Analytical Solution of
the Diffusivity Equation
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Introduction
Analytical Solution
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Learning Objectives
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Learning objectives in this module:
Analytical Solution
Linear Systems
Radial Systems
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1. Develop problem solution skills using computers and numerical
methods
2. Review flow equations and methods for analytical solution the
equations
3. Develop programming skills using FORTRAN
No new FORTRAN elements are introduced in this
module, you should, from what you have learnt earlier,
be able to solve this problem without any problems
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Introduction
Learning Objectives
Introduction
Analytical Solution
Linear Systems
Radial Systems
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In analysis of fluid flow in petroleum reservoirs, we need partial
differential equations that describe the fluids flowing and the
reservoir they are flowing in. Then we need to be able to solve the
equations for the conditions of flow that we are interested in.
Derivation of the equations normally involves the following
elements:
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



Continuity equations
Darcy’s equations
PVT relationships for the fluids
Compressibility of reservoir rock
Examples of such equations are the simplest forms of the diffusivity
equations for linear and radial flow
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Introduction
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Introduction
Analytical Solution
Below, the geometries of the two simple reservoir systems and the
corresponding partial differential equations are shown:
Linear Systems
Radial Systems
 2P
c P
(
)
k t
x 2
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x=L
Q
Linear flow
x=0
1  P
c P
(r ) 
r r r
k t
r
Radial flow
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Analytical Solution
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Introduction
Analytical Solution
Linear Systems
Radial Systems
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In order to solve the partial differential equations shown earlier, we
need to have initial conditions, i.e. initial pressure distribution in the
system, and boundary conditions, i.e. rates or pressures at for
instance left and right sides of the systems. We will examine two of
the most common sets of conditions and analytical solutions for
these
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Linear System
Radial System
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Linear System
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Analytical Solution
Linear Systems
For the linear system, we have a horizontal porous rod, where fluid
is being injected into the left face at a flow rate Q. The injected fluid
will be transported through the rod and eventually be produced out
of the right face of the rod.
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The one-phase partial differential equation (PDE) for this system, in
it’s simplest form, is called the linear diffusivity equation. It is valid
for one-dimensional flow of a liquid in a horizontal system, where it
is assumed that porosity (), viscosity (), permeability (k ) and
compressibility (c ) all are constants.
PR
PL
Qout
x=L
Qin
x=0
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The linear diffusivity equation may be written as:
Analytical Solution
Linear Systems
Radial Systems
Program Exercise
 2 P c  P
(
)
2
x
k t
(1)
Continue
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If the initial pressure of the rod is PR , and we assume constant
pressures at the end faces, PL and PR for left and right faces,
respectively, we have the following analytical solution:
x 2  1
n2  2 k
n x 
P(x,t )  PL  (PR  PL ) 
exp(  2
t )sin(
)
L

n
L

c
L




n 1

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(2)
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Linear System
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Introduction
Analytical Solution
Linear Systems
The pressure solution is dependent on position, x, as well as time, t.
As time increases, the exponential term becomes smaller, and
eventually the solution reduces to the steady-state form:
Radial Systems
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x 2  1
n2  2 k
n x 
P(x,t )  PL  (PR  PL ) 
exp(  2
t )sin(
)
L

n
L

c
L




n 1

(2)
(3)
? the
Click
to expression
see what the
to as time increases
which is
forequation
a straightreduces
line
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Linear System
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Introduction
Analytical Solution
The corresponding steady state differential equation is obtained by
setting the right hand side of Eq. (1) equal to zero:
Linear Systems
d 2P
0
dx 2
Radial Systems
Program Exercise
(4)
Graphically, the solution may be presented as:
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P
Left side
pressure
Steady state
solution
Transient
solution
Initial and
right side
pressure
x
As can be observed from the
figure, the pressure will increase
in all parts of the system for
some period of time (transient
solution), and eventually
approach the final distribution
(steady state), described by a
straight line between the two
end pressures
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Radial System
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Introduction
Analytical Solution
Linear Systems
Radial Systems
For the radial system below (one-dimensional cylindrical
coordinates), we have a horizontal porous disk, where fluid is being
injected at the outer boundary and produced at the center. The onephase one-dimensional (radial) flow equation (PDE) in this
coordinate system becomes:
Continue
Program Exercise
1  1 P  c P

r r r r   k  r
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(5)
rw
r
For an infinite reservoir at an initial pressure Pi and with P(r∞)=Pi
Continue
and well rate q from a well in the center (at r=rw) the analytical
solution is:
 cr2 
q
P  Pi 
Ei

