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Ho Chi Minh City University of Technology
Faculty of Geology & Petroleum Engineering
Modeling & Simulation Division
Presenter: D r . D o Q u a n g K h a n h
Email: [email protected]
Website: www.hcmut.edu.vn
PARTIAL DIFFERENTIAL EQUATIONS(PDEs)
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Partial Differential Equations (PDEs)
Physical meaning of PDEInitial and boundary conditions
Classification
parabolic vs. hyperbolic linear vs. nonlinear
Solution Methods
Analytical, numerical, transformation methods
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Classification
Laplace steady-state Elliptic
Diffusitivity
(heat cond) transient Parabolic
Wave transient Hyperbolic
2
2
2
2 0T
x
T
y+ =
2
2
p
x
p
t=
2
2
22
2 1
t
y
cx
y
=
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Example: Diffusivity Equation
t
p
k
cp t
=
2
Expresses conservation of mass
Slightly compressible fluid
Porous media
Single phase flow
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Linearity
If does not depend on p theequation is linear.
If does depend on p the
equation is nonlinear.
k
ct
k
ct
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Coordinate Systems
Cartesian
Radial
t
p
k
c
y
p
x
p t
=
+
2
2
2
2
t
p
k
c
r
p
rr
p t
=
+
1
2
2
These equations are for 2D problems. To go to 3D
add .2
2
z
p
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Transient versus Steady-State
Neeeds initial
and boundary conditionsNeeeds only
boundary conditions
Late-time solution
at least with const pressures
at the ends
2
2
p
x
p
t=
2
2 0p
x=
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Derivatives
The diffusivity equation includesderivatives of pressure in space and intime.
To solve the diffusivity equationnumerically we must find ways torepresent these derivatives.
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Taylor Series
p(x)
p(x+ x)
xx
x+ x
p
( ) ( ) ( )
( )
( )( )
( ) +
+
+
++=+
xp!n
x
xp!2
x
xpxxpxxp
nn
2
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First Derivative (Forward)
( ) ( ) ( )
( )
x
ppp
xx
xpxxpxp
i1ii
+
+=
+
writtenusuallyisThis
rearrangeandtermsecondafterseriesTruncate
O
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First Derivative (Backward)
( ) ( ) ( )
( )
x
ppp
x
x
xxpxpxp
1iii
+
=
writtenusuallyiswhich
Similarly,
O
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First Derivative (Central)
( ) ( ) ( )
( )
x2ppp
xx2
xxpxxpxp
1i1ii
2
+
+=
+
writtenusuallyiswhich
seriesbackwardandforwardtheSubtract
O
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Second Derivative (Central)
( ) ( ) ( ) ( )
( )
( )
( )21ii1i"
i
2
2
xpp2pp
x
x
xxpxp2xxpxp
+
+
++=
+
writtenusuallyiswhich
seriesbackwardandforwardtheAdd
O
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Accuracy
Getting the derivatives approximatedcorrectly is an important part of getting anaccurate numerical solution.
How accurate are these Taylor seriesforms?
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Experiment with the spreadsheet tolearn for yourself how xand t
affect the accuracy of the derivative.
How accurate is a second derivativeterm (space) ?
How accurate is a first derivativeterm (time)?
Accuracy
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You can study the accuracy of thederivatives (as a function of x) byplotting the absolute error between thetrue f(x) and the calculated f(x). [Also
f(x)]
You may want to consider using
logarithmic axes on these plots.
Accuracy
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Accuracy
You should have found that the spacederivative is O(x2) and the timederivative is O(t).
Knowing this if you wanted to improve theaccuracy of your solution and could halveeither x or t which would you choose?
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Linear Reservoir
Constant Pressure Boundaries
0 L
p const
(left)p const
(right)
x
ft
daypsi
1/psi
md
cp
2
20 00633
p
x
c
k
p
t
t=.
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Uni_Names
(ft^3/day)*cp/(md*ft*psi) 158.00836
(md*ft*psi)/(cp*ft^3/day) 0.00633
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Diffusivity: Numerical Solution
First discretize the region oninterest over both space and time.
xi Xi+1 Xi+2Xi-1Xi-2
k
cwhere
t
p
x
p t
00633.02
2 =
=
...,t,t,tt 321=
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Diffusivity: Numerical Solution
Replace the analytical derivatives by numericalapproximations.
Central difference in space
Backward difference in time
Right now we are not stating what timestep (n orn+1) the terms on the left are being evaluated.
