trefftz-herrera collocation method for combustion fronts in oil

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TrefftzTrefftz--Herrera Collocation Herrera Collocation Method for Combustion Fronts in Method for Combustion Fronts in

Oil Reservoirs: ApplicationOil Reservoirs: Application

M. A. DíazM. A. Díaz--Viera, D. A. LópezViera, D. A. López--Falcón Falcón Instituto Mexicano del Petróleo, Instituto Mexicano del Petróleo,

I. HerreraI. HerreraInstituto de GeofInstituto de Geofíísica, UNAMsica, UNAM

7th World 7th World CongressCongress onon ComputationalComputational MechanicsMechanicsHyattHyatt RegencyRegency CenturyCentury Plaza HotelPlaza Hotel

Los Los AngelesAngeles, California, CaliforniaJulyJuly 16 16 -- 22, 200622, 2006

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IntroductionIntroduction

Recently, some authors (H. Recently, some authors (H. HoteitHoteit andand FiroozabadiFiroozabadi, 2005) , 2005) in the specialized oil reservoir literature have suggested the in the specialized oil reservoir literature have suggested the application of Discontinuous application of Discontinuous GalerkinGalerkin Method to capture Method to capture sharp moving fronts in modeling sharp moving fronts in modeling multicomponentmulticomponent fluid fluid flow in fractured mediaflow in fractured media.

In the present work we will consider, as a very competitive In the present work we will consider, as a very competitive alternative method, the alternative method, the TrefftzTrefftz--Herrera approach in Herrera approach in conjunction with collocation to obtain the numerical conjunction with collocation to obtain the numerical solution of a system of boundary value problems with solution of a system of boundary value problems with prescribed jumps (prescribed jumps (BVPJ’sBVPJ’s) which models the dynamics of ) which models the dynamics of combustion fronts in porous media.combustion fronts in porous media.

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TrefftzTrefftz--Herrera Collocation Method was introduced in Herrera Collocation Method was introduced in previous papers: previous papers:

–– Herrera, I.; Diaz, M. “Indirect Methods of Collocation: Herrera, I.; Diaz, M. “Indirect Methods of Collocation: TrefftzTrefftz--Herrera Herrera Collocation”. Numerical Methods for Partial Differential EquatioCollocation”. Numerical Methods for Partial Differential Equations. ns. 1515(6) 709(6) 709--738, 1999.738, 1999.

–– Herrera, I., Yates R. and Diaz M. “General Theory of Domain Herrera, I., Yates R. and Diaz M. “General Theory of Domain Decomposition: Indirect Methods”. Numerical Methods for Partial Decomposition: Indirect Methods”. Numerical Methods for Partial Differential Equations, Vol. 18, No. 3, pp. 296Differential Equations, Vol. 18, No. 3, pp. 296--322, may. 2002.322, may. 2002.

–– Diaz M. and I. Herrera, “THDiaz M. and I. Herrera, “TH--Collocation for the Biharmonic Collocation for the Biharmonic Equation”, Equation”, Advances in Engineering SoftwareAdvances in Engineering Software,, Volume 36, Issue 4, Volume 36, Issue 4, Pages 243Pages 243--251, April 2005.251, April 2005.

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The distinguishing feature of the The distinguishing feature of the TrefftzTrefftz--Herrera Herrera procedure is the use of specialized test functions, that procedure is the use of specialized test functions, that could be in general discontinuous, becomes it in a very could be in general discontinuous, becomes it in a very attractive choice for solving problems with prescribed attractive choice for solving problems with prescribed jumps and with discontinuous coefficients.jumps and with discontinuous coefficients.

In particular, the numerical modeling of a planar In particular, the numerical modeling of a planar combustion front for the steady state under adiabatic combustion front for the steady state under adiabatic conditions is implemented. Some preliminary numerical conditions is implemented. Some preliminary numerical results and its comparison with the analytical solutions results and its comparison with the analytical solutions are presented.are presented.

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Collocation MethodsCollocation Methods

Collocation is known as an efficient and highly accurate Collocation is known as an efficient and highly accurate numerical solution procedure for partial differential equations.numerical solution procedure for partial differential equations.

