Chapter 7 - Steady-state Laminar Flow of Homogeneous Fluids
7.1 Horizontal, 1D, Steady-State Flow
Steady state condition
The steady state flow regime can be characterized by pressure and fluid velocity
distributions that are independent of time; therefore, the continuity equation can be
simplified to:
 
  v  0
(7.1)
An application of Eq. (7.1) is the measurement of permeability in laboratory samples.
For linear, horizontal flow of an incompressible fluid through an isotropic porous media
the diffusivity equation becomes;
k

    0


(7.2)
Consider the example shown in figure 7.1. A long cylindrical core of length, L, and
cross-sectional area, A, has applied pressure of p  p(0)  p( L) . The flow rate, q, is
constant and follows Darcy’s Law. Also allow permeability to be a function of position
as well as pressure.
P0
PL
q
X=0
X=L
Figure 7.1 Schematic of flow through a horizontal core
The solution is:
q
k A p
 L
(7.3)
where
L L dx
 
k 0 k(x)
where k is the harmonic mean permeability of the sample.
7.1
(7.4)
Chapter 7 - Steady-state Laminar Flow of Homogeneous Fluids
Ideal Gas Flow
As a second example, let’s use gas as the flowing fluid; therefore, we must
account for both compressibility and gas-slip effects (Collins,1961). The diffusivity
equation becomes
 b  dp 
d 
k  p1     0
dx 
 p  dx 
(7.5)
where the Klinkenberg Effect is given by
kk
 b
1  
( x )  p 


(7.6)
Note the equivalent liquid permeability is a function of distance, x. Integrating twice
results in

b  p
k  p1  
c
p L

where c is a constant, p 
(7.7)
p(0)  p( L)
and average permeability is given by Eq. (7.4)
2
with k  replacing k . Comparing with Darcy’s Law yields, c 
q p
, where the flow
A
rate is evaluated at the mean pressure.
Radial Geometry
The equivalent form of the continuity equation in radial geometry for steady state
condition is,
 
1 
r v  0
r r
(7.8)
Assuming an incompressible fluid, isotropic media, and constant fluid properties results
in the following continuity equation.
1    
r
0
r r  r 
(7.9)
 (r )  C1 ln(r )  C2
(7.10)
The general solution is
7.2
Chapter 7 - Steady-state Laminar Flow of Homogeneous Fluids
where C1 and C2 are constants to be determined from the boundary conditions.

if assume Pwf at rw and Pe at re then obtain Darcy’s steady state flow equation.
q


2kh ( re   rw )
 r

 ln e r 
w

if source located at (a,b), in Cartesian coordinates,
2
q 
 ( x, y ) 
ln x  a 2  ( y  b)   C
4kh 

(7.11)
if anisotropic,
 ( x, y ) 

k
2
q
ln x  a 2  x ( y  b)   C

ky
4 k x k y h 


(7.12)
7.2 Streamlines, Isopotentials, source/sink
A streamline indicates the instantaneous direction of flow for a fluid particle
throughout the system. For steady state, this direction of flow is constant for all time,
however, for transient conditions the streamline is valid only for an instant in time.
Streamlines must be orthogonal to the equipotential lines throughout the region of flow.
An exception to this rule will be shown later for anisotropic media. Isopotential or
equipotential lines indicate lines of constant potential. A streamtube defines the area
between two adjacent streamlines.
A Flow net is a set of equipotential lines and
streamlines. A simple example is shown in Figure 7.2.
Figure 7.2 Flow net for a simple flow system (Freeze and Cherry, 1979)
Constant hydraulic head, h, of 100 m and 40 m, exists at the left and right boundaries,
respectively. No flow boundaries occur at the top and bottom. As can be seen, the
7.3
Chapter 7 - Steady-state Laminar Flow of Homogeneous Fluids
flowlines indicate the direction of flow from higher potential to lower, i.e., to the right.
The equipotential lines (dashed lines) are orthogonal to these flowlines.
For
homogeneous, isotropic media, flow nets must obey the following rules.
a. flowlines and equipotential lines are normal to each other
b. equipotential lines must intersect impermeable boundaries at right angles
c. equipotential lines must parallel constant potential boundaries
Notice the boundary conditions are special forms of flowlines or equipotential lines. For
a closed boundary the fluid velocity normal to the boundary is zero (See figure 7.3).
Since there is no flow across the boundary, the flowlines adjacent to the boundary must
be parallel to it, and therefore the equipotential lines must meet the boundary at right
angles (See fig. 7.2 as an example).
ln
Flux normal to boundary is zero
Figure 7.3 Schematic of a closed boundary
Mathematically, this boundary can be expressed as:
vn  0

0
ln
(7.13)
where n is the unit vector normal to the boundary and ln is the distance measured parallel
to n.
In the case of an open boundary, flux is continuous, therefore, the boundary is an
isopotential line, (= constant) and no flux exists tangential to the boundary. Figure 7.4
represents this condition.
7.4
Chapter 7 - Steady-state Laminar Flow of Homogeneous Fluids
Flux normal to boundary is
constant
Figure 7.4 Schematic of an open boundary
Effect of anisotropy on flow net
An example of an anisotropic system is illustrated in Figure 7.5. The vertical
section represents flow from a surface pond at h = 100 toward a drain at h = 0. The drain
is considered to be one of many parallel drains set at a similar depth oriented
perpendicular to the page.
h=100
h=0
Figure 7.5 Flow problem in a homogeneous, anisotropic system.
The vertical impermeable boundaries are "imaginary"; they are created by the
symmetry of the overall flow system. The lower boundary is a real boundary; it
represents the base of the surficial soil, which is underlain by a soil or rock formation
with a conductivity several orders of magnitude lower. If the vertical axis is arbitrarily set
with z = 0 at the drain and z = 100 at the surface, then from h = p + z, and the h values
given, we have p = 0 at both boundaries. The soil in the flow field has an anisotropic
conductivity.
The influence of anisotropy on the nature of groundwater flow nets is illustrated
in Figure 7.6 for the previous problem. The streamlines are shown as the solid lines, the
lines of equipotential are dashed. Note, the most important feature of the anisotropic
flow nets is the lack of orthogonality.
7.5
Chapter 7 - Steady-state Laminar Flow of Homogeneous Fluids
kx
ky
kx
ky
kx
ky

