and Series Flow Equations Slide — 4

Horizontal Linear Flow: Series Flow (1/2)Flow Equilibrium:

321 qqqqt ===

Pressure Balance:

321 ppppt Δ+Δ+Δ=Δ

Distance (L)

Layer Pressure Drop:

tj

jj q

whB

kL

p μ=Δ

Total Pressure Drop:

tt qwhB

kLp μ

=Δ

PETE 311 (07C) — Darcy's Law — Parallel and Series Flow Equations Slide — 5

Horizontal Linear Flow: Series Flow (2/2)

Substituting Layer Pressure Drop Relations:

Cancelling Like Terms:

j

jn

jnn

kL

kL

kL

kL

kL

122

11 ...

=∑=+++=

Defining the Average Permeability:

j

jn

j kL

Lk

1

=∑

= This result is ONLY valid for the horizontal linear flow-in-series case.(Completely different result for radial flow-in-series case).

Pressure Balance:

nt pppp Δ++Δ+Δ=Δ ... 21

tnn

ttt qwhB

kLq

whB

kLq

whB

kLq

whB

kL ...

22

11 μμμμ

+++=

PETE 311 (07C) — Darcy's Law — Parallel and Series Flow Equations Slide — 6

Radial Flow: Parallel Flow (1/2)Flow Balance:

321 qqqqt ++=

Pressure Equilibrium:

321 ppppt Δ=Δ=Δ=Δ

Layer Rates:

twe

jjj p

rrBhk

q Δ= )/ln(

2 μ

π

Total Rates:

twe

t prrB

hkq Δ= )/ln(

2 μ

π

pw

pe

Logarithm of Radial Distance [ln(r)]

rw re

PETE 311 (07C) — Darcy's Law — Parallel and Series Flow Equations Slide — 7

Radial Flow: Parallel Flow (2/2)Flow Balance:

nt qqqq +++= ... 21Substituting Layer Rate Relations:

Cancelling Like Terms:

jj

n

jnn hkhkhkhkhk

12211 ...

=∑=+++=

Defining the Average Permeability:

jj

n

jhk

hk

1 1

=∑=

This result is valid for BOTH the horizontal radial and linear flow-in-parallel cases.

twe

nnt

wet

wet

wep

rrBhk

prrB

hkprrB

hkprrB

hkΔ++Δ+Δ=Δ

)/ln(2...

)/ln(2

)/ln(2

)/ln(2 2211

μπ

μπ

μπ

μπ

PETE 311 (07C) — Darcy's Law — Parallel and Series Flow Equations Slide — 8

Radial Flow: Series Flow (1/2)Flow Equilibrium:

321 qqqqt ===

Pressure Balance:

321 ppppt Δ+Δ+Δ=Δ

Layer Pressure Drop:

tj

jjj q

krr

hBp

)/ln(

21 1−=Δ

μπ

Total Pressure Drop:

twe

t qk

rrh

Bp)/ln(

21 μπ

=Δ

pw

pe

Logarithm of Radial Distance [ln(r)]

r0=rw r3=rn=rer1 r2

PETE 311 (07C) — Darcy's Law — Parallel and Series Flow Equations Slide — 9

Radial Flow: Series Flow (2/2)

Substituting Layer Pressure Drop Relations:

Cancelling Like Terms:

j

jjn

jnnnwwe

krr

krr

krr

krr

krr )/ln(

)/ln(

...)/ln()/ln(

)/ln( 1

1

12

121

1 −

=

− ∑=+++=

Defining the Average Permeability: (rn=re and r0=rw)

j

jjn

j

we

krr

rrk)/ln(

)/ln(

1

1

−

=∑

= This result is ONLY valid for the radial flow-in-series case.

Pressure Balance:

nt pppp Δ++Δ+Δ=Δ ... 21

tnnn

ttw

twe q

krr

hBq

krr

hBq

krr

hBq

krr

hB

)/ln(

21... )/ln(

21)/ln(

21

)/ln(

21 1

212

11 −++++=

μπ

μπ

μπ

μπ