HT-splitting method for the Riemann problem of multicomponent two

phase flow in porous media

Anahita ABADPOUR Mikhail PANFILOV

Laboratoire d'Énergétique et de Mécanique Théorique et Appliquée (LEMTA)

Introduction

• Compositional model

• Diagrammatical representation of the split thermodynamics

• Reimann problem in terms of Ht-split model

• Results for three and four components problem

Compositional model

Hydrodynamic Equations • Mass conservation of each component

• Momentum balance for each phase

Where

))(())1(( )()()()( gradpkckcdivscsct l

kllg

kgg

kll

kgg

Nk ,...,1

pgradkK

Vl

ll

pgradkK

Vg

gg

0g

jj

),(

/

/00 lgj

K

K

gj

gjj

0P

Pp

L

Xx ii

Thermodynamics Closure relations

• Chemical potential equilibrium equations

• Equations of phase state

• Normalizing equations

• Rheological equations of state

Nkccpccp Nll

kl

Ngg

kg ,...,1,),...,,(),...,,( )()1()()()1()(

lgjccp Njjjj ,,),...,,( )()1(

lgjccp Njjjj ,,),...,,( )()1(

11

)(

N

k

kgc 1

1

)(

N

k

klc

Main Parameters• Relative phase mobility

• Perturbation ratio

,

Splitting of Hydrodynamics & Thermodynamics (S. Oladyshkin, M. Panfilov -2006 )

= +

00

00

lg

gl

*

*

t

t )/( 002* PKLt g

05.001.0~,

Compositional flow model(2N+3 equations)

Thermodynamicindependent model(2N+1 equations)

Hydrodynamicindependent model

(2 equations)

Ht-split form of compositional model

• Differential thermodynamic equations (DTE)

• Hydrodynamic equations

where

1,...,2,011

)(

)(

)(

)(

Nk

dp

dc

cdp

dc

c

Ng

k

kg

k

))(( pgradkkdivt llgg

2

)(

)(1

)(),(x

p

pk

x

pk

xt

pps

t

sN

ggglll

lgjpd

cd

cdp

d Nj

k

Nj ,,

1)(

)(

)(

dp

ds

dp

dsps

Ng

g

Nl

l

)()(

)1(),(

First integral of the HT-split model

• Steady-state pressure

• Equation of saturation transport

Where

Fractional mass flow function of gas phase

llgg kkx

p

pkk

psF

t

pps

x

psF

t

sN

g

llggl

)(),(

),(),(

llgg

gg

kk

kpsF

),(

Diagrammatical Representation of the

Split Thermodynamics

Phase diagrams and tie-lines

P Oversaturated Gas

Equilibrium Liquid + Gas

0S

C – total concentration of the light component

Equilibrium Gas

1S

1S

Equilibrium Liquid

UndersaturatedLiquid

0S

0 1S

Concept of a P-surface

Reimann problem in terms of Ht-split

model

Problem formulation

• Initial state of gas saturation

• Pressure boundary condition

• Initial condition of phase composition

)(

)(

0

0

tss

xssinj

x

init

)(0 tpp injx

2,...,1

2,...,1

),()(

),()(

Nkcc

Nkcc

injkgpp

k

g

inikgpp

kg

inj

ini

)(ts inj

)(xs ini

Lack of discontinuity conditions

• Hugoniot condition for transport equation

• Entropy condition ( Lax inequality )

• These two conditions are unfortunately largely insufficient, as a simultaneous shock of saturation and concentrations is determined by N + 1 parameters from one side of the shock: the shock velocity, N-1 concentrations and 1 saturation at the shock, where N is the number of components.

),( psFVs fl

)(),()( ssss

ssss s

psF

s

psF

s

psF ),(),(),(

Degenerating Hugoniot conditions

• Compact form of the Compositional model

Where

• New Hugoniot conditions

lgjx

pkWscsc jjj

kll

kgg

k ,,,)1()()()(

NkcWcWxt

kll

kgg

k ,...,1,0)( )()()(

lgf WWV

NkcWcWV kll

kggf

k ,...,2,)()()(

Pure Saturation shocks

• Eliminating liquid velocity

• Eliminating gas velocity

• Adding up together

0

000 ,)(

l

glllgggflg VVVs

NkVcVcVscc lllk

lgggk

gfk

llk

gg ,...,2,)( 0)(0)()()(

gggk

fk

g VcVsc 0)()(

lllk

fk

l VcVsc 0)()(

0000 VVVs

V

s

Vlg

gg

lg

Total System of Hugoniot Conditions

FV

V ggf

0

1,...,2)(

)()(

)(

)()(

Nkc

c

c

cN

g

Ng

N

kg

kg

k

Nk

c

cFVV k

gk

kg

ggf ,...,2,)()(

)(0

Where:

)1( ss lg

Intermediate P-surfaces

One of the significant qualitative results of the classic theory of the

Riemann problem announces that in an N-component two-phase

system that does not change the number of phases, the phase

concentrations should follow (N−1) different tie lines including two

”external” tie lines that correspond to the initial and the injection

states and (N−3) ”intermediate” or ”crossover” tie-lines.

In the case of variable pressure we assume the same result to be

valid, so in a N-component two-phase system that does not change

the number of phases the phase concentrations should follow (N−1)

different P-surfaces including two ”external” P-surfaces that

correspond to two boundary pressures and (N−3) ”intermediate” or

”crossover” P-surfaces.

This means that the concentrations can have (N−2) internal shock.

Algorithm of solving the Riemann problem

• Front tracing :

Determination of the backward and forward concentrations at all the CS-shocks:

solution to the transcendent system of : Degenerating Hugoniot relations at the shocks Differential thermodynamic equations between the shocks Chemical potential equilibrium equations

Determination of the shock saturations and velocities :solution to the transcendent system of : Remaining Hugoniot Relations Entropy Condition

• Solution to the differential transport equation :

Solution to the saturation transport equation while taking into account the priori determined parameters and placing of all the shocks

Determining intermediate

concentrationsFrom:

With some arithmetic calculations:

Eliminating saturations and densities:

))(1( Ng

kl

Ng

kl

kg

Nl

kg

Nl

kg

Ng

kg

Ngl ccccccccccccs

1,...,2)(

)()(

)(

)()(

Nkc

c

c

cN

g

Ng

N

kg

kg

k

1,...,2,))(1( Nkccccccccccccs kg

Nl

kg

Nl

kg

Ng

kg

Ng

Ng

kl

Ng

kll

222222

222222

gNlg

Nlg

Ngg

Ng

Ngl

Ngl

Ngl

Nglg

Nlg

Nlg

Ngg

Ng

cccccccccccc

cccccccccccc

1,...,3,

Nkcccccccccccc

cccccccccccckg

Nl

kg

Nl

kg

Ng

kg

Ng

Ng

kl

Ng

kl

Ng

kl

Ng

kl

kg

Nl

kg

Nl

kg

Ng

kg

Ng

Unknown phase concentrations and

equations at sc-shock

Shock number Unknown concentrations

First (from injection P-surface)

2N

Second up to (N-3)th 4N

Last (to initial P-surface) 2N

Total 4N(N-3)

Type of equation Number of utilization

Total number

(N-3) Hugoniot Conditions (N-2) (N-3)*(N-2)

N chemical potential equilibrium

2(N-3) 2N(N-3)

2 normalizing equations 2(N-3) 4(N-3)

N-2 differential thermodynamics equations

(N-3) (N-2)(N-3)

Total 4N(N-3)

• Unknown phase concentrations:

• Equations at sc-shocks :

Determination of the sc-shocks saturations

Nk

c

cFFk

gk

kg

gg ,...,2,)()(

)(

After finding the concentrations on each P-surface, we are able to use just 2 of these N-1 equations to find the saturations before and after each concentrations shock:

Here the total velocity was assumed to be constant. It is significant that this assumption is not exact, but it is much weaker than the assumption of constant total velocity all over the problem in the classic theory.

Results for three and four components problem

Three Components Problem

0 0.1 0.2 0.3 0.4 0.5 0.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

s

Four Components Problem

0 0.1 0.2 0.3 0.4 0.5 0.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

s

Advantage of the developed method

• Concentrations at the shocks can be determined explicitly as the solution to an algebraic system of equations coupled with the thermodynamic block.

• For a sufficiently low number of components, this system can be solved in the analytical way.

• Saturations at the discontinuity then could be determined using the intermediate concentrations.

parameters of the shocks can be determined before constructing the solution to the Riemann problem:

Thank you for your attention!