An overview of Computational FluidDynamic applied to Petroleum Reservoir Simulations

An overview of Computational Fluid Dynamic applied to Petroleum Reservoir

Simulations

Simone Ribeiro

Universidade Federal de Uberlândiaverão de 2010

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Wednesday, January 6, 2010

Outline

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• Historical overview of CFD• Euler equations• Navier-Stokes equations

• The Conservation Laws• Applications• Multiphase flow problem• Solutions to conservation laws

• The two-phase flow model• Numerical approximation• Numerical results

• Final considerations


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History of CFD

• Heraclitus postulated that “Everything Flows”

• Archimedes initiated the fields of Hydrostatic: the mesure of densities and volume of objects

• Leonardo da Vinci planned and supervised canal and harbor works over a large part of Italy.

• Isaac Newton made contributions to fluid mechanics with his Second Law: F = ma.


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18th-19th centuriesHistory of CFD

• Bernoulli stated that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluidʼs potential energy.

• Euler proposed the Euler equations, which describe the conservation of momentum for an inviscid fluid, and conservation of mass.

• Claude Navier and George Stokes introduced viscous transport into the Euler Equations, which resulted in the now famous Navier-Stokes Equations.


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History of CFDThe Navier-Stokes Equations

The Navier-Stokes equation are a set of equations that describe the movement of fluids such as gases or liquids. They establish that changes in momentum and acceleration of a particle results from changes in pressure and viscous forces inside the fluid.

They are obtained from the basic principles of Conservation of Mass, Momentum and Energy.

Navier Stokes


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History of CFDThe earliest CFD calculations

Lewis Fry Richardson developed the first weather prediction system: he divided the physical space into grid cells and used the finite difference approximations.

The calculation of weather of a 8-hour period took 6 weeks of real time and ended in failure.

For efficiency in calculations, he proposed the “forecast factory”. This is the earliest ideas of CFD calculations and parallel computing.


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The Conservation LawsIn one space dimension, the equations take the form:

The main assumption underlying this equation is that knowing the value of u(x,t) at a given point and time allows us to determine the rate of flow, or flux, of each state variable at (x,t)

!u(x, t)!t

+!f(u(x, t))

!x= 0

f : Rm ! Rm

u : R! R " Rm

(x, t) #" (u1(x, t), . . . , um(x, t))


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The Conservation Laws

Applications


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The Conservation LawsApplications

Euler Equations of Gas Dynamics

!

!t

!

"""vE

#

$ +!

!x

!

""v

"v2 + pv(E + p)

#

$ = 0

!

v

!v

Ep

density functionvelocity

momentumenergypressure

!!!!!


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Aerodynamics


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The Dambreak Problem


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The Dam Break Problem


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• Meteorology

• Astrophysical

• The study of explosions

• The flow of glaciers


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The Conservation LawsThe Multiphase Flow Problem


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The Mathematical and Numerical difficulties


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The Conservation LawsThe Difficulties

• Discontinuous solutions do not satisfy the PDE in the classical sense at all points, since the derivatives are not defined at discontinuities.

• A finite difference discretization of the PDE is inappropriate near discontinuities.


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Initial Condition: u(x, 0) = 0.5 + sin xGrid: 100 points


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The Conservation LawsThe Dicfficulties

ut + (u2/2)x = 0

Initial Condition: u(x, 0) = 0.5 + sin x

Grid: 100 points

ut + ux = 0


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x

u(x, 0)u(x, 0)

u(x, 0) =!

1, x < a0, x ! a

a a


1

0

For Discontinuous Galerking Method and

Godunovʼs Method


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Oscillations arising in a shock computed with Godunovʼs method

The oscillations Amplifications


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Which features a good numerical method

should have?


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• At least second order accuracy on smooth region of a solution.

• Sharp resolution of discontinuities without excessive smearing.

• The absence of spurious oscilations in the computed solution.

• Nonlinear stability bounds that, together with consistency, alow us to prove convergence as the grid is refined.

• Convergence to the physically correct solution.

• Computational Efficiency

The Conservation LawsFeatures of Numerical Methods


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The Multiphase Flow ModelOne phase flows

Henry Darcy (1844) described the flow of a fluid through a porous media. This mathematical description is known as Darcy’s Law.

Darcyʼs Law

v = !k

µ"P

The Darcy’s Law tell us that the flow of a fluid is proportional to the pressure

gradient.


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The Multiphase Flow ModelTwo phase flows

Two phase flows characterize the displacement of two immiscible fluids, such as water and oil, through a porous media.

Muskat generalized the Darcy’s Law for two phase flows introducing the concepts of effective permeability and relative permeability.

G. Chavent. A new formulation of diphasic incompressible flows in porous media. Volume 503 Lecture Notes in Mathematics, Springer


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The Multiphase Flow ModelTwo phase flows

When two fluids fills a porous media, the ability of a fluid to flow is called effective permeability of that fluid. It is denoted by where indicates the fluid

Relative permeability is a dimensionless measure of the effective permeability of each fluid or phase.

K! !

v! = !K!

µ!"P!Kr! =

K!

K=!