4 kh  4kt 
(6)

where
e u
Ei( x)  
du
u

is the exponential integral
x
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Radial System
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Analytical Solution
Linear Systems
A steady state solution does not exist for an infinite system, since
the pressure will continue to decrease as long as we produce from
the center. However, if we use a different set of boundary
conditions, so that:
Radial Systems
Program Exercise
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P(r  rw )  Pw
P(r  re )  Pe
(7)
we can solve the steady state form of the equation: Continue
1 d 1 dP 
0
r dr r dr 
(8)
By integrating twice, the steady state solution becomes:
Continue
P  Pw 
Pe  Pw 
ln r / rw 
ln re / rw 
(9)
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Introduction
Analytical Solution
Linear Systems
Radial Systems
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This programming exercise involves the construction of a
reservoir simulation program, although in a very simple form.
The following steps should be carried out:
1. Make a FORTRAN program that computes the analytical
solutions of Eqs. (2) and (6). When the program is started, it
should ask on the screen which geometry should be used, LIN or
RAD, and the name of the input data file (where all parameters
are to be read from)
2. Read from the screen which values of x (or r) and t the solution
should be computed for.
3. The results should be written to the screen as well as to an
output file
Data set for linear system
Here
Data set for radial system
Here
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Data Set for Linear System
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PL=2 atm
 = 0.25
k=1.0 Darcy
P0=PR=1 atm
L=100 cm
A=10 cm3
=1.0 cp
c=0.0001 atm-1
t-intervals:
t=10-3 , 10-2 , 10-1 s
x-intervals:
x=5, 50 cm
k = permeability [Darcy]
L = length (of rod) [cm]
=viscosity [cp]
= porosity
c = compressibility [atm-1]
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Data Set for Radial System
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h=1000 cm
 = 0.25
k=1.0 Darcy
c=0.0001 atm-1
rw=25 cm
q=104 cm3/s
=1.0 cp
t-intervals:
t= 1E06, 5E06, 10E06 s
r-intervals:
r=100, 1000, 5000 cm
k = permeability [Darcy]
rw=wellbore radius [cm]
=viscosity [cp]
= porosity
c = compressibility [atm-1]
q=flowrate [cm3/s]
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Introduction
Analytical Solution
Linear Systems
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Introduction to Fortran
Fortran Template
here
The whole exercise in a printable format
here
Web sites
 Numerical Recipes In Fortran
 Fortran Tutorial
 Professional Programmer's Guide to Fortran77
 Programming in Fortran77
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General information
Learning Objectives
Introduction
Analytical Solution
Linear Systems
Title:
Analytical Solution of the Diffusivity Equation
Teacher(s):
Professor Jon Kleppe
Assistant(s):
Per Jørgen Dahl Svendsen
Abstract:
Provide a good background for solving problems within
petroleum related topics using numerical methods
4 keywords:
Diffusivity Equation, Linear Flow, Radial Flow, Fortran
Radial Systems
Program Exercise
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Topic discipline:
Level:
2
Prerequisites:
None
Learning goals:
Develop problem solution skills using computers and
numerical methods
Size in megabytes:
0.7 MB
Software requirements:
MS Power Point 2002 or later, Flash Player 6.0
Estimated time to complete:
Copyright information:
The author has copyright to the module and use of the
content must be in agreement with the responsible author
or in agreement with http://www.learningjournals.net.
About the author
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No questions have been posted yet. However, when questions are
asked they will be posted here.
Remember, if something is unclear to you, it is a good chance that
there are more people that have the same question
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For more general questions and definitions try these
Dataleksikon
Webopedia
Schlumberger Oilfield Glossary
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References
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See for instance:
H. S. Carslaw and J. C. Jaeger: Conduction of Heat
in Solids, 2nd ed., Oxford, 1985
Program Exercise
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Numerical Recipes in Fortran in pdf format online:
Numerical Recipes in Fortran
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Summary
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Introduction
Analytical Solution
Subsequent to this module you should...
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

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
be able to keep track of loops and conditional statements
have no problems handling output and input data
have obtained a better understanding on solving problems
in Fortran
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