CVEEN 7920: Carbon Capture and Storage
Wednesday, 20 October 2010
Topic: Solving PDE’s (solutions to mathematical models)
1
Models of CCS = Models of Fluid Flow & Other Processes
1. Reprise: the continuity equation and groundwater
(fluid) flow equation.
1. What do you need to solve the groundwater flow
equation?
2. Taylor Series - the backbone of the FDM
3. Building an FDM Approximation
2
Groundwater Flow Equation
dh
Left side of continuity equation has q, this is driven by
dx
Let’s try to relate the left side to h
Assume the axes of
Then
qX   K X
K are parallel to x,y,z
h
h
h
, qY   KY
, qZ   K Z
x
y
z
 h 
h 
h 





K
 K X 
 Y y   K Z 
h
x 
z 

 
 
 SS
x
y
z
t
  Kh  S S
h
t
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If K axes not aligned with x, y, z, then we must use   Kh  S S
h
t
Groundwater Flow Equation
Possible combinations of (in)homogeneity and (an)isotropicity
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(Freeze and Cherry, 1979)
Groundwater Flow Equation
 h 
h 
h 


 K X   KY   K Z 
y 
h
x 
z 

 
 
 SS
x
y
z
t
 h   K h 
 h 
 K 
 y   K 

x

 
   z   S h
S
x
y
z
t
heterogeneous, anisotropic medium
heterogeneous, anisotropic media
 2h
 2h
 2h
h
K x 2  K y 2  K z 2  SS
x
y
z
t
heterogeneous, anisotropic media
  2h  2h  2h 
h
K  2  2  2   S S
z 
t
 x y
heterogeneous, anisotropic media
h
K h  S S
t
2
5
Groundwater Flow Equation
 h 
h 
h 


 K X   KY   K Z 
y 
h
x 
z 

 
 
 SS
x
y
z
t
All transient conditions:
heterogeneous, anisotropic medium
 h   K h 
 h 
 K 
 y   K 

x

 
   z   S h
S
x
y
z
t
heterogeneous, isotropic media
 2h
 2h
 2h
h
K x 2  K y 2  K z 2  SS
x
y
z
t
homogeneous, anisotropic media
  2h  2h  2h 
h
K  2  2  2   S S
z 
t
 x y
homogeneous, isotropic media
h
K h  S S
t
2
6
Groundwater Flow Equation
 h0
2
homogeneous, isotropic, steady-state
7
Models of CCS = Models of Fluid Flow & Other Processes
1. Reprise: the continuity equation and groundwater
(fluid) flow equation.
1. What do you need to solve the groundwater flow
equation?
2. Taylor Series - the backbone of the FDM
3. Building an FDM Approximation
8
Solving the Groundwater Flow Equation
Three primary approaches:
(1) Analytical (applied more often than you might think)
(2) Graphical (flow nets)
(3) Numerical (most common)
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Solving the Groundwater Flow Equation
The ‘Boundary Value Problem’ (BVP):
1)Governing equation
2)Region of flow (e.g., geometry, dimensions, etc.)
3)Material properties, or parameterization
4)Boundary conditions
5)Initial conditions (transient problems)
6)Method of solution
10
11
Models of CCS = Models of Fluid Flow & Other Processes
1. Reprise: the continuity equation and groundwater
(fluid) flow equation.
1. What do you need to solve the groundwater flow
equation?
2. Taylor Series - the backbone of the FDM
3. Building an FDM Approximation
12
Backbone of FDM: Taylor Series Expansion
If the value of a function f(x) can be expressed in a region of x close to x=a
by the infinite pow er series
( x  a )2
( x  a )3
( x  a )n ( n )
f ( x )  f ( a )  ( x  a ) f '( a ) 
f ''( a ) 
f '''( a )...
f ( a )...
2!
3!
n!
then f(x) is said to be analytic in the region near x=a, and the series above
is unique and called the Taylor series expansion of f(x) in the
neighborhood of x=a.
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Backbone of FDM: Taylor Series Expansion
If the Taylor series exists, then knowing f(a) and all of the derivatives of f at
x = a, we can find the value of f(x) at some x different from a, as long as
we remain Òfairly closeÓto x = a.
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Backbone of FDM: Taylor Series Expansion
How w ell can one approxima te f(x) near x = a by taking only a few terms of
the Tay lor series (rather than in infinite number of terms)?
Examp le: suppose you wish to find f(b)
1 term:
f ( b)  f ( a )
2 terms :
f ( b)  f (a )  ( b  a ) f '( a )
3 terms :
( b  a )2
f ( b)  f ( a )  ( b  a ) f '( a ) 
f ''(a )
2!
Three terms will obviously get you closer to f(b) than only one or tw o
terms. Eac h additional term gets you closer to an infinite number of terms
and thus improv es the accuracy of the approximat ion.
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Backbone of FDM: Taylor Series Expansion
Regard ing the truncation of the series: the terms that are truncated
compr ise the error,  .
f ( x )  f ( a )  ( x  a ) f '( a )   ( x  a )2
The terminology used here is that the error imposed by truncating the
series after the term with (x-a)n is Òof ht e order of (x-a)n+1Ó.
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Backbone of FDM: Taylor Series Expansion
Thus , the error imposedby truncating the series after the fourth term of the
series is
( x  a )2
( x  a )3
f ( x )  f ( a )  ( x  a ) f '( a ) 
f ''( a ) 
f '''( a )   ( x  a )4
2!
3!
17
Models of CCS = Models of Fluid Flow & Other Processes
1. Reprise: the continuity equation and groundwater
(fluid) flow equation.
1. What do you need to solve the groundwater flow
equation?
2. Taylor Series - the backbone of the FDM
3. Building an FDM Approximation
18
The FDM
Fundame ntal to the finite difference approach is the concept of
discretization, where a continuous domai n D is represented by a number of
subareas. The basic idea of the FDM is to replace derivatives at a point by
ratios of the changes in appropriate variables over a sma ll but finite
interval.
h dh
h h

 lim


x

0
x dx
x x
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The FDM
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Construction of an FDM Approximation
Co nsider a function f(x) that is smoot h and continuous over some specified
interval. Then we can appr oximate f by expansion into a Taylor series
about x:
in the positive direction,
f ( x )2  2 f ( x )3 3 f
...


f ( x  x )  f ( x )  x
3
2
3! x
2! x
x
solving for
f
gives
x
f ( x  x )  f ( x )
f
  ( x )

x
x
remaining terms in series
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Construction of an FDM Approximation
Now the Òfirst forward differenceÓapproximation to the derivative of f ( x ) by
is obtained by dropping the rema ining terms  ( x ) ,
f
f (x  x)  f (x)

x
x
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