Usually this kind of methods are applied using Usually this kind of methods are applied using splinessplines. However, . However, a more general point of view is obtained when they are a more general point of view is obtained when they are formulated using the approach proposed by Herrera, in spaces formulated using the approach proposed by Herrera, in spaces of fully discontinuous functions; i.e., spaces in which the of fully discontinuous functions; i.e., spaces in which the functions and their derivatives may have jump discontinuities. functions and their derivatives may have jump discontinuities.

In the case of elliptic equations of second order, it is standarIn the case of elliptic equations of second order, it is standard d requiring continuity of both, the function and its derivative. requiring continuity of both, the function and its derivative. However, these conditions can be relaxed when However, these conditions can be relaxed when TrefftzTrefftz--Herrera Herrera method is applied. method is applied.

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Classification of Collocation Classification of Collocation MethodsMethods

Direct and indirect methodsDirect and indirect methods: Direct methods are those in : Direct methods are those in which collocation is used to construct the solution directly, whwhich collocation is used to construct the solution directly, while ile indirect methods are those in which collocation is applied to indirect methods are those in which collocation is applied to construct construct specialized test functionsspecialized test functions..

Overlapping and nonOverlapping and non--overlapping methodsoverlapping methods: depending on : depending on whether the whether the subregionssubregions used in the construction of the solution used in the construction of the solution are disjoint or overlapping.are disjoint or overlapping.

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Diagram of Collocation MethodsDiagram of Collocation Methods

Direct-non-overlapping(Standard Collocation)

Indirect-non-overlapping Direct-overlapping Indirect-non-overlapping(TH-Collocation)

Collocation Methods

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Boundary Value Problem with Boundary Value Problem with Prescribed Jumps (BVPJ)Prescribed Jumps (BVPJ)

LLeett uuss ccoonnssiiddeerr tthhee oonnee--ddiimmeennssiioonnaall eelllliippttiicc ddiiffffeerreennttiiaall eeqquuaattiioonn ooff sseeccoonndd oorrddeerr iinn aann iinntteerrvvaall [ ]0, ssuubbjjeecctteedd ttoo DDiirriicchhlleett bboouunnddaarryy ccoonnddiittiioonnss aanndd jjuummpp ccoonnddiittiioonnss::

( )

( ) ( )0

0 1

;

0 ;

, 1,..., 1i i iii

d du du a bu cu fdx dx dx

u u and u u

duu j and a j at x i Edx

Ω⎛ ⎞≡ − + + =⎜ ⎟⎝ ⎠

= =

= = = −

L

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Standard Collocation MethodThe collocation approximate solution is represented by :

( ) ( ) ( ){ }0 1

0

ˆ ˆ ˆE

i i i ii

u x u h x u h x′

=

= +∑

where ( )0ih x and ( )1

ih x are piecewise Hermite cubic polynomials

with support in the interval ( )1 1, i ix x− +

The approximate solution ( )u x must fulfill the collocation

equations:

{ }ˆ 0; 1,..., , 1, 2ejx

u f e E jΩ− = = =L

where ( ), 1, 2ejx j = are the Gaussian points in the interval ( )1, e ex x− .

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THTH--CollocationCollocation

The resulting system of equations using Trefftz-Herrera approach

is an (E-1) by (E-1) system, since the only unknowns are ˆiu , for

i=1,..., E-1. It is:

* ˆ, , ; 1,..., 1k kS u w f g j w k E− = − − = −or, more explicitly

1

1

ˆ , ; 1,..., 1kE

k ki

i i

dwa bw u f g j w k Edx

−

=

⎡ ⎤− + = − − = −⎢ ⎥

⎣ ⎦∑

where [ ] u u u+ −= − and ( ) 2u u u+ −= +

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Weighting FunctionsWeighting FunctionsThe basic strategy of TH-formulation is to concentrate all the

information about the sought solution at the internal nodes. To this

end specialized test functions are developed. They satisfy:

( ) ( )

* 0

0 0

0

i ii i

i i

i

i

d dw dww a b cwdx dx dx

w w

w

⎛ ⎞≡ − − + =⎜ ⎟

⎝ ⎠= =

=

L

iw

1ix − ix 1ix +

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Construction of Weighting FunctionsConstruction of Weighting Functions

Then define ( ) ( ) ( ) ( ) ( )111, 1,ˆ i

i i i i iw x x m x P x− −= +

( ) ( ) ( ) ( ) ( )22, 1 1,ˆ i

i i i i iw x x m x P x− −= +

where ( ) 1, 1

1

ii i

i i

x xxx x

−−

−

−=

− , ( )1,1

ii i

i i

x xxx x−

−

−=

− , ( ) ( ) ( )1, 1, , 1i i i i i im x x x− − −=

( ) ( )1iP x and

( ) ( )2iP x are polynomials of degree G-2. The G-1 coefficients of

each one of these polynomials can be determined by orthogonal collocation; that

is, it is required to satisfied the following system of equations

( )ˆ* 0; 1, 2 1,..., 1 1,...,i ejw x j G e Eα α= = = − =L

where ejx are the G-1 Gaussian points of the interval ( )1, e ex x− .

