NOTE: Darcys flow
1 Horizontal flow
The relationship between flow in the horizontal direction and pressure loss is expressed
by Darcys equation
v
k dp
 dx
(1)
There
v: flow velocity, flow rate divided by cross-sectional area
k: permeability
m: viscosity
p: pressure
x: flow direction
The pressure gradient here is due to friction when fluid flows through pore channels. The
friction gradient follows from equation (1)
dp f
dx


k
v
(2)
2 Non-horizontal flow
When the direction is not horizontal, the height changes along the flow lines. Then it is
necessary to add the change in static pressure, similar to flow in pipes. With static
gradient due to gravity and friction gradient expressed by (2), the flow equation becomes
dp
dh 
 g
 v0
dx
dx k
(3)
When we know the flow direction, we can find the pressure / rate by integrating the flow
equation (3) above
Example: Gravitational flow
To find permeability of fine sand, we may measure the flow through a column, Figure 1
filled with sand. (Devices to maintain constant water level at the inlet and outlet are
needed, but not shown in Figure 1)
Figure 1: Column for measuring permeability
Task
a) Find the pressure profile: p (x), x = (0 - 1 m) in closed off column ( no water flow)
b) Find the pressure profile with water flow (atmospheric inlet and outlet pressure)
c) flow were measured at 0,5 ml / s. Find the permeability.
---------------------------
3 Hydraulic potential
By introducing hydraulic potential: , the flow equation (3) may be re-written as
k d
 ds
  p  gh
v
(4)
(5)
The pressure in Darcys equation for horizontal flow (1) is replaced by potential (5), it
becomes equal to equation (4) above. This means that the flow formulas are derived for
the horizontal reservoir also apply equally to inclined reservoir, as long as we replace
pressure by potential.
If the pressure measured at the reference depth (h = constant), the change in hydraulic
potential (5) correspond to pressure changes: d=dp. Thus, potential may be
interpreted as "pressure at the reference depth", and is often reported as such.
The hydraulic potential represents the sum of potential energy due to pressure and
elevation. In a static water column pressure rises with depth, such that the hydraulic
potential (5) remains constant. During fluid flow, friction transfers potential energy into
heat. This reduces the hydraulic potential.
The properties of fluids in the reservoir are related to local pressure and temperature. For
example, in a tilted reservoir the pressure in higher lying sections may decrease such that
gas is released, while pressure at some reference depth (for example at the well) may be
above saturation. The hydraulic potential requires single-phase fluid, and will then no
longer be valid.
4. Multi dimensional flow
In many cases it is difficult to determine the flow lines in advance. They have to be
derived from solutions and not vice versa. This can be done by first finding the potential
distribution. Flow components along axes directions may be estimated by (4) and the
flow direction equals the vector sum of these

k  k 
v  vx  v y  

 x  y
(6)
The vector properties of Darcys law has been experimentally visualized the by Kozeny:
Wassekraft und Wasserwirtschaft; referred by Muskat: The Flow of Homogenous Fluid
Through Porous Media / 1946 /.
5 Source-sink
We will as an example to examine the effect of locating a horizontal well near an
impervious layer, as sketched in Figure 2. We assume that the total layer height: h is large
(h>> b) so that other boundaries will not affect by the flow. (The case is similar for a
vertical well near a tight faulting.)
Figure 2. Well located in the distance b from an impervious layer.
Fluid can not flow through the layer. If two similar wells are located in an infinite
reservoir, we can expect an equivalent hydraulic boundary between the wells.
Pressure distribution at the stationary influx to a vertical well in an infinite reservoir has
been derived previously (2-3). By substituting the potential instead of pressure and well
length (Lw) instead of layer height, we obtain
 r    w 
qo Bo  o r
ln
2 k Lw rw
(7)
The potential will thus change in proportion to the logarithm of distance: lnr. By
substituting press with hydraulic potential, the potential may be expressed as
 r   C ln r   pw  C ln rw 
(8)
The terms in the parenthesis are constants. Since we will consider potential differences,
these will ultimately cancel. To simplify the notation, these are normally omitted in
potential calculations.
Potential Distribution with 2 wells
Figure 3 shows a well placed in x = 0, y = b with a "mirror well" at: x=0, y=-b. Since
Darcys equation is linear, solutions can be added. An arbitrary point (x, y) is considered:
the distance from the well, and mirrors of well is: r1 and r2. By adding up the potential
contributions for each of the wells: (x,y)=(r1)+(r1) and convert to Cartesian
coordinates the total potential is expressed as
 x , y   C  ln x 2   y  b 2  ln x 2   y  b 2 


(9)
Figur 3 Two radial sources, in the Cartesian coordinate system
Figure 4 shows the potential distribution (9) with the well 2m from the boundary layer.
Figure 4 Potential distribution and flow lines
Task
If the potential (9) is correct, the x-axis act as a flow barrier. Use the flow equation (6) to
examine if this is true
k  x , y 
vy

