Flow in porous media: physical, mathematical and numerical aspects

19/04/23 - Stavanger-CFD Workshop

Peppino Terpolilli TFE-Pau

Flow in porous media:

physical, mathematical and

numerical aspects

- CFD Stavanger 219/04/23

OUTLINE

• Darcy lawDarcy law• Mathematical issuesMathematical issues• Some models: Black-oil, Dead-oil,Buckley-Some models: Black-oil, Dead-oil,Buckley-

Leverett………Leverett………• Numerical approachNumerical approach


Darcy law

• Navier-Stokes equations:Navier-Stokes equations:

• Darcy law:Darcy law:

• is the matrix of permeability: porous is the matrix of permeability: porous

media characteristicmedia characteristic

1vv v p v f

t

pK

q

K


Darcy law

• Continuum mechanics:Continuum mechanics:

at a REV located at :at a REV located at :

porosity: ratio of void to bulk volume porosity: ratio of void to bulk volume

permeability: Darcy lawpermeability: Darcy law

REVREV

( )K x

( )x

x


Darcy law

• Darcy law:Darcy law:

empirical law (Darcy in 1856)empirical law (Darcy in 1856)

• theoretical derivation: theoretical derivation:

Scheidegger, King Hubbert, Matheron (heuristic)Scheidegger, King Hubbert, Matheron (heuristic)

Tartar (homogeneization theory)Tartar (homogeneization theory)

Stokes Darcy lawStokes Darcy law

G


Darcy law

• Poiseuille flow in a tube:Poiseuille flow in a tube:

single-phase, horizontal flowsingle-phase, horizontal flow

steady and laminarsteady and laminar

no entrance and exit effectsno entrance and exit effects

mean velocitymean velocity

radius radius

lengthlength

pressure gradientpressure gradient

2

8

R pv

L

R

p

L

v


Darcy law

• Poiseuille flow in a tube:Poiseuille flow in a tube:

unit: darcy unit: darcy

2

8

RK

2 12 210m m


Darcy law

• Different scale:Different scale:

pore level: Stokes equationspore level: Stokes equations

lab: measureslab: measures

numerical cell: upscalingnumerical cell: upscaling

field: heterogeneityfield: heterogeneity

Darcy law Darcy lawDarcy law Darcy lawG


Black-oil model

• Extended Darcy law:Extended Darcy law:

• relative permeability of phase prelative permeability of phase p

• the depththe depth

( )rpp p p

p

Kkq p g D

rpk

D


Darcy law

• Continuum mechanics:Continuum mechanics:

at a REV located at :at a REV located at :

saturation: fraction of pore volumesaturation: fraction of pore volume

relative permeabilityrelative permeability

capillary pressure capillary pressure

REVREV

( )cp S

( )rk S

x

, ,o w gS


Kr-pc


Math issues

• For single-phase flows Darcy law leads to linear For single-phase flows Darcy law leads to linear equation:equation:

• For multi-phase flow we recover nonlinear For multi-phase flow we recover nonlinear equtions: hyperbolic, degenerate parabolic etc…..equtions: hyperbolic, degenerate parabolic etc…..

( ( ). )p

C div K x p ft


Math issues

• The mathematical model is a system of PDE with The mathematical model is a system of PDE with appropriate initial and boundary conditions appropriate initial and boundary conditions

• the coefficients of the equations are poorly known the coefficients of the equations are poorly known stochastic approach stochastic approach

• geology + stochastic = geostatisticgeology + stochastic = geostatistic

( , )K x


A field….


Math issues

Data:Data:

• wells : core, well-logging, well test wells : core, well-logging, well test

• extension: geophysic, geologyextension: geophysic, geology

• scale problems and uncertainty scale problems and uncertainty

(geostatistic) (geostatistic)


Uncertainty

• SPDE: SPDE:

• These problems are difficult:These problems are difficult:

experimental design approachexperimental design approach

‘ ‘ Grand projet incertitude ’Grand projet incertitude ’

Industrial tools Industrial tools

( ( , ). )pdiv K x p f

t


Black-oil model

• Hypotesis: Hypotesis:

three phases: 2 hydrocarbon phases andthree phases: 2 hydrocarbon phases and

waterwater

hydrocarbon system: 2 componentshydrocarbon system: 2 components

a non-volatile oila non-volatile oil

a volatile gas soluble in the oil phasea volatile gas soluble in the oil phase


Black-oil model

• Hypotesis:Hypotesis:

components phasescomponents phases

oil oiloil oil

gas oilgas oil

gas gasgas gas

water waterwater water


Black-oil model

phases:phases:

water: wetting saturation water: wetting saturation

oil : partially wetting saturationoil : partially wetting saturation

gas : non wetting saturationgas : non wetting saturation

wS

oS

gS


Black-oil model

• Validity of the hypothesis:Validity of the hypothesis:

dry gasdry gas

depletion, immiscible water or gas injectiondepletion, immiscible water or gas injection

oil with small volatility oil with small volatility


Black-oil model

• PVT behaviour: formation