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Oil Reservoir Simulation General Math & Num Models Computational Approach References
Computational Methods for Oil RecoveryPASI: Scientific Computing in the Americas
The Challenge of Massive Parallelism
Luis M. de la Cruz Salas
Instituto de Geofsica
Universidad Nacional Autonoma de Mexico
January 2011
Valparaso, Chile
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Oil Reservoir Simulation General Math & Num Models Computational Approach References
Mathematical and Computational Group (GMMC)
Natural Resources Dept.Dr. Ismael Herrera Revilla
Dra. Graciela Herrera Zamarron
Dr. Luis M. de la Cruz Salas
Dr. Guillermo Hernandez Garca
Dr. Norberto Vera Guzman
Students
Esther Leyva
Antonio Carrillo
Ivan Contreras
Alberto Rosas
Emilio Zavala
Ricardo Flores
Computational Group
Daniel A. Cervantes Cabrera
Alejandro Salazar SanchezDaniel Monsivais Velazquez
Renato Leriche Vazquez
Hector U. Barron Garca
Ismael Herrera Zamarron
Eduardo Murrieta Leon
http://www.mmc.geofisica.unam.mx
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Oil Reservoir Simulation General Math & Num Models Computational Approach References
Table of contents
1 Oil Reservoir SimulationMotivation
2 General Math & Num Models
Axiomatic FormulationNumerical MethodsFinite Volume Method
3 Computational Approach
TUNA
4 References
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Oil Reservoir Simulation General Math & Num Models Computational Approach References
Motivation
Table of contents
1 Oil Reservoir SimulationMotivation
2 General Math & Num Models
Axiomatic FormulationNumerical MethodsFinite Volume Method
3 Computational Approach
TUNA
4 References
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Oil Reservoir Simulation General Math & Num Models Computational Approach References
Motivation
Oil Reservoir Projects
Funded by PEMEX
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O l R S l G l M h & N M d l C l A h R f
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Oil Reservoir Simulation General Math & Num Models Computational Approach References
Motivation
Oil Reservoir Projects
Funded by PEMEX
Collaboration with IMP and CIMAT.
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Oil R i Si l ti G l M th & N M d l C t ti l A h R f
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Oil Reservoir Simulation General Math & Num Models Computational Approach References
Motivation
Oil Reservoir Projects
Funded by PEMEX
Collaboration with IMP and CIMAT.
1 WAG injection.
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Oil R s i Sim l ti G l M th & N m M d ls C m t ti l A h R f s
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Oil Reservoir Simulation General Math & Num Models Computational Approach References
Motivation
Oil Reservoir Projects
Funded by PEMEX
Collaboration with IMP and CIMAT.
1 WAG injection. 2 AIR injection.
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Oil Reservoir Simulation General Math & Num Models Computational Approach References
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Oil Reservoir Simulation General Math & Num Models Computational Approach References
Motivation
Oil Reservoir Projects
Funded by PEMEX
Collaboration with IMP and CIMAT.
1 WAG injection. 2 AIR injection. 3 SLS method.
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Oil Reservoir Simulation General Math & Num Models Computational Approach References
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Oil Reservoir Simulation General Math & Num Models Computational Approach References
Motivation
Oil Reservoir Projects
Funded by PEMEX
Collaboration with IMP and CIMAT.
1 WAG injection. 2 AIR injection. 3 SLS method.
Oil reservoir simulation is a grand challenge.
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Oil Reservoir Simulation General Math & Num Models Computational Approach References
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Oil Reservoir Simulation General Math & Num Models Computational Approach References
Motivation
The major goal of reservoir simulation is to predict futureperformance of reservoir and find ways and means of optimizing
the recovery of some of the hydrocarbons under various operatingconditions.
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O Rese vo S u at o Ge e a at & u ode s Co putat o a pp oac Re e e ces
Motivation
The major goal of reservoir simulation is to predict futureperformance of reservoir and find ways and means of optimizing
the recovery of some of the hydrocarbons under various operatingconditions.It involves four main interrelated modeling stages:
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p pp
Motivation
The major goal of reservoir simulation is to predict futureperformance of reservoir and find ways and means of optimizing
the recovery of some of the hydrocarbons under various operatingconditions.It involves four main interrelated modeling stages:
And requires a combination of skills of physicists, mathematicians,
reservoir engineers, and computer scientists.Comp EOR LMCS 6 / 43
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Motivation
Software Engineering (IEEE Comput Societys Software Eng. Body of Knowledge)
Application of a systematic, disciplined, quantifiable approach to the development,operation, and maintenance of software, and the study of these approaches; that is, theapplication of engineering to software.
