PETE 4052
Lecture 2
Well Testing
Darcy’s Law for Linear and Radial Flow
Spring 2002
January 25
Darcy’s Law for Linear and Radial Flow
Outline
0.
1.
2.
3.
4.
5.
Synopsis
The Differential Form of Darcy’s Law
Darcy’s Law for Anisotropic Permeability
Integrated Forms of Darcy’s Law for Incompressible Fluids
Integrated Forms of Darcy’s Law for Slightly Compressible Fluids
Integrated Forms of Darcy’s Law for Gases
Related reading
Amyx et al., p. 72-73
Craft and Hawkins: p.218-221
Dake: Sections 4.3 and 4.4, p. 107-113; Section 4.7, p. 115-117 (don’t worry about skin yet; stop
at Fig. 4.5).
2.0
Synopsis
In the previous lecture, we discussed Darcy’s Law over finite lengths. In this leture, we will give
the differential form of Darcy’s Law. The differential form will be used to compute flow relations
for several different geometries, and for several different types of fluids.
2.1
The Differential Form of Darcy’s Law
Darcy’s Law was based on experiments with the potential (similar to pressure) drop p measured
over a finite length, L, in a sandpack of pemeability, k, and cross-sectional area, A. For linear flow
of an incompressible fluid of viscosity  through this sand pack, the flow rate is related to other
factors by
k Ap
q
............................................................. (2.1)
L
q
in consistent units. The superficial velocity is v  . Also, we recognize that the limit of the
A
quotient of the pressure difference and the length of flow is minus the pressure gradient (or its
p( x  x)  p( x) dp
derivative with respect to length, lim
. This gives us the differential form

x 0
x
dx
of Darcy’s Law:
k dp
u  vsup  
....................................................... (2.2)
 dx
The change in sign is necessary because the direction of flow is opposite to the direction of
pressure change.
We can integrate Eqn. (2.2) to compute forms of Darcy’s Law for various flow geometries and
fluid types. Before proceeding to that, however, we will first consider a generalization of Darcy’s
Law for anisotropic permeability.
Page 1 of 8
PETE 4052
Lecture 2
Well Testing
Darcy’s Law for Linear and Radial Flow
Spring 2002
January 25
2.2 Darcy’s Law for Anisotropic Permeability
In many reservoir rocks, the permeability depends on a direction. Often, the vertical permeability
is much less than the horizontal permeability because of shale layers or other sedimentary
features. The permeability may vary horizontally because of tectonic stresses, fracturing, or
depositional processes. The result is that we must consider different permeabilities for flows in
these different directions. In a three dimensional system, the components of velocity are
1  p
p
p 

u x    k xx
 k xy
 k xz
  x
y
z 
1
p
p
p 
 k yx
 ......................................... (2.3)
 k yy
 k yz
  x
y
z 
1  p
p
p 

u z    k zx
 k zy
 k zz
  x
y
z 
In anisotropic systems, permeability is a tensor. You are familiar with tensors from your statics
class; stress is a tensor. Physically, elements like kxy give the flow that occurs in the x-direction
due to pressure gradients in the y-direction. You see, anisotropic porous media can act like
“louvres”, with flow being “shunted” toward the direction of highest permeability (parallel to the
louvers) even if the pressure gradient is not in that direction. For porous media, kij = kji, and one
can always define a coordinate system in which the off-diagonal terms are zero. This is the
principal coordinate space, which is defined in exactly the same way the principal stress space
k p
was defined in statics. In the principal coordinate space (X, Y, Z), u X   X
, and so on.
 X
uy  
The points you should recall from this discussion are conceptual rather than detailed:
 Permeability may vary with direction in porous media.
 Permeability can be represented as a symetric tensor. One can always find a space in which
the tensor is diagonal.
 In general, 6 factors must be specified to describe permeability in three dimensions (kxx, kxy,
kxz, kyy, kyz, and kzz or kX, kY, kZ, and three angles).
 The vertical permeability is often much less than the horizontal permeability.
 Reservoir engineers seldom have enough data to know what the permeability tensor is, nor do
most solution methods include it. However, engineers often use a lower vertical permeability
compared to horizontal.
2.3
Integrated Forms of Darcy’s Law for Incompressible Fluids
General
There is a very important point about Darcy’s Law that we haven’t emphasized until this point:
unlike your favorite snack food items, it sells by volume, not weight. That is, the pressure
gradient is proportional to the volume flow rate, not the mass flow rate. For steady-state flow, the
mass flow rate is constant. For incompressible flow, the volume flow rate is also constant.
However, if the fluid is compressible, the volume flow rate varies and the integrated form of
Darcy’s Law will look different (as we shall see below).
We will consider linear, radial, and spherical flow (Fig. 2.1).
Page 2 of 8
PETE 4052
Lecture 2
Well Testing
Darcy’s Law for Linear and Radial Flow
Spring 2002
January 25
Figure 2.1 – Illustration of three different flow geometries and their
variations in cross-sectional area for flow.
We now introduce the formation volume factor, B. The formation volume factor converts a mass
quantity (stock tank barrels) to a volume quantity (reservoir barrels). Therefore, we can rewrite
the differential form of Darcy’s Law (Eqn. 2.2) as
k A dp
q
........................................................... (2.4)
B dx
Linear Flow
The equation for linear flow over a finite length can be obtained by rearranging and integrating
Eqn. (2.4):
L
p ( L ) kA
0 qdx   p ( 0) B dp
kA p( L)  p(0)
........................................... (2.5a)
q  ( L  0)  
B
kA p( L)  p(0)
q
BL
Page 3 of 8
PETE 4052
Lecture 2
Well Testing
Darcy’s Law for Linear and Radial Flow
Spring 2002
January 25
in consistent units. In field units,
k A p ( L)  p (0)
.......................................... (2.5b)
BL
When the pressure is greater at 0 than at L, the difference in the numerator on the right hand side
is negative, and flow goes from 0 to L. On the other hand, when pressure is greater at L than at 0,
flow goes from L to 0. The sign convention makes sense. Note that Eqn. (2.5) is very similar to
our ealier form of Darcy’s Law [Eqn. (1.1)], except that we have now included B and are being
much more careful about the sign of the pressure drop and the direction of flow.
q  0.001127
Radial Flow
The derivation for radial flow is very similar, except the area is a function of r (Fig. 2.1). The
equation to be integrated is
k dp
u
 dr
....................................................... (2.6a)
q
k dp

