Chapter 5
Darcy’s Law and Applications
§ 4.1 Introduction
Darcy' s law
qreservoir 
Note : Time scale
is
dN p
dt
added
1
§ 4.2 Darcy’s Law; Fluid Potential
h1  h2
h
uK
K
l
l
Different Sand Pack → Different K
2
Darcy’s experimental law was proved to
be independent of the direction of flow in
the earth’s gravitational field.
 p  pa  g (h  z)
The pressure at any point in the flow path in fig 4.2
p  gh  gz  g (h  z)
From Eq.(4.1)
uK
 hg  (
dh
dl
p

 gz ) (4.2)
( where K is a const.) (4.3)
3
Eq(4.2) & Eq(4.3)
•
 K d  p

d 1  p
u  K    gz  
  gz 
dl  g  
 g dl  

K d

(4.4)
g dl
  Fluid
potential
p
d  d (  gz )
where


p
p 1atm



p
dp
p 1atm


p
dp
pb


dp


z
z 0
gdz
 gz
 g ( z  zb )
z  0, p  1atm (4.5)
z  zb , p  pb (4.6)
4
The reason for this (Eq(4.6)) is that fluid flow between two points A and B is
governed by the difference in potential between the points, not the absolute
potentials, i.e.
 A  B  
PA
Pb
 PB dp
 A dp
dp
 g ( z A  z b )  
 g ( z B  zb )    g ( z A  z B )

 Pb 
 PB 
P
d  d (

p

 gz
p

 gz )
(???)
5
Experiments performed with a variety of different liquids revealed that
the law can be generalized as
k d
u
 (4.8)
 dl
where k= permeability
μ = viscosity
ρ = density of fluid
Note :
u
K d
(4.4)
g dl
d  d (
p

 gz )
(k/μ) = K/g -- > k/μ = K/g  K = (k/μ) g
k dp
u
 dl
6
§ 4.3 Sign convention
Φ1
Linear flow
Φ2
u
l1
u
l2
k d
(4.9)
 dl
Radial flow
Φ1
rw
k d
u
(4.10)
 dl
u
r
Φ2
7
§ 4.4 Units: Units Conversion
Darcy’s equation
k d
u
 dl
L
( M / L3 ) ( L2T 2 )
 []k
 k[]L2
T
( M / LT ) ( L)
or
k dp
u
 dl

L
1
[]k
T
( M / LT )
In cgs units
In SI units
(M
L 1
)
2
2
T L  k[]L2
( L)
k[=]cm2
k[=]m2
d  d (
 
p

p

 gz )
 gz
8
In Darcy units
k dp
u
 dl
k  1 Darcy
Darcy unit



when
u  1cm / sec


  1 cp , and
hybrid units
dp / dl  1 atm / cm 
9
10
Darcy equation
In absolute units (cgs units)
dl
dz
θ
u
k d
k dp
dz
  (  g )  (4.12)
 dl
 dl
dl
u[]cm / sec
k[]cm
2
gm
[ ]
[] poise
cm  sec
p[]dyne / cm 2
g[]cm / sec 2
l , z[]cm
dyne / cm 2
dyne  cm
[]
[

]
gm
gm / cm 3
[]gm / cm
3
cm 3
cm
q(
)  u ( )  A(cm 2 )
sec
sec
11
Darcy equation
In Darcy units
dl
dz
k dp
g
dz
u ( 
)
6
 dl 1.0133  10 dl
u[]cm / sec
k[]Darcy
p[]atm
l , z[]cm
[]cp
[]gm / cm 3
θ
g[]cm / sec 2
cm 3
cm
q(
)  u ( )  A(cm 2 )
sec
sec
12
Darcy equation
In field units
dl
dz
θ
q  1.127  10 3
kA  dp
dz 

0
.
4334



  dl
dl 
or
q  1.127  10
where
3
kA  dp


0
.
4334

sin



  dl

g
q[]bbl / D
p[] psi
 [ ]
k[]md
l , z[] ft
[]cp
A[] ft 2
( g ) water
13
Note: In this text
1 Darcy = 1.0133×10-8 cm2
or 1 Darcy

10
-8
cm2
or 1 Darcy

10-12 m2 = 1 m 2
In other book
1 Darcy = ０.986927×10-8 cm2
1 Darcy = ０.986927×10-12 m2
1 Darcy
1
m 2
14
Conversion of Permeability (md) and Coefficient of
Permeability (m/s)
1 md  7.324  10
8
( ft 3 / s)cp
ft 2 ( psi / ft )
 H O  62.4 lbm / ft 3
2
 H O  1 cp
2
lbm ( ft / s 2 )
g c  32.174
lb f
P=K=Coefficient of Permeability
k  g 
  
  gc 
 1 md  9.676  10
9
m
s
15
§ 4.5 Real Gas potential
p
dp
pb


The fluid potential function
 gz (4.6)
For an incompressible fluid (   consant )
p
dp
pb


pb

 gz (4.7)
pM
zRT


p
  f ( p, T ) )
For a compressible fluid (
p

 gz
zRT
RT
dp  gz 
pM
M

p
pb
zdp
 gz (4.19)
p
RT z

dp  gdz 

M p
 (4.20)
dp

or d 
 gdz


or d 
16
§ 4.6 Datum Pressure
From eq(4.20)
d  dp  gdz  d
From eq(4.12)
q
kA d
kA d

(4.22)
 dl
 dl
Note: Ψ potential is frequently refered to as the “datum pressure”
17
§ 4.7 Radial Steady State Fluid; Well Stimulation
Assume
(1)Steady state