Unified Process (UP, Boochet al. [2])
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Motivation
Software Engineering (IEEE Comput Societys Software Eng. Body of Knowledge)
Application of a systematic, disciplined, quantifiable approach to the development,operation, and maintenance of software, and the study of these approaches; that is, theapplication of engineering to software.
Unified Process (UP, Boochet al. [2])
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Motivation
Software Engineering (IEEE Comput Societys Software Eng. Body of Knowledge)
Application of a systematic, disciplined, quantifiable approach to the development,operation, and maintenance of software, and the study of these approaches; that is, theapplication of engineering to software.
Unified Process (UP, Boochet al. [2])
Requirements
EfficiencyAccuracyAbstraction
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Motivation
Software Engineering (IEEE Comput Societys Software Eng. Body of Knowledge)
Application of a systematic, disciplined, quantifiable approach to the development,operation, and maintenance of software, and the study of these approaches; that is, theapplication of engineering to software.
Unified Process (UP, Boochet al. [2])
Requirements
EfficiencyAccuracyAbstraction
We get a software:
ModularMantainableReliableEfficientProductive
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Motivation
Oil production stages
First stage of oil reservoir production, primary recovery, the oilis extracted by natural drive mechanism.
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Motivation
Oil production stages
First stage of oil reservoir production, primary recovery, the oilis extracted by natural drive mechanism.
The reservoir pressure can be maintained, using techniques such asgas or water injection. This is known as secondary recovery.
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Motivation
Oil production stages
First stage of oil reservoir production, primary recovery, the oilis extracted by natural drive mechanism.
The reservoir pressure can be maintained, using techniques such asgas or water injection. This is known as secondary recovery.
Tertiary or Enhanced Oil Recovery (EOR) is a generic term
that embraces several techniques used to increase the amount ofcrude oil that can be extracted from an oil field.
These techniques are based on the injection of materials notnormally present in the reservoir, and is the most advanced stage of
the exploitation of a reservoir.
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Motivation
Oil production stages
First stage of oil reservoir production, primary recovery, the oilis extracted by natural drive mechanism.
The reservoir pressure can be maintained, using techniques such asgas or water injection. This is known as secondary recovery.
Tertiary or Enhanced Oil Recovery (EOR) is a generic term
that embraces several techniques used to increase the amount ofcrude oil that can be extracted from an oil field.
These techniques are based on the injection of materials notnormally present in the reservoir, and is the most advanced stage of
the exploitation of a reservoir.Primary recovery techniques produce 10 15 % of the reservoirsoil content. Combining the processes of secondary andtertiary recovery techniques, it is possible to produce 30
60 % of the reservoirs total oil content.
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Axiomatic Formulation
Table of contents
1 Oil Reservoir SimulationMotivation
2 General Math & Num Models
Axiomatic FormulationNumerical MethodsFinite Volume Method
3 Computational Approach
TUNA
4 References
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Axiomatic Formulation
Extensive and Intensive Properties
In the physical sciences, intensive property(also called a bulkproperty, intensive quantity, or intensive variable), is a physicalproperty of a system that does not depend on the system size or
the amount of material in the system: it is scale invariant.Density
By contrast, an extensive property(also extensive quantity,extensive variable, or extensive parameter) of a system is directlyproportional to the system size or the amount of material in the
system.Mass
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Axiomatic Formulation
Axiomatic Formulation, (Herrera et al. [3, 4]) I
1 To find extensive Eand intensive properties :
E(t) =
B(t)
(x, t)dx
2 To establish balances:
dE
dt
= d
dt
B(t)
(x, t)dx= B(t)
q(x, t)dx+ B(t)
(x, t) ndS (1)
where q(x, t)y (x, t)are the source term in B(t)and the fluxvector through the boundary B(t), respectively
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Axiomatic Formulation
Axiomatic Formulation, (Herrera et al. [3, 4]) II
Global balance
B(t)
t + (v)
dx=
B(t)
qdx+
B(t)
dx (2)
Local balance
t + (v) =q+ (3)
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Axiomatic Formulation
Conservative form
Defining a flux function (see [5]) as f=v we get:
t
B(t)
dx+
B(t)
fdx=
B(t)
qdx (4)
and therefore
t + f=q (5)
Equivalently (4) can be written as follows
t
B(t)
dx+
B(t)
f ndS=
B(t)
qdx (6)
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Numerical Methods
Table of contents
1 Oil Reservoir SimulationMotivation
2 General Math & Num Models
Axiomatic FormulationNumerical MethodsFinite Volume Method
3 Computational Approach
TUNA
4 References
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Oil Reservoir Simulation General Math & Num Models Computational Approach References
N i l M th d
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Numerical Methods
In general, the equations governing a mathematical model of areservoir cannot be solved by analytical methods.