2rh
 dr
Rearranging and integrating,
r q dr
p 2k h
r r   p B dp
2k h
qln( re )  ln( rw )  
( p e  p w ) ...................................... (2.6b)
B
2k h( p e  p w )
q
r 
B ln  e 
 rw 
This definition makes sense, in that flow is negative, that is against r, when the pressure increases
with r. However, because most wells are production wells, and reservoir engineers don’t like
dragging minus signs around all the time, the radial flow equation is almost always defined as
(Craft et al., p. 226):
2kh( pe  p w )
...................................................... (2.7)
q
 re 
B ln  
 rw 
in consistent units. In field units,
kh( pe  p w )
q  0.00708
r 
B ln  e 
 rw 
This is just a convention, a matter of convenience. As long as one remembers the convention,
either of Eqns. (2.6b) or (2.7) can be used. I will generally use Eqn. (2.7).
e
e
w
w
Spherical Flow
For spherical flow (Fig. 2.1), the equation to be integrated is
k dp
u
 dr
....................................................... (2.8a)
q
k dp

4r 2
 dr
Page 4 of 8
PETE 4052
Lecture 2
Well Testing
Darcy’s Law for Linear and Radial Flow
which yields
Spring 2002
January 25
pe 4k
q dr


rw r 2 pw B dp
re
1 1 
4k
 q    
( pe  p w ) ........................................ (2.8b)
B
 re rw 
4k ( pe  p w )
q
1 1
B   
 rw re 
Again, this sign convention is often reversed to give the more convenient form,
q
4k ( pe  pw )
...................................................... (2.9)
1 1
B   
 rw re 
Applications
The integrated flow equations [Eqns. (2.5), (2.7), and (2.9)] can be used to predict steady-state
flow of incompressible fluids.
Linear flow occurs in core floods. It may also be a good approximation to flow far from wells in
reservoirs, in aquifers, or between wells in a pattern flood.
Radial flow occurs around most wells.
Spherical flow occurs in wells that do not penetrate the entire formation, at the tips of horizontal
wells, and near perforations.
Observations
The pressure is linear versus a different function of distance for each flow geometry:
Linear:
Radial:
Spherical:
2.4
x
ln(r)
1/r
Integrated Forms of Darcy’s Law for Slightly Compressible
Fluids
Compressibility Model
The formation volume factor is the ratio of the volume of the fluid at reservoir conditions to its
volume at standard (or reference , R, or stock-tank, sc) conditions. Thus,
V
B
VR
........................................................... (2.10a)
R


where the fluid density is . For a fluid of small and constant compressibility (Craft et al., p. 214),
   R exp c( p  p R ) ............................................... (2.10b)
Page 5 of 8
PETE 4052
Lecture 2
Well Testing
Darcy’s Law for Linear and Radial Flow
Spring 2002
January 25
1 
. For oil and water the product of fluid
 p
compressibility and pressure change is usually small: typical water compressility is 3  10-6/psi
and (p-pR) is seldom more than 104, so the maximum product is about 10-2 or less. For such
systems, the exonential can be very accurately approximated as
   R expc( p  p R )
............................................... (2.10c)
  R 1  c( p  p R )
Finally, we get an expression of B by combining Eqns. (2.10a) and (2.10c):
B  1  c( p  p R ) ..................................................... (2.11)
where c is the isothermal compressibility, c 
Linear Flow
Now, Eqn. (2.11) for B can be substituted into the differential form of Darcy’s Law [Eqn. (2.4)]
to get an expression for linear flow of an slightly compressible fluid:
k A dp
q
B dx
............................................... (2.12)
kA
dp