 0?
y 0
 y y 0
---------------------------
6 Potential losses from the reservoir to the well
Figure 4 indicate that when the distance from the well is large, the potential lines
approach the half circles. Along the y-axis (x = 0) the potential (9) far from the well may
be approximated as
 r  r b  Clnr  b  lnr  b  2C ln r
(10)
The potential of the well, we find from (9) by setting: x = rw, y = b
 w   rw ,b   C  ln rw  ln rw 2  2b 2 

(11)

Subtraction of (10) - (11) gives a potential change from an outer radius to the well
 2b 
q B
r q B
 r    w  o o o ln  o o o ln 1   
 kh
rw 2 k h
 rw 
2
(12)
The first part after the equallity sign in (12) indicates potential change due to radial flow
over a semi-circle sector: , (180o). (This we can be seen by comparing (12) with (7),
which represents the radial influx of full-sector opening: 2 .). The second part may be
interpreted as potential change due to deviations from radial inflow, near the well. This
can be seen as skin loss
2
qo Bo o 1   2b  
 s  
ln 1   
 k h 4   rw  


(13)
Negative skin means that the pressure loss will be less if well will be placed further away
from the borders
7 Productivity of pseudo-stationary production
Pseudo Stationary production involves reservoir with close to outer boundary. If the
reservoir geometry is "fairly square" we can represent this by a semicircle with the
equivalent radius equal to the drainage cross-section. The influx is considered as radial
over semicircle sector, with local skin quantified by (13). By analogy to previous results
(2-14), we can set up the productivity index
qo
 kh
J

(14)
2
 R  w
 re 3 1 

b
Bo  o  ln   ln 1  4 2  
rw  
 rw 4 4 
Then inflow Efficiency becomes
ln re rw   0.75
(15)
  2b  2 

ln re rw   0.75  0.25 ln 1  
rw  



Figure 6 illustrates the efficiency factor as a function of distance between well and
reservoir. Distance to the outer boundary and the well radius is: re=200, rw =0.1m. With
well 5m from the outer boundary (re/rw=50), the inflow efficiency as 1.5, ie 50% better
than for a well at the border. With the increase of distance from the boundary layer, the
efficiency of the well will increase "without flow obstruction", i.e. with radial influx of
full circular opening
E
Figure 6 The flow efficiency as a function of distance from the flow boundary
Script
% Well and image well in infinite resevoir
clear
% øvre grenser
xe=10;
ye=10;
% nedre grenser
x=-xe;
y= -ye ;
% brønn
b=2;
rw=0.1;
%diskretiseringsintervaller
dy=0.10;
dx=0.10;
xlengde=xe -x;
ylengde=ye-y;
nx=xlengde/dx;
ny=ylengde/dy;
%------------------------- utrekningssløyfe
fak=1;
% forhold kilde/sluk
for m=1:ny
% y-retning
for k=1:nx
% x-retning
x=x+dx;
p(m,k)=log((x^2 +(y-b)^2)^0.5)+ fak*log((x^2 +(y+b)^2)^0.5);
% potensialfunksjon
strom(m,k)=atan2(x,(y-b))+fak*atan2(x,(y+b));
%
strømningsfunksjon
xp(k)=x;
end
x=-xe;
yp(m)=y;
y=y+dy;
end
% Plotte
clf
%
undersøk hvilken
lengde som begrenser plottet
if ylengde<xlengde/1.35
% x-lengden begrenser plottet, set y-lengde i forhold
bredde=0.9;
hoyde=bredde*ylengde/xlengde*1.35;
else
% y-lengden begrenser plottet, set x-lengde i forhold
hoyde=0.6*0.9;
bredde= 2*hoyde*xlengde/ylengde/1.35;
end
axes('position',[ 0.05 0.05 bredde hoyde ])
%
strømnngslinjer (
strømningsfunksjon --- koteinndeling)
maxstrom=fix( max(max(strom)) )
minstrom=fix(min(min(strom)))
delstrom=(maxstrom-minstrom)/20;
vstrom=minstrom:delstrom:maxstrom;
[C,H]= contour(xp,yp,strom,vstrom);
%
potensial ---koteinndeling
maxp=max(p);
minp=log(rw)+ log((rw^2 +(b+b)^2)^0.5);
delp=(maxp-minp)/20;
v=minstrom:delstrom:maxstrom;
%
om tallfesting av kotene er ønsket clabel(C,H)
hold on
[C,H]= contour(xp,yp,p,v,'LineColor',[ 0 0 1]);
axis([-10 10 0 10])
% brønn
xw=[0 0];
yw=[0 0];
plot(0,b,'ko')
hold off
grid