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N i l M th d
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Numerical Methods
In general, the equations governing a mathematical model of areservoir cannot be solved by analytical methods.
Instead, a numerical model can be produced in a form that isamenable to solution by digital computers.
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Numerical Methods
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Numerical Methods
In general, the equations governing a mathematical model of areservoir cannot be solved by analytical methods.
Instead, a numerical model can be produced in a form that isamenable to solution by digital computers.
Since the 1950s, numerical models have been used to predict,understand, and optimize complex physical fluid flow processes inpetroleum reservoirs.
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Oil Reservoir Simulation General Math & Num Models Computational Approach References
Numerical Methods
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Numerical Methods
In general, the equations governing a mathematical model of areservoir cannot be solved by analytical methods.
Instead, a numerical model can be produced in a form that isamenable to solution by digital computers.
Since the 1950s, numerical models have been used to predict,understand, and optimize complex physical fluid flow processes inpetroleum reservoirs.
Recent advances in computational capabilities have greatlyexpanded the potential for solving larger problems and hence
permitting the incorporation of more physics into the differentialequations.
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Numerical Methods
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Numerical Methods
1 Finite Diferences Method (FDM)
The FDM can be very easy to implement.Faster than FEM.High accuracy difference schemes can be constructed.In its basic form is restricted to handle only rectangular shapes.Introduce considerable geometrical error and grid orientation effects.Curvilinear coordinates can be used.
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Numerical Methods
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Numerical Methods
1 Finite Diferences Method (FDM)
The FDM can be very easy to implement.Faster than FEM.High accuracy difference schemes can be constructed.In its basic form is restricted to handle only rectangular shapes.Introduce considerable geometrical error and grid orientation effects.Curvilinear coordinates can be used.
2 Finite Element Method (FEM)The FEM cand handle complicated geometries.Reduce the grid orientation effects.Solid theoretical foundations.Can manage local grid refinement.
The quality of a FEM approximation is often higher than in thecorresponding FDM approach.FEM is the method of choice in structural mechanics.It is not easy to implement and is slower than FDM.
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Numerical Methods
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3 Finite Volume Method (FVM)
Values are calculated at control volumes.
Conservative method: the flux entering a given volume is identicalto that leaving the adjacent volume.Can easily be formulated to allow for unstructured meshes.Used in many computational fluid dynamics packages.FVM is in between FDM and FEM: faster and easier to implementthan FEM; and more accurate and versatile than FDM.
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Numerical Methods
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3 Finite Volume Method (FVM)
Values are calculated at control volumes.
Conservative method: the flux entering a given volume is identicalto that leaving the adjacent volume.Can easily be formulated to allow for unstructured meshes.Used in many computational fluid dynamics packages.FVM is in between FDM and FEM: faster and easier to implementthan FEM; and more accurate and versatile than FDM.
In oil-reservoir problems we usually require a large number of cells(105 106), therefore cost of the solution favors simpler and lowerorder approximation within each cell.
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Numerical Methods
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3 Finite Volume Method (FVM)
Values are calculated at control volumes.