Darcy’s law for the radial flow of single phase oil
p
 0  pe and pressure profile  const. with time
t
q
kA dp
(4.23)
 dr
(2)Reservoir is homogeneous
(3)Well is perforated across the entire formation thickness.
For q=const. across any radial area A=2π rh
q
k 2  rh dp
q
 dp 
dr

dr
2  rhk
q r 1
pwf dp  2kh rw r dr
q
r
p  pwf 
ln (4.24)
2kh rw
p
at r  re
pe  pwf 
q
r
ln e (4.25)
2kh rw
18
• In drilling a well
• Pwellbore>Pformation  mud flow from wellbore to formation
•
• mud particle suspended in the mud plug the pore spaces
•
• Reducing k → Damage zone (rw < r < ra )
p skin 
q
S  (4.26)
2kh
S  skin factor , dim ensionless number
19
Eq(4.25) and Eq(4.26)
Pe  Pwf 
r
q
(ln e  S )  (4.27)
2kh
rw
In field unit, Eq(4.27) is
qBo
re
Pe  Pwf  141.2
(ln  S ) (4.28)
kh
rw
oil rate ( STB / D)
q
7.08  10 3 kh
 PI 


r
Pe  Pwf
pressure drawdown
Bo (ln( e )  S )
rw
PI = productivity index of a well (a direct measure of the well performance)
20
The various methods of stimulation
(a) Removal of skin (S)
- Determination of skin(S) － pressure buildup test
- If S>0 ,  acid treatment
－ Hydrochloric acid if formation limestone
－ Mud acid if formation is sandstone
(b) Increasing the effective permeability (k)
(a) hydraulic fracturing － sandstone
(b) fracture-acidising － carbonate reservoir
(c) Viscosity reduction (μ)
o high  PI  low
•
Tstimulation
  o  fig. 4.6 p119
The
(Steam Soaking)
21
(d) Reduction of the oil formation volume e
factor (Bo )
•
Bo =(RB/STB) can be minimized by choosing
the correct surface separator ,or combination
of separator.
(e) Reduction in the ratio re/rw
－ it is seldom considered as a means of well
stimulation
•
(f) Increasing the well penetration (h)
22
§ 4.8 Two-Phase Flow：Effective and
Relative Permeabilities
• Darcy equation
In Single phase flow
k  dp
g
dz 


 in Darcy units
  dl 1.0133  10 6 dl 
k  dp

u  1.127  10 3 
 0.4334 sin   in field
  dl

u
units
Two-phase flow---oil and water (or oil and gas, or gas and water)
--- > Effective permeability in Darcy equation
Effective permeability = f (fluid saturation)
k effevtive  k absolute
k o , k w : effective permeabili ty
23
S w  connate or irreducible water saturation
S or  residual oil saturation
k o (S w )
k
k (S )
k rw ( S w )  w w
k
k ro ( S w ) 
and
Normalized curve
Shape of curves
 Wettability of
rock surface
24
Alternative manner of normalizing the effective permeability
kk
k o ( S w  S wc )
kk
k o (S w ) 
k o ( S w  S wc ) 
 (4.32)
k w (S w ) 

K rw ( S w ) 
k o ( S w  S wc ) 
K ro ( S w ) 
ko
kw
In the displacement of one
immiscible fluid by another
k rw K rw

k ro K ro
Note : kr = f ( sw, h)
h= capillary & gravity forces
25
§ 4.9 The Mechanics of Supplementary Recovery
•
•
•
•
•
•
•
•
--Supplementary recovery
—by displacing the hydrocarbons towards the producing well with
some injected fluid .
injected fluid --water because - - availability
- - low cost
- - high specific gravity
--Mobility of any fluid (λ)
λ = k* kr/μ
Relative mobility = kr/μ
•
--Water displaces oil
26
-- Water displaces oil
Ideal or piston-like displacement
Injecting well
Water zone
Interface
( water front)
Producing well
Oil zone
Water zone
r 
k rw ( S w  1  S or )
w

k rw
'
w
r 

k ro ( s w  s wc )
o
k ro
'
o
The favourable type of displacement
k rw
'
k ro
'
w
 M  1  piston  like displaceme nt
o
where M = end point mobility ratio
k ro and k rw  endpo int relative permeabili ties
'
'
 const.
Oil recovered
= water injected in a
linear reservoir block
27
Non-ideal displacement
Injecting well
Producing well
Non  ideal displaceme nt
Oil re cov ered  1 or 1 water injected
5
6
Common M  1
28
Mobility Control
k rd
Mobility ratio  M 
'
d
Mobility of the displacing fluid

'
Mobility of the displaced fluid
k ro
o
Where d in krd’ is displacing fluid
Mobility control  to reduce M
  polymer
flooding (increase  d )
polymers, such as polysaccha ride, are dissolved in the injection water
29
 Thermal method (decrease
For very viscous crudes
o
o
o
d )
 w ~ order of 1000
 w can be drasticall y reduced by increa sin g the temperatur e
 hot water injection
 steam injection
 in  situ combustion
Thermal method   Secondary re cov ery processes with some tertiary side effects
1) mobility control
2) distillati on of the crude
3) exp ansive of the oil on heating
30
Tertiary flooding  re cov ering the oil remaining in the reservoir after a
convention al sec ondary re cov ery project .
After sec ondary re cov ery  trapping of oil droplets (because of surface tension )
k ro
After secondary
recovery
To displace the oil with a
fluid which is soluble in it
B
So
A
Sor
Sw
.
A
So
Sor
To flood with a fluid which is miscible or partially miscible with the oil
→ Eliminating surface tension
31
 Miscible ( LPG) flooding
LPG  Liquid Petroleum Gas
 Carbone dioxide flooding
1.CO2  oil
k ro 
( pt , A  pt , C )
2.
o 
(mobility control )
3.
extracting light H .C. from oil
 Surfac tan t flooding (Micellar soluton flooding )
32