solution methods - cairo university · 2020-03-11 · 12/26/2017 dr.helmy sayyouh 2 one-dimensional...
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Solution Methods
Dr. Helmy Sayyouh
Petroleum Engineering
Cairo University
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One-dimensional flow system
One-dimensional system composed of five blocks.
A no-flow boundary condition is imposed on the left end, while a non-zero pressure gradient is specified on the right.
A well with a flow rate qs is located in block #2. We assume that the flowing fluid is slightly compressible.
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The characteristic equation for any block
Pressure is not known in any of these five blocks, and so we must write
Equation for each of them (i.e., i = 1,..., 5):
D P E P F P Qi
n
i
n n
i
n
i
n
i
n
i
1
1
1
1
1
1
i P E P F P Q
i P E P F P Q
i P E P F P Q
i P E P F P Q
i P E P F P Q
n n n n n n
n n n n n n
n n n n n n
n n n n n n
n n n n n n
1
2
3
4
5
1 0
1
1 1
1
1 2
1
1
2 1
1
2 2
1
2 3
1
2
3 2
1
3 3
1
3 4
1
3
4 3
1
4 4
1
4 5
1
4
5 4
1
5 5
1
5 6
1
5
:
:
:
:
:
D
D
D
D
D
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Moving to block #5, the imposed constant pressure gradient
implies that
or
P P
xC
n n
6
1
5
1
1
P P C xn n
6
1
5
11
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In implementing the flow rate specified at the well block, we invoke the following definition of Q2:
Q qV c
tPs
b n
2
2
25615
.
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The 5x5 coefficient matrix is a tri-diagonal coefficient matrix, which is normally written in a more compact form as follows:
O O
O
O O
O
O O
0
0
F
E
D
i
i
i
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Two-dimensional flow system
The reservoir under consideration has three no-flow boundaries and one constant-pressure boundary.
A well is located in block (2,2), at which sand face pressure is specified as Psf.
Although there are nine blocks in the system, the constant pressure specification on the boundary blocks (1,1), (2,1), and (3,1) means that only six blocks have unknown pressures.
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We need to write Equation for just these six
blocks:
i = 1, j = 2: B1,2
n
i = 2, j = 2: B2,2
n
2,2
i = 3, j = 2: B3,2
n
3,2
Pn
Dn
Pn
En
Pn
Fn
Pn
Hn
Pn
Q
Pn
Dn
Pn
En
Pn
Fn
Pn
Hn
Pn
Q
Pn
Dn
Pn
En
P
1 1
1
1 2 0 2
1
1 2 1 2
1
1 2 2 2
1
1 2 1 3
1
1 2
2 1
1
2 2 1 2
1
2 2
1
2 2 3 2
1
2 2 2 3
1
2 2
3 1
1
3 2 2 2
1
3
, , , , , , , , , ,
, , , , , , , , ,
, , ,
, , , , , ,
, , , , , , , , , ,
, , , , , , , , ,
2
1
3 2 4 2
1
3 2 3 3
1
3 2
1 2
1
1 3 0 3
1
1 3 1 3
1
1 3 2 3
1
1 3 1 4
1
1 3
2 2
1
2 3 1 3
1
2 3 2 3
1
2 3 3 3
1
2 3 2 4
1
2
nF
nP
nH
nP
nQ
Pn
Dn
Pn
En
Pn
Fn
Pn
Hn
Pn
Q
Pn
Dn
Pn
En
Pn
Fn
Pn
Hn
Pn
Q
i = 1, j = 3: B1,3
n
i = 2, j = 3: B2,3
n
,
, , , , , , , , , ,
3
3 2
1
3 3 2 3
1
3 3 3 3
1
3 3 4 3
1
3 3 3 4
1
2 3i = 3, j = 3: B3,3
nP
nD
nP
nE
nP
nF
nP
nH
nP
nQ
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The matrix in this case is a penta-diagonal coefficient matrix, and it is
normally written in the more compact form shown below:
O O O
O O
O O O
O O
O O O
O O
O O O
0
0
0
0
Hi j
Fi j
Ei j
Di j
Bi j
,
,
,
,
,
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Three-dimensional flow system
We start with the flow equation in its finite difference
form:
Z P B P D P E P F P
H P S P Q
i j k
n
i j k
n
i j k
n
i j k
n
i j k
n
i j k
n
i j k
n
i j k
n
i j k
n
i j k
n
i j k
n
i j k
n
i j k
n
i j k
n
i j k
, , , , , , , , , , , , , , , , , , , ,
, , , , , , , , , ,
1
1
1
1
1
1 1
1
1
1
1
1
1
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A three-dimensional single-phase problem leads to
a hepta-diagonal coefficient matrix whose general form is as follows:
O O O O
O O O
O O O O O
O O O
O O O O
O O O
O O O O
O O O
O O O O
O O O
O O O O O
O O O
O O O O
0
0
0
0
0
0
Si j k
Hi j k
Fi j k
Ei j k
Di j k
Bi j k
Zi j k
, ,
, ,
, ,
, ,
, ,
, ,
, ,
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Irregularly bounded reservoirs
The tri-, penta-, and hepta-diagonal matrix structures
obtained for the three previous examples are quite
standard. However, discretization of irregularly-
bounded reservoirs may yield other matrix structures.