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Comparative TableMethod

Standard Collocation TH Collocation

Classification

Direct method Indirect method

Unknowns Function and its first derivatives at the nodes

Function u at the nodes

Continuity

Function and its first derivatives

In general may be fully discontinuous

Weighting Functions

Hermite cubic polynomials Polynomials of degree G=2,3...

Matrix

2(E+1) by 2(E+1)

Tridiagonal (E-1) by (E-1)

Expected Error

O(h4)

O(h2N), where (N=G-1)

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The problem of a planar The problem of a planar combustion front in porous mediacombustion front in porous media

⇒ Since the high complexity of the processes involved during

air injection, it is convenient to consider certain particular

cases which allow the analysis of fundamental features of

combustion process.

⇒ Here, we will focus in the study of the planar combustion

front dynamics for the steady state under adiabatic

conditions (no heat losses).

⇒ In particular, we are interested in modeling the propagation

of the oxidation front produced into the porous medium due

to fuel oxidation in a combustion tube.

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BVPJ for the temperatureBVPJ for the temperature

The corresponding boundary value problem with prescribed jumps

(BVPJ) resulting from the total energy balance in terms of

temperature is as follows

( ) ( )

0

1 0; in

; ; on

0; v ; on

s s g g g

f o

f

Tc T c u T Bt

T T T T B

T T n Q n

φ ρ λ ρ

λ ρ−∞ +∞

Σ Σ

∂− −∇⋅ ∇ + ⋅∇ =

∂= = ∂

= ∂ ∂ = − ⋅ Σ

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11--D BVPJ for the temperatureD BVPJ for the temperature

When we are considering the case of a planar front the previous

problem can be reduced to a one dimension. And, for convenience, a

coordinate transformation is introduced. The new coordinate system

moves with the front velocity, vx tξ Σ= − , where ξ is the new

coordinate and vΣ is the front velocity. Therefore, the problem

becomes

( )( )2

2

00

0

1 v 0

0; v ;

;

g g g s s

f

f o

T Tc u c

TT Q

T T T T

ξξ

λ ρ φ ρξ ξ

λ ρξ

Σ

Σ==

−∞ +∞

∂ ∂− + − − =

∂ ∂

∂= = −

∂

= =

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Analytic solution of the BVPJ for Analytic solution of the BVPJ for the temperaturethe temperature

Using the continuity conditions of the BVPJ, we obtain the generalsolution

( )( )

0

0

; 0

vexp ; 0

f

fT

T

T

T QT A

A

ξ

ξ ρξ ξ

λΣ

<⎧⎪

= ⎨− >⎪

⎩

where ( )1 v

g g g s s

T

c u cA

ρ φ ρλ

Σ− −=

The value fT is obtained from the jump condition

0T = (continuity of T at 0ξ = ), such that

( )( )0

0 1 vf

f g g g s s

Q vT T

c u cρ

ρ ρ φΣ

Σ

= −− −

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BVPJ for the oxygen mass fraction BVPJ for the oxygen mass fraction

In a similar way we can proceed with the BVPJ resulting from the

oxygen mass balance

( ) ( ) ( )

( ) 0

0; in

1; ; on

0; v ; on

gg g g

M

b

gM g f

YD Y Y u B

tY Y Y B

Y D Y n Y n

ρφ ρ ρ

ρ μ μ ρ−∞ +∞

Σ Σ

∂− ∇ ⋅ ∇ + ∇ ⋅ =

∂= = ∂

= ∂ ∂ = + ⋅ Σ

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11--D BVPJ for the oxygen mass D BVPJ for the oxygen mass fraction fraction

Making the same considerations the previous BVPJ becomes

( )

( )