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The Multiphase Flow ModelSimplified Hypothesis

• Gravity is not considered

• Constant porosity

• Incompressible flow

• The reservoir is saturated


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The Multiphase Flow ModelThe Mathematical Model

• Darcy’s Law

• Convective Transport Equation

• The reservoir is saturated

! · v = 0v = !K(x)!"P

! !!tsw +! · (vfw) = 0

sw + so = 1.0Z. Chen. Computational methods for multiphase flows in porous media. SIAM


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The Multiphase Flow ModelNumerical Strategy

Numerical Approximation


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Operator Splitting!

v = !Kµ"P, " · v = 0

!!sw!t +" · (vfw) = 0


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Operator Splitting

!

!tsw +! · (vfw) = 0v = !K(x)!"P

! · v = 0

Advantages:

• Different time steps for each problem

• Appropriate numerical method for each equation

!v = !K

µ"P, " · v = 0!!sw

!t +" · (vfw) = 0


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Numerical ApproximationVelocity-pressure equation

! · v = 0v = !K(x)!"P

• Mixed Finite Element Method

• Raviart-Thomas Element

• Preconditioned Conjugate Gradient Method

• G. Chavent and J. Roberts. A unified physical presentation of mixed, mixed-hybrid finite •elementsfor determination of velocities in waterflow problems. IRIA, Chesnay, 1989.


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Numerical ApproximationSaturation equation

! !!tsw +! · (vfw) = 0

• Second order central schemes

• Semi-discrete Godunov type schemes

• Second order Runge-Kutta

Ribeiro S., Pereira F., Abreu E. Central schemes for porous media flows. Journal of Computational and Applied Mathematics, 2008


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Historical development... in 1954


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• Lax-Friedrichs numerical scheme (1954)


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• centered finite differencing


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• simplicity


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• simplicity

• time step restricted to a CFL condition


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• simplicity


• High numerical diffusion and inversely proportional to the time step.


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Lax-Friedrichs scheme


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Historical development... in 1961 • Rusanovʼs numerical scheme (1954)


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• based on Lax-Friedrichs method


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• first order approximation


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• simplicity


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• simplicity

• time step no longer restricted to a CFL condition


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• simplicity

• time step no longer restricted to a CFL condition

• Lower numerical diffusion independent of the time step.


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Rusanovʼs scheme


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Historical development... in 1990 • Nessyahu and Tadmor numerical scheme

J. Comp. Phys. 1990


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J. Comp. Phys. 1990

• second order extension of LxF method


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J. Comp. Phys. 1990


• central scheme based on REA algorithm from Godunov, Mat. Sb., 1959


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J. Comp. Phys. 1990



• simplicity


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J. Comp. Phys. 1990



• simplicity


• Sharp resolution without spurious oscillation


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J. Comp. Phys. 1990



• simplicity


• Sharp resolution without spurious oscillation

• It captures the entropic solution


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NT scheme


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• Kurganov and Tadmor numerical scheme J. Comp. Phys. 2000


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• second order extension of Rusanovʼs method with REA algorithm


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• simplicity


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• simplicity

• time step is not restricted to a CFL condition


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• simplicity


• Much better resolution with longer time steps


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• simplicity


• Much better resolution with longer time steps

• Numerical diffusion is independent of time step


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KT scheme


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Finite Volume Strategy

• Construction Step (Leveque, Finite volume method for hyperbolic problem)

• Divide the domain in a collection of finite control volumes with fixed size

• (Evolution Step) Integrate the conservation law over each control volume.

!

!ts +

!

!x(xvf(s)) +

!

!y(yvf(s)) = 0

! ! !

V

"

!Snj,k(x, y) = S

nj,k + (Sx)n

j,k · (x! xj) + (Sy)nj,k · (y ! yk)

dt


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• Construction Step (Leveque, Finite volume method for hyperbolic problem)

• Divide the domain in a collection of finite control volumes with fixed size

• (Evolution Step) Integrate the conservation law over each control volume.

!

!ts +

!

!x(xvf(s)) +

!

!y(yvf(s)) = 0

! ! !

V

" !dV

!Snj,k(x, y) = S

nj,k + (Sx)n

j,k · (x! xj) + (Sy)nj,k · (y ! yk)

dt


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!

Lx

Ly

Two spatial dimensions


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Finite Volume StrategyTwo spatial dimensions

xj!1/2 xj+1/2

yk+1/2

yk!1/2

!Y

!XWednesday, January 6, 2010

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xj!1/2 xj+1/2

yk+1/2

yk!1/2

Contruction Step

S̄nj,k

!Snj,k(x, y) = S

nj,k + (Sx)n

j,k · (x! xj) + (Sy)nj,k · (y ! yk)


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xj!1/2 xj+1/2

yk+1/2

yk!1/2

Sn+1j,k

Evolution and Average Step


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d

dtSjk(t) = !