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E1 P1 + F1 P2 + H1 P5 = Q1 D2P1 + E2P2 = Q2
E3P3 + F3P4 + H3P6 = Q3
D4P3 + E4P4 + F4P5 + H4P7 = Q4
B5P1 + D5P4 + E5P5 + H5P8 = Q5
B6P3 + E6P6 + F6P7 = Q6
B7P4 + D7P6 + E7P7 + F7P8 + H7P10 = Q7
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Multi-Phase Flow Equations: Solution Methods
Simulating multiphase fluid flow in porous media involves solving a system of coupled non-linear partial differential equations.
The various solution techniques differ with respect to how we manipulate the governing partial differential equations.
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IMPES method
Obtain a single equation in which the sole unknown is the pressure of one of the phases.
We achieve this by combining the partial differential equations for each phase in such a way as to eliminate the saturation derivatives.
Assume capillary pressure to be constant during any time step.
Obtain just one partial differential equation, with a phase pressure as the only unknown (this is usually the water-phase pressure).
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Obtain the appropriate characteristic equation.
Generate a system of linear algebraic equations.
The coefficients appearing in this system of equations are functions of the pressures and saturations; therefore, they are estimated using the information available at the previous iteration level.
When the solution for the phase pressure (e.g., Pw) distribution is obtained, the next step is to solve explicitly for that phase saturation distribution, Sw from the partial differential equation describing the flow of that phase.
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At this stage, we know the Pw and Sw distributions.
This enables us to determine the oil phase pressure
distribution using the capillary pressure relationship
between the oil and water phases.
Similar to the determination of Sw, after obtaining
the Pw distribution, we explicitly solve the oil-phase
partial differential equation for the oil-phase
saturation, So.
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With the values of So and Sw calculated, we can
easily determine Sg (Sg = 1 - So - Sw).
Finally, using the capillary pressure relationship
between the oil and gas phases, we obtain the gas
phase pressure (Pg) distribution.
This completes one iteration; we then repeat the
whole procedure until we achieve convergence.
At the beginning of each iteration, all the pressure
and saturation dependent terms are updated using
the most recent information available.
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Flow chart highlighting the major steps involved in the
IMPES method
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Solution of Matrix Equations
In setting the computer model, we face the problem of
solving a set of n equations relating n unknowns,
which are expressed in the form of
A11 x1 + a12 x2 + a13 x3 + ... +a1n xn = C1
A21 x1 + a22 x2 + a23 x3 + ... +a2n xn = C1
....................................................
An1 x1 + an2 x2 + an3 x3 + ... +ann xn = Cn
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We can put the left-hand members of the previous equation into a square array of the coefficients, known as the coefficient matrix,
and the unknown vector,
a a a
a a a
a a a
n
n
n n nn
11 12 1
21 22 2
1 2
X
x
x
x n
1
2
....
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Standard ordering by rows
Ordering by rows for grid blocks in a 5x2 model, and the resulting coefficient matrix.
The non-zero elements of the coefficient matrix are indicated by x; positions, while zero elements are left blank.
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Standard ordering by columns
The band width for
the coefficient
matrix is five
(2x2+1=5).
This ordering
scheme will
significantly reduce
computational
requirements.
Figure shows the standard ordering by columns of the same reservoir.
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Checkerboard (Cyclic-2) Ordering In ordering the grid
blocks, we number
the shaded blocks
first and then the
unshaded blocks.
The resulting
coefficient matrix
offers the advantage
that, during the
forward solution
stage of Gaussian
elimination, only a
small portion of the
matrix needs to be
worked for
triangularization
Consider a reservoir model numbered in a
checkerboard fashion.
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Checkerboard (D-4) Ordering
In checkerboard D-4 ordering, we number grid blocks along alternate diagonals.
This alternate diagonal ordering leads to substantial savings in computational overhead (CPU time and storage).
Typical coefficient matrix generated
from D-4 ordering of a 4x5 matrix.
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(D-2) Ordering
The D-2 ordering scheme is
also known as diagonal
ordering.
The resulting coefficient
matrix has a relatively small
bandwidth
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System of linear equations solution methods
Our goal is always to end up with a system of linear
equations, regardless of whether the original equations
are linear or non-linear.
Two main categories of solution methods for the
resulting system of linear algebraic equations: direct
and iterative.
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Direct solution methods
For a direct solution, we assume that the machine
performing the computations is capable of carrying an
infinite number of digits (i.e., there is no round-off
error).
A direct solution method will yield an exact solution
after a finite number of elementary arithmetical
operations.
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The Gaussian elimination technique
Is the fundamental algorithm used by direct
solvers.
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Thomas’ Algorithm
Is designed for tri-diagonal coefficient
matrices.
Simply avoid performing any arithmetic
operations on the zero elements of the
coefficient matrix, thereby saving a
significant amount of time.
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Iterative methods
Iterative solution methods involve making an initial
guess, or approximation of the solution vector, and
then improving on this guess by successfully
implementing the algorithm.
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Iterative methods offer two important
advantages: less storage requirements and
less computational work.
The simplest and best-known iterative
methods for solving systems of linear
algebraic equations are grouped under the
name fixed point iterative methods.
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Use a common multiplier, ωopt (the optimum acceleration parameter), whose value is always lies between 1 and 2.
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Comparison of convergence paths for the three
algorithms.
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Comparison of direct and iterative
methods