 1  c( p  p R ) dx
For linear flow, this can be rearranged and integrated as follows:

L
0
q dx   
p( L)
p (0)
q ( L  0)  
q
kA
dp
 1  c( p  p R )
kA
ln 1  c( p L  p R )  ln 1  c( p0  p R )................... (2.13a)
c
kA  1  c( p L  p R )  
ln

cL   1  c( p0  p R )  
in consistent units. In field units,
q  0.001127
kA  1  c( p L  p R )  
ln
 ............................. (2.13b)
cL  1  c( p0  p R )  
Assumptions
 Compressibility is constant; needed for Eqn. (2.10b).
 Compressibility is small; needed for Eqn. (2.10c).
 Viscosity and permeability are constant; needed to integrate Eqn. (2.13).
Applications
Craft et al. (p. 220) show that this effect is small for most oil and gas systems; that is not
surprising given the calculations we sketched out above. Most steady-state liquid inflow
calculations can safely neglect compressibility and use Eqns. (2.5), (2.7) and (2.9) as appropriate.
You will have an opportunity to assess the importance for yourself in the homework.
Amyx et al. (p. 72) get a different form because they assume that a constant density can be used
to convert the flow rate to a mass rate; this is less accurate than the approach used in Craft et al.
Page 6 of 8
PETE 4052
Lecture 2
Well Testing
Darcy’s Law for Linear and Radial Flow
Spring 2002
January 25
Other Flow Geometries
Beginning with the differential form of Darcy’s Law for slightly compressible flow [Eqn. (2.12)],
you can compute the inflow equations for other geometries using the same methods used above.
These calculations are relatively easy, because the variable are separable and we can solve these
ordinary differential equations by integrating with respect to p and r or x separately.
2.5
Integrated Forms of Darcy’s Law for Gases
For a gas, the formation volume factor is
B
p sc Tz
......................................................... (2.14a)
pTsc
in consistent units. In field units,
p sc Tz
..................................................... (2.14b)
5.615 pTsc
We will go ahead and solve this problem in consistent units, and then convert to field units at the
end. Substituting the expression for B [Eqn. (2.14a)] into the differential form of Darcy’s Law
[Eqn. (2.4)],
k A dp
q
B dx
k A dp

..................................................... (2.15)
p Tz dx
 sc
pTsc
B
k Tsc Ap dp
p sc Tz dx
This can be integrated as we have done before:
L
p kTsc A
0 q dx  p p Tz p dp ............................................. (2.15b)
sc
This integral (on the right-hand side) is (as mathematicians say) nontrivial --  and z are functions
of pressure. Later, we will actually do this integral to improve our analysis of gas flow. For now,
we will follow Craft et al. (p. 221) and assume they are approximately constant and move them
outside the integral:
kTsc A p
L
0 q dx   p Tz p p dp
sc

L
0
L
0
kTsc A  1 2
 p L  p02  ........................................... (2.16a)

p sc Tz  2

2
2
kT A p L  p0 
q   sc
2 p sc TzL
qL  
in consistent units. In field units,
kTsc A p L2  p02 
......................................... (2.16b)
p sc TzL
The constant given in Craft et al. is very slightly wrong.
q  0.003124
The viscosity, , and gas deviation factor, z, should be evaluated at the mean pressure,
p  p0
.
p L
2
Page 7 of 8
PETE 4052
Lecture 2
Well Testing
Darcy’s Law for Linear and Radial Flow
Spring 2002
January 25
Assumptions
 z, k, Tconstant
  and z evaluated at the mean pressure.
Discussion
Craft et al. (p. 221) discuss how to better
approximate the integrals evaluated above based
on the behavior at relatively low and relatively
high pressure (Fig. 2.2, from Craft et al.). The
relatively simple low- and high-pressure behavior
can be exploited to derive corresponding
approximations for the flow equations.
Figure 2.2 – Behavior of  as a function of
pressure. At low pressure (below 1000 psi),
zis approximately constant; above 6000 psi
z/p is approximately constant. The
intermediate behavior is more complex.
Later, we will take a more accurate, more general,
and more reliable approach based on a transform
called real gas pseudopressure. As we will see,
this is the preferred method for analyzing gas flow
– it involves a few more computations, but at the
end of the day analysis is simpler because you
don’t have to use different approximations at high
and low pressure.
For an ideal gas, z is one and  does not depend on
pressure. Thus, Eqn. (2.16) could be used to
accurately analyze gases whose PVT behavior
near ideal.
At low pressure, flow of gases is much different than flow of liquids – flow rate is proportional to
the difference of the squares of pressure rather than the difference of pressure itself. At high
presssure, the gas compressibility becomes small and high-pressure gas flow is similar to liquid
flow.
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