Hxj+1/2,k(t)!Hx

j!1/2,k(t)!X

!Hy

j,k+1/2(t)!Hyj,k!1/2(t)

!Y

• Second order Runge-Kuttaʼs method

• Time step restricted to a stability condition

max!!tRK

!Xmax

s|xv(t) · (f(s))x|, !tRK

!Ymax

s|yv(t) · (f(s))y|

"! TRK


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Finite Volume StrategyConservative Formulation of Semi-discrete

central scheme -- SD2Dd

dtSjk(t) = !

Hxj+1/2,k(t)!Hx

j!1/2,k(t)!X

!Hy

j,k+1/2(t)!Hyj,k!1/2(t)

!Y

• Second order Runge-Kuttaʼs method

• Time step restricted to a stability condition

max!!tRK

!Xmax

s|xv(t) · (f(s))x|, !tRK

!Ymax

s|yv(t) · (f(s))y|

"! TRK


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SD2D numerical scheme

xj!1/2 xj+1/2

yk!1/2

yk+1/2


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Numerical Flux in X-direction

Hxj+1/2,k(t) =

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!xvj+1/2,k+1/2(t)

"f(S+!

j+1/2,k+1/2(t)) + f(S!!j+1/2,k+1/2(t))#

+xvj+1/2,k!1/2(t)!f(S++

j+1/2,k!1/2(t)) + f(S!+j+1/2,k!1/2(t))

"#

!cxj+,k

2

!S+

j+,k(t)! S!j+,k(t)"

xj!1/2 xj+1/2

yk!1/2

yk+1/2


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xj!1/2 xj+1/2

yk!1/2

yk+1/2



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xj!1/2 xj+1/2

yk!1/2

yk+1/2

Hyj,k+1/2(t) =

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!yvj+1/2,k+1/2(t)

"f(S!+

j+1/2,k+1/2(t)) + f(S!!j+1/2,k+1/2(t))#

+yvj!1/2,k+1/2(t)!f(S++

j!1/2,k+1/2(t)) + f(S+!j!1/2,k+1/2(t))

"#

!dy

j,k+1/2

2

!S+

j,k+1/2(t)! S!j,k+1/2(t)"

Numerical Flux in Y-directionSD2D numerical scheme


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SD2D numerical schemeThe velocity field

xj+1xj

yk

yk+1

xvj+1/2.k+1/2


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Numerical Results in 2D

Slab Geometry


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Geometria Slab

Lx

Ly


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Geometria Slab

Geology model


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The permeability fieldScalar Log-Normal Permeability Field

Only rock heterogeneities drive macroscopic fluid mixing

where is GaussianK = K (x) = K0 e ! "(x) !(x)


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CV = 0.5 SD2DGrid: 256 x 64


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CV = 0.5 SD2DGrid: 256 x 64

NT2DGrid: 256 x 64


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CV = 0.5 SD2DGrid: 256 x 64


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CV = 0.5 SD2DGrid: 256 x 64

NT2DGrid: 512 x 128


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CV = 0.5 SD2DGrid: 256 x 64


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CV = 0.5 SD2DGrid: 256 x 64

NT2Dmalha: 1024 x 256


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CV = 0.5 SD2DGrid: 256 x 64

NT2Dmalha: 1024 x 256

3.4 min

24 h


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LxF 2DGrid: 256 x 64 cells

NT2DGrid: 256 x 64 cells

SD2DGrid: 256 x 64 cells


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Numerical Results in 2D

Permeability Field SD2D

Gradient pressure fixed


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Advantages and Disadvantages of the

numerical approximation


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Advantages and Disadvantages• Very easy formulation and implentation

• It is very easy to extend to system of conservation laws.

• It depends on the geometry

• It needs an adaptivity strategy to refine the mesh only near discontinuities.

• Ongoing work: develop lagrangian schemes which have self adaptable mesh


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Mesh with adaptivity strategy

Courtesy from Steve Dufour generated using Discontinuous Galerkin method


Bibliography

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• Ribeiro S., Pereira F., Abreu E. Central schemes for porous media flows. Journal of Computational and Applied Mathematics, 2008

• Ribeiro S., Francisco A, Pereira F. Water-Air simulation in porous media with one-phase pressure boundary condition. XI EMC

• Ribeiro S., Pereira, F. A new two-dimensional second order non-oscillatory central scheme. ArXiv

• E. Abreu, F. Furtado, and F. Pereira. Three-phase immiscible displacement in heterogeneouspetroleum reservoirs. Mathematics and computers in simulation


Bibliography

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• G. Chavent. A new formulation of diphasic incompressible flows in porous media. Volume 503 Lecture Notes in Mathematics, Springer

• F. Furtado, F. Pereira. Scaling analysis for two-phase immiscible flows in heterogeneous porous media. Comp. Appl. Math, 17, 1998.

• J. Glimm, B. Lindquist, F. Pereira, and R. Peierls. The multi-fractal hypothesis and anomalous diffusion. Math. Appl. Comput. 11, 1982.

• G. Chavent and J. Roberts. A unified physical presentation of mixed, mixed-hybrid finite elementsfor determination of velocities in waterflow problems. IRIA, Chesnay, 1989.


Bibliography

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Bibliography


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Obrigada!


An overview of Computational FluidDynamic applied to Petroleum Reservoir Simulations

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