2

2

-

00 0

0,

1,

0, v

gM

b

gM g b f

Y YD u v

Y Y Y

Y D Y Yξ ξ

φξ ξ

ρ ξ μ μ ρ

Σ

∞ +∞

Σ= =

∂ ∂− + − =

∂ ∂= =

= ∂ ∂ = +

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Analytic solution of the BVPJ for Analytic solution of the BVPJ for the oxygen mass fraction the oxygen mass fraction

Finally, the solution for the whole region is as follows

( ) ( ) ( )1 1 exp ; 0; 0

b Y

b

Y AY

Yξ ξ

ξξ

− − <⎧⎪= ⎨>⎪⎩

Where ( )vg

YM

uA

Dφ Σ−

=

And the value bY can be obtained using the jump condition,

( )( )

0

0

v

v

g g gf

b g g gg f

uY

u

ρ φρ μρ

ρ φρ μ ρΣ

Σ

− +=

− −

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Data of the BVPJ´sData of the BVPJ´sThe following data which are typical values for an in-situ combustion

process are using in the numerical experiments

Parameter Value Parameter Value g gc ρ 1.2339 kJ/m3K 0

fρ 19.2182 kg/m3

φ 0.3 gρ 1.22516 kg/m3

λ 8.654x10-4

kW/mK gμ 1

( )1 s scφ ρ− 2.02x103 kJ/m3K μ 3.018

Q 39542 kJ/kg MD 2.014 x 10-6

m2/s

(I. Yucel Akkutlu and Yanis C. Yortsos, Combustion and Flame, 134 (2003) 229-247)

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Example 1: Temperature ProfileExample 1: Temperature Profile

-0.6 -0.4 -0.2 0 0.2 0.4 0.6ξ [m]

300

400

500

600

700

800

T [K

]

VΣ=1 m/day, Vg=200 m/dayTH-Collocation (cubics)Exact Solution

Tf=763.66K

T0=373.150K

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Example 1: Oxygen fraction profileExample 1: Oxygen fraction profile

-0.06 -0.04 -0.02 0 0.02 0.04 0.06ξ [m]

0

0.4

0.8

1.2

Y

VΣ=1 m/day, Vg=200 m/dayTH-Collocation (cubics)Exact Solution

Y0=1.0

Yb=0.164

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-0.6 -0.4 -0.2 0 0.2 0.4 0.6ξ [m]

300

400

500

600

700

800

T [K

]

VΣ=0.6 m/day, Vg=100 m/dayTH-Collocation (cubics)Exact Solution

Tf=761.20K

T0=373.150K

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-0.06 -0.04 -0.02 0 0.02 0.04 0.06ξ [m]

0

0.4

0.8

1.2

Y


Y0=1.0

Yb=0.036

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-0.6 -0.4 -0.2 0 0.2 0.4 0.6ξ [m]

300

400

500

600

700

800

T [K

]


Tf=767.41K

T0=373.150K

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-0.06 -0.04 -0.02 0 0.02 0.04 0.06ξ [m]

0.2

0.4

0.6

0.8

1

1.2

Y


Y0=1.0

Yb=0.303

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-0.6 -0.4 -0.2 0 0.2 0.4 0.6ξ [m]

300

400

500

600

700

800

T [K

]


Tf=761.55K

T0=373.150K

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-0.06 -0.04 -0.02 0 0.02 0.04 0.06ξ [m]

0

0.4

0.8

1.2

Y


Y0=1.0

Yb=0.057

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ConclusionsConclusionsHere, we have presented some preliminary numerical results that, as was expected, have shown a good agreement with the analytical solutions.

In this sense, TH-Collocation method has proved to be a very efficient framework to obtain highly accurate numerical solutions of the BVPJ’s which describe the dynamics of a planar combustion front for the steady state case under adiabatic conditions.

This is only the starting point in the numerical simulation of the more general case which is a subject of future work.

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Future WorkFuture WorkWe intend to consider the following cases:

Non Adiabatic ConditionsNon Linear problems (non constant gas phase density)Transient problems. 2D and 3D problems (reservoir scale)More general front geometries.

Combustion front tracking: Local Grid Refinement Methods or Level Set Methods (Osher & Sethian)

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!!!Thank you, for you attention!!!Thank you, for you attentionWe are open to hear We are open to hear

suggestions!!!suggestions!!!

trefftz-herrera collocation method for combustion fronts in oil

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