Transcript Document

Average Permeability
Flow in Layered Systems
Some figures were taken from Amyx, Bass and Whiting, Petroleum Reservoir
Engineering (1960).
Average Permeability
• If permeability is not a constant function of space (heterogeneity),
we can calculate the average permeability
– Common, simple flow cases are considered here
•
•
•
•
Linear, Parallel (cores, horizontal permeability)
Linear, Serial (cores, vertical permeability)
Radial, Parallel (reservoirs, horizontal layers)
Radial, Serial (reservoir, damage or stimulation)
• Average permeability should represent the correct flow capacity
– For a specified flow rate, average permeability results in same pressure drop
(and vice versa)
• Review Integral Averages (Self Study, e.g. Average Velocity)
Linear Flow, Pressure Profile
• Review, Darcy’s Law:
– horizontal flow (F=p)
q
k  dp 
vs 
  
A
μ  ds 
kA
q ds  
dp
μ
L
A
q
L
1
2
kA
q  ds  
μ
0
p2
 dp
p1
kA
p1  p2 
q
μL
Linear Flow, Pressure Profile
• We can determine how
pressure varies along the
flow path, p(x), by
considering an arbitrary
point, 0x  L
– Integral from 0x
x
kA
q  ds  
μ
0
p(x)
 dp
p1
qμ x
p(x) p1 
kA
OR, equivalently
L
A
q
L
2
1
– We could also integrate
xL
p2
kA
q  ds  
dp

μ p(x)
x
q μ (L  x)
p(x) p2 
kA
Linear Flow, Pressure Profile
• Pressure profile is a linear
function for homogeneous
properties
p1
p
– slope depends on flow rate
A
q
L
2
p(x)
p2
1
0
0
x
x
L
Linear, Parallel Flow
• Permeability varies across several
horizontal layers (k1,k2,k3)
– Discrete changes in permeability
h  h1  h2  h3  hi
– Same pressure drop for each layer
p1 - p2  Δp  Δp1  Δp2  Δp3
– Total flow rate is summation of flow rate for all layers
q  q1  q2  q3  qi
– Average permeability results in correct total flow rate
kwh
q
Δp ; A  w  h
μL
Linear, Parallel Flow
• Substituting,
kwh
k1 w h1
k2 w h2
k3 w h3
q
Δp 
Δp 
Δp 
Δp
μL
μL
μL
μL
• Rearranging,
k

k
i
 hi
h
• Average permeability reflects flow capacity of all layers
Linear, Serial Flow
• Permeability varies across several
vertical layers (k1,k2,k3)
– Discrete changes in permeability
L  L1  L2  L3   Li
– Same flow rate passes through each layer
q  q1  q2  q3
– Total pressure drop is summation of pressure drop across layers
p1  p2  Δp1  Δp2  Δp3  Δpi
– Average permeability results in correct total pressure drop
qμ L
p1 - p2 
; A  wh
kwh
Linear, Serial Flow
• Substituting,
qμ L
q μ L1 q μ L2
q μ L3
p1 - p2 



k w h k1 w h k 2 w h k 3 w h
L
k
Li
k
i
• If k1>k2>k3, then
– Linear pressure profile in each layer
p1
k
p
• Rearranging,
p2
0
0
L
x
Radial Flow, Pressure Profile
• Review, Darcy’s Law:
– horizontal flow (F=p)
q
k  dp 
vs 
  
A
μ  ds 
q
k
dr  dp
2π rh
μ
rw
q
rw
re
1
2π kh
q  dr 
r
μ
re
pw
 dp
pe
2π kh
pe  p w 
q
μ ln(re /rw )
Radial Flow, Pressure Profile
• We can determine how
pressure varies along the
flow path, p(r), by
considering an arbitrary
point, rwr  re
– Integral from r  rw
rw
1
2π kh
q  dr 
r
μ
r
pw
 dp
p(r)
q μ ln(r/rw )
p(r)  p w 
2π k h
OR, equivalently
OR
r
– Integral from rer
1
2π kh
q  dr 
r
μ
re
p(r)
 dp
pe
q μ ln(re /r)
p(r)  p e 
2π k h
Radial Flow, Pressure Profile
• Pressure profile is a linear
function of ln(r) for
homogeneous properties
pe
– slope depends on flow rate
p
p(r)
pw
q
0
rw
r
ln(r) 
rw
re
re
Radial, Parallel Flow
• Permeability varies across several
(3) horizontal layers (k1,k2,k3)
– Discrete changes in permeability
h  h1  h2  h3  hi
– Same pressure drop for each layer
pe - pw  Δp  Δp1  Δp2  Δp3
– Total flow rate is summation of flow rate for all layers
q  q1  q2  q3  qi
– Average permeability results in correct total flow rate
2π k h
q
Δp
μ ln(re /rw )
Radial, Parallel Flow
• Substituting,
2π k h
q
Δp
μ ln(re /rw )
2π k1 h1
2π k 2 h 2
2π k 3 h 3

Δp 
Δp 
Δp
μ ln(re /rw )
μ ln(re /rw )
μ ln(re /rw )
• Rearranging,
k

k
i
 hi
h
• Average permeability reflects flow capacity of all layers
Radial, Serial Flow
• Permeability varies across two vertical
concentric cylindrical layers
[k(rwrr2) = k1, k(r2rre = k2]
– Discrete changes in permeability
re  rw  Δr1  Δr2  Δri
– Same flow rate passes through each layer
q  q1  q 2
– Total pressure drop is summation of pressure drop across layers
pe  pw  Δp1  Δp2  Δpi
– Average permeability results in correct total pressure drop
q μ ln(re /rw )
pe - p w 
2π k h
R1 of this
figure is r2 of
equations
Radial, Serial Flow
• Substituting (rw=r1, r2 ,re=r3),
q μ ln(re /rw ) q μ ln(r2 /rw ) q μ ln(re /r2 )
pe - p w 


2π k h
2π k1 h
2π k 2 h
• Rearranging,
ln(re /rw )
k
(ln(ri 1/ri )

ki
All Layers
Radial, Serial Flow
• Damage: k1<k2
pe
p
– Shown in sketch to the left
– Permeability is damaged near the
wellbore
0
• Reactive fluids
• Fines migration
k
pw
rw
re
pe
ln(r) 
k1  
– Shown in sketch to the right
– Permeability is improved near
the wellbore
• Acid stimulation
p
• Stimulation k1>k2
k
pw
0
rw
re
ln(r) 
Integration of Darcy’s Law
• Beginning with the differential form of Darcy’s Law
q
• Previous lecture on gas flow
– gas properties are functions of pressure
• include gas properties in the dp integral
k  dp 
vs 
  
A
μ  ds 
• In this lecture
– parallel flow (permeability varies over cross sectional area)
• integrate over area (integrated average value)
– serial flow (permeability varies along flow path)
• integrate over flow path (leave k in ds integral)
• This approach can be extended to other cases (order of precedence
as shown)
– Any term that varies as a function of pressure can be included in the dp
integral
– Any term that varies along flow path can be included in the ds integral
– Any term that varies over cross sectional area can use an integrated average
value (integrated over cross sectional area, e.g. parallel flow)