Conservative method: the flux entering a given volume is identicalto that leaving the adjacent volume.Can easily be formulated to allow for unstructured meshes.Used in many computational fluid dynamics packages.FVM is in between FDM and FEM: faster and easier to implementthan FEM; and more accurate and versatile than FDM.
In oil-reservoir problems we usually require a large number of cells(105 106), therefore cost of the solution favors simpler and lowerorder approximation within each cell.
About 80% 90% of the total simulation time is spent on thesolution of linear systems.
Fast linear solvers are crucial to solve sparse, highly nonsymmetric,and illconditioned systems.Krylov subspace (preconditioned) algorithms are preferred.
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Finite Volume Method
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Table of contents
1 Oil Reservoir SimulationMotivation
2 General Math & Num Models
Axiomatic FormulationNumerical MethodsFinite Volume Method
3 Computational Approach
TUNA
4 References
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Finite Volume Method
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Finite volume methods are derived on the basis of the conservativeform of the balance equations [3, 4, 5].
t
B(t)
dx+
B(t)
f dx=
B(t)
qdx or
t
B(t)
dx+
B(t)
f ndS=
B(t)
qdx
FVM is conservative.
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Finite Volume Method
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Conservative formof general balance equation:
t
B(t)
dx+
B(t)
f dx=
B(t)
qdx
Integrating ont and taking B(t) V:
t
t
V
dV dt+ t
V
fdV dt= t
V
qdV dt
xz
y
E
W
N
S
F
B
Pee
ww
nn
ss
ff
bb
V = xyz
Notation:
NB id
P i ,j,kE i+ 1, j , k
W i 1, j , kN i, j+ 1, kS i, j 1, kF i ,j,k+ 1B i ,j,k 1
nb id
e i+ 12 , j , kw i 1
2, j , k
n i, j+ 12
, k
s i, j 12
, k
f i ,j,k+ 12
b i ,j,k 12
t n; t+ t n+ 1
t
g dt
n+1n
g dt
V
g dV
f
b
ns
ew
g dx dy dz
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Finite Volume Method
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Approximation of the integrals:
n+1
n
t
V
dV dt
n+1P n
P
V
n+1n
V
f dV dt
n+1n
F( fnb
)dt
t
V
qdV dt
n+1n
QV dt
xz
y
E
W
N
S
F
B
Pee
ww
nn
ss
ff
bb
V = xyz
Theta scheme:
n+1
n
Fdt=
Fn+1
+ (1 )Fn
t, 0 1
Explicit (ForwardEuler) = 0 Fnt.Implicit (BackwardEuler) = 1 Fn+1t.CrankNicolson = 1/2 (Fn + Fn+1)t/2.
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Recall that f =v :
F( fnb)
V
f dV =
ew
ns
f
b
(vx x)x
+(vy y)
y +
(vz z)
z
dxdydz
xz
y
E
W
N
S
F
B
Pee
ww
nn
ss
ff
bb
V = xyz
Discretized flux function:
F( fnb
) = (vx x)e (vx x)wAx+(vy y)n (vy y)s
Ay+
(vz z)f (vz z)b
Az
where Ax = y z, Ay = x z, Az = x y, represents the area ofthe faces.
Advectivev and Diffusive terms need to be approximated on the faces.