1
4
1
4
Figure 7.6 Flow distributions for various permeability anisotropies
(Freeze and Cherry, 1979)
7.3 2D flow, superposition
Consider horizontal (xy) plane flow in a homogeneous media, with upper and
lower impermeable boundaries as illustrated in Figure 7.7. Permeability is anisotropic
and aligned with the principal directions. In this example, no vertical flow exists; i.e., vz
= 0.
impermeable
z
x
impermeable
y
vz = 0
Figure 7.7 Schematic of horizontal flow in a homogeneous media
7.6
Chapter 7 - Steady-state Laminar Flow of Homogeneous Fluids
From the definition of potential,
p dp
 
 gz
po 
it is evident that pressure and density are dependent on the vertical direction, z, therefore
dp
 g
dz
(7.14)
The continuity equation for this problem becomes,
  k x  p    k y

    
x    x  y  

 p 
    0
 y 
(7.15)
where the horizontal components of velocity are given by Darcy’s Law.
For the ideal liquid case the pressure becomes, p
( x, y,z)
p
( x, y,0)
 gz , where z=0
occurs at the bottom of the flowing layer.
To simplify the problem, apply the following coordinate transformations,
x  x
kx
y 
ky
y
(7.16)
The result is a Laplace-type equation,
2
2
 p  p

0
2
2
x
y
(7.17)
This solution is valid on the xy plane at a given depth, z. However, pressure is a function
of z as well, therefore to account for this dependency, introduce a new variable U’
(Collins, 1961).
U 
kxk y

p  gz 
(7.18)
The solution is:
2
2
 U  U

0
2
2
x
y
7.7
(7.19)
Chapter 7 - Steady-state Laminar Flow of Homogeneous Fluids
Both Eqs. (7.17) and (7.19) are Laplace-type equations. Numerous solution techniques
have been presented in the literature for these equations and the associated boundary
conditions.
Multiwell systems
The Superposition principle defines the net potential at any point in the multiwell
system as the sum of the potential contributions from each well computed as if it were
alone in the system. The linear combination of n source/sink terms in an isotropic and
homogeneous media can be expressed as;
 ( x, y ) 


2

 n
 qi ln  x  xi 2  ( y  yi )   C
4kh i 1


(7.20)
For example, Figure 7.8 illustrates a two well system, a source and a sink. The desired
location for potential is at (x,y).
Sink
(x2,y2)
Source
+ (x,y)
(x1,y1
)
Figure 7.8 Schematic of a two well system for the principle of superposition
The flux in the x and y directions can be determined by using Darcy’s Law,
vx  
k 
?
 x
(7.21)
Flow potential near a discontinuity
As a special 2D problem consider a horizontal layer of uniform thickness and
infinite in areal extent.
This layer is separated by a vertical plane; with differing
permeability on opposite sides. Figure 7.9 illustrates the problem and the differential
equations on the surface of the discontinuity and in the two regions.
incompressible…thus density and viscosity are invariant.
7.8
The fluid is
Chapter 7 - Steady-state Laminar Flow of Homogeneous Fluids
kb
2
 
2
 
(-d,0)
x
y
cq
b 
2
b 0
2
ka
(d,0)
d
q
2
2
 a  a

0
2
2
x
y
At discontinuity:
 a (0,0)   b (0,0)
 a
 b
ka
x
 kb
x
Figure 7.9 Schematic of a 2D problem with a linear discontinuity
A well of fixed point source strength q is located at (d,0). To satisfy the boundary
conditions at the discontinuity, another point source of strength cq is located at (-d,0) in
medium (b). This procedure is known as the method of images. From the principle of
superposition, the system of equations becomes,
 a ( x, y) 
q  
2
2 
2
2

ln x  d   y   c ln x  d   y  




4ka h 
 b ( x, y ) 
2 
q  
2
ln x  d   y    D
4kbh  

where c and D are constants to be determined.
Applying the boundary conditions at the discontinuity (y=0) we have
k
1 C  D a
kb
and
1 C  D
combine and substitute in Equations (7.22) and (7.23), results in
7.9
(7.22)
(7.23)
Chapter 7 - Steady-state Laminar Flow of Homogeneous Fluids
 a ( x, y ) 
q 
 

2   1  kb ka  ln x  d 2  y 2  
2
ln x  d   y   








4ka h 

 1  kb ka 


 b ( x, y) 
2    2 kb ka 
q  
2
ln x  d   y    
4k h  
   1  kb ka 
b
(7.24)
(7.25)
Two limiting examples illustrate the utility of the above equations. If kb = 0, there exists
no potential in region (b) and therefore the discontinuity reduces to an impermeable
boundary. If kb  , then the discontinuity becomes an equipotential boundary.
7.10