( ) tpp
x
ppp n
i
n
iiii
=
+ +
+
1
2
11 2
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Implicit SolutionTemplate
t
t
t
t
xxxx
i-1 i i+1
1
n
n+1
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Implicit - Time Level of Terms
An implicit solution scheme means thatthe pressures at the nodes will beevaluated at the new timestep (n+1) insecond derivative (space) term.
( ) tpp
x
ppp n
i
n
i
n
i
n
i
n
i
=
+ ++
++
+
1
2
1
1
11
1 2
( ) ( )
t
x
k
c
t
x t
=
=
22
00633.0
n
i
n
i
n
i
n
i
n
i ppppp =+ ++
++
+
11
1
11
1 2
n
i
n
i
n
i
n
i pppp =++ +
++
+
1
1
11
1 )2(Dr. Do Quang Khanh23
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Boundary Conditions
We must specify the conditions of the left and rightboundaries
constant pressure pi= const (I = 1 or n)
constant rate
constxppor
constxppexamplefor
boundaryflownoaisx
p
constx
prateSpecity
NN =
=
=
=
1
12
,
,
""0
:
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( ) ni
n
i
n
i
n
i pppp =++ +
+++
1
1
11
1 2
valueleftgivenpn 11 =+
valuerightgivenpnnx 1
=+
=
5
4
3
2
1
5
4
3
2
1
55
444
333
222
11
d
d
d
d
d
x
x
x
x
x
ba
cba
cba
cba
cb
Thomas algorithm
System of Equations
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Thomas Algorithm
=
5
4
3
2
1
5
4
3
2
1
55
444
333
222
11
d
d
d
d
d
x
x
x
x
x
ba
cba
cba
cba
cb
The problem Ax=b can be solved very efficientlywhen A is a tridiagonal matrix. The matrix itself is not
stored. Only three vectors a,b and c are stored.
These hold the values on the matrix diagonals.
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Sub Thomas(a() As Double, b() As Double, c() As Double, _d() As Double, x() As Double)
'Tridiagonal system of equationsDim n As Integer, i As Integern = UBound(b)ReDim w(n) As Double, g(n) As Doublew(1) = b(1)g(1) = d(1) / w(1)For i = 2 To n
w(i) = b(i) - a(i) * c(i - 1) / w(i - 1)g(i) = (d(i) - a(i) * g(i - 1)) / w(i)Next ix(n) = g(n)For i = n - 1 To 1 Step -1x(i) = g(i) - c(i) * x(i + 1) / w(i)
Next i
End Sub
n : int, inp
a(n) : real, inp b(n) : real, inp c(n) : real, inp d(n) : real, inp
x(n) : real, out
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Sub VBA111()' Linear reservoir: constant pressures at the two end pointsDim nx As Integer, nt As Integer, ipr As Integer
'Input
ReDim a(nx) As Double, b(nx) As Double, c(nx) As DoubleReDim d(nx) As Double, p(nx) As Double'
dx = xlen / (nx - 1)dt = tend / ntalpha = phi * mu * ct / (0.00633 * k) * dx ^ 2 / dt
'Initializationt = 0
'Time stepsFor it = 1 To nt
'make a, b, c, d,Call Thomas(a, b, c, d, p)Next it
End With
End Sub
ProgramS
tructure
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Explicit Solution Scheme
Discretize the region of interest
Discretize the diffusivity equation
Central difference in space Forward difference in time
( ) ( )
k
cwhere
ppt
xppp
t
ninininini
00633.0
2 1
2
11
=
=+ +
+
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Finite Difference Template
t
t
t
t
xxx x
i-1 i i+1
1
n
n+1
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Explicit Solution
Explicit equations
Left boundary (i = 1)
Interior points (i = 2,,nx-1)
Right boundary (i = nx)
valueleftgivenp1
=(or no flow)
( ) 111 2 ++
+ ni
n
i
n
i
n
i pppp
valuerightgivenp 1nnx =+ (or no flow)
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Stability
Implicit: stableExplicit: at less than 2 it is unstable!
Ex: What is the maximum timestep thatwe can take with the explicit FDEfor thefollowing data:
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Linear Reservoir with Const. Press. Boundaries:
Suppose we have a sand packed core container 20ftlong. We pressure it with air at 100 psia. Then we openvalve on both ends to the atmospheric pressure, 14.7psia. Consider the finite difference equation for this
flow problem in the following form:
Suppose we are using 5 grid points with a grid point oneach end. Fill in the table of the matrix coefficients and
right hand side for the first timestep. Include theproper boundary conditions. Use exact values whereyou can, but use the math symbols elsewhere.
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ASSIGNMENTS, TEST PROBLEMS
)(2 11111
1
n
i
n
i
n
i
n
i
n
i ppppp =+ ++
++
+