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Finite Volume Method
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Diffusive terms
Central differences: e.g. (x)e=D
x
e
=D
P
E
xeAdvective terms
Average : (vx)e= (vx)e(E+ (1 )P), where = xexP
xe
Upstream:
Upwind
if ((vx)e >0) thene =P
else
e =Eend if
QUICK (second order
upstream)
if ((vx)e >0) thene =h(W, P, E)
else
e =h(P, E, EE)end if
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Finite Volume Method
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Implicit, non-linear:
an
+1P n
+1P =an
+1E n
+1E +an
+1W n
+1W +an
+1N n
+1N +an
+1S n
+1S +
an+1F
n+1F
+an+1B
n+1B
+qnP
Implicit linear:
anP
n+1P
= anE
n+1E
+anW
n+1W
+anN
n+1N
+anS
n+1S
+anF
n+1F
+anB
n+1B
+qnP
Explicit:an
Pn+1
P =an
En
E +a
Wn
W+an
Nn
N+an
Sn
S +an
Fn
F +an
Bn
B +qn
P
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Oil Reservoir Simulation General Math & Num Models Computational Approach References
Finite Volume Method
S
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Linear Systems
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TUNA
T bl f
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Table of contents
1 Oil Reservoir SimulationMotivation
2 General Math & Num Models
Axiomatic FormulationNumerical MethodsFinite Volume Method
3 Computational Approach
TUNA
4 References
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TUNA
T l t U it f N i l A li ti
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Template Units for Numerical Applications
Natural convection in box-shaped containers
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TUNA
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f1(t) =
0.5sin2(4t) for 0 t < /81 for /8 t
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Navier-Stokes equations
Mass balance:
ujxj = 0
Momentum balance (Navier-Stokes):
0 ui
t
+ujui
xj =
p
xi+
2ui
xjxj+bi
Energy balance:T
t +uj
T
xj=
2T
xjxj
Equations of state:
= 0[1 (T T0)] , = 1
0
T
T=T0
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TUNA
Template Units for Numerical Applications I
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Template Units for Numerical Applications I
TUNA use several C++ template techniques (Blitz++).
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TUNA
Template Units for Numerical Applications II
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Template Units for Numerical Applications II
http://mmc.geofisica.unam.mx/(then go to my homepage).
Examples : programs that use TUNA
|--- tuna-cfd-rules.in => rules to compile the examples
|--- 01StructMesh => uniform structured meshes
|--- 02NonUniformMesh => non uniform structured meshes
|--- 03Laplace => Solution of Laplace equation
|--- 04HeatDiffusion => Solution of heat conduction problems
|--- 05ConvDiffForced => Solution of forced convection
|--- 06ConvDiff => Solution of natural convection problems
|--- 07ConvDiffLES => Solution of turbulent natural convection
|--- README.pdf => Explanation of the examples of each directory
1) Unpack TUNA and change to the TUNA dir:
% tar zxvf TUNA.tar.gz% cd TUNA
2) Blitz++: http://www.oonumerics.org/blitz/
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Oil Reservoir Simulation General Math & Num Models Computational Approach References
TUNA
Template Units for Numerical Applications III
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Template Units for Numerical Applications III
- Unpack with: tar zxvf blitz-09.tar.gz
- Change to blitz-0.9 with: cd blitz-0.9- Config blitz with: ./configure --prefix=$PWD/../BLITZ
- Compile and install blitz with: make install
These instruction will install Blitz in the TUNA/BLITZ directory
3) Run the examples
- Change to the Examples directory: cd Examples
- Edit the files tuna-cfd-rules.in
Change the environment variable BASE according to your paths.
(e.g. BASE = /home/luiggi/TUNA)
- Then, e.g. change to the 06ConvDiff dir:
% cd 06ConvDiff% make
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Example: Natural Convection I
#include "Meshes/Uniform.hpp"#include "Storage/DiagonalMatrix.hpp"
#include "Equations/ScalarEquation.hpp"
#include "Schemes/CDS_CoDi.hpp"
#include "Equations/Momentum_XCoDi.hpp"
#include "Schemes/CDS_XCoDi.hpp"
#include "Equations/Momentum_YCoDi.hpp"
#include "Schemes/CDS_YCoDi.hpp"
#include "Equations/Momentum_ZCoDi.hpp"
#include "Schemes/CDS_ZCoDi.hpp"
#include "Equations/PressureCorrection.hpp"
#include "Schemes/Simplec.hpp"
#include "Solvers/TDMA.hpp"
#include "Utils/inout.hpp"#include "Utils/num_utils.hpp"
#include "Utils/GNUplot.hpp"
using namespace Tuna;
typedef TunaArray::huge ScalarField3D;
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TUNA
Example: Natural Convection II
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Example: Natural Convection II
DiagonalMatrix A(num_nodes_x, num_nodes_y, num_nodes_z);
ScalarField3D b(num_nodes_x, num_nodes_y, num_nodes_z);
StructuredMesh > mesh(length_x, num_nodes_x,
length_y, num_nodes_y,
length_z, num_nodes_z);
ScalarField3D T(mesh.getExtentVolumes());
ScalarField3D p(mesh.getExtentVolumes());
ScalarField3D u(mesh.getExtentVolumes()); // u-velocity
ScalarField3D v(mesh.getExtentVolumes()); // v-velocity
ScalarField3D w(mesh.getExtentVolumes()); // w-velocity
Range all = Range::all();T(T.lbound(firstDim), all, all) = left_wall; // Left
T(T.ubound(firstDim), all, all) = right_wall; // Rigth
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TUNA
Example: Natural Convection III
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Example: Natural Convection III
ScalarEquation energy(T, A, b, mesh.getDeltas());
energy.setDeltaTime(dt);
energy.setNeumann(TOP_WALL);
energy.setNeumann(BOTTOM_WALL);
energy.setDirichlet(LEFT_WALL, left_wall);
energy.setDirichlet(RIGHT_WALL, right_wall);
energy.setNeumann(FRONT_WALL);
energy.setNeumann(BACK_WALL);
energy.setUvelocity(us);energy.setVvelocity(vs);
energy.setWvelocity(ws);
energy.print();
Momentum_XCoDi mom_x(us, A, b, mesh.getDeltas());
Momentum_YCoDi mom_y(vs, A, b, mesh.getDeltas());
Momentum_ZCoDi mom_z(ws, A, b, mesh.getDeltas());
PressureCorrection press(pp, A, b, mesh.getDeltas());
Comp EOR LMCS 36 / 43
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TUNA
Example: Natural Convection IV
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a p e Natu a Co vect o V
template
class CDS_CoDi : public ScalarEquation< CDS_CoDi< Tprec, Dim > >{
public:
typedef Tprec prec_t;
typedef typename TunaArray::huge ScalarField;
CDS_CoDi() : ScalarEquation() { }
~CDS_CoDi() { };
inline void calcCoefficients1D();
inline void calcCoefficients2D();
inline void calcCoefficients3D();
inline void printInfo() { std::cout
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p
template
inline void CDS_CoDi::calcCoefficients2D() {
prec_t Gdy_dx = Gamma * dy / dx, Gdx_dy = Gamma * dx / dy;
prec_t dxy_dt = dx * dy / dt;
aE = 0.0; aW = 0.0; aN = 0.0; aS = 0.0; aP = 0.0; sp = 0.0;
for (int i = bi; i
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p
for(iteration = 1; iteration tol_simplec) && (counter < 20) ) {
energy.calcCoefficients();
Solver::TDMA3D(energy, tolerance, max_iter);
errorT = energy.calcErrorL2();
energy.update();
mom_x.calcCoefficients();
Solver::TDMA3D(mom_x, tolerance, max_iter);
errorX = mom_x.calcErrorL2();
mom_x.update();
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TUNA
Example: Natural Convection VII
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p
mom_y.calcCoefficients();Solver::TDMA3D(mom_y, tolerance, max_iter);
errorY = mom_y.calcErrorL2();
mom_y.update();
mom_z.calcCoefficients();
Solver::TDMA3D(mom_z, tolerance, max_iter);
errorZ = mom_z.calcErrorL2();mom_z.update();
press.calcCoefficients();
Solver::TDMA3D(press, tolerance, max_iter);
press.correction();
sorsum = fabs( press.calcSorsum() );
++counter;
}
return sorsum;
}
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TUNA
Example: Natural Convection VIII
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Comp EOR LMCS 41 / 43
Oil Reservoir Simulation General Math & Num Models Computational Approach References
References I
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[1] R.A. Tapia and C. Lanius,Computational Science: Tools for a Changing World
http://ceee.rice.edu/Books/CS/chapter1/intro52.html, 2001.
[2] I. Jacobson and G. Booch and J. RumbaughPrimer,The Unified Software Development Process,Addison Wesley, 1999.
[3] I. Herrera and M. B. Allen and G. F. Pinder,Numerical modeling in science and engineering,John Wiley & Sons., USA, 1988.
[4] I. Herrera and G. F. Pinder,General principles of mathematical computational modeling,John Wiley, in press.
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References II
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[5] R.J. Leveque,
Finite Volume Method for Hyperbolic Problems,Cambridge University Press, 2004.
Comp EOR LMCS 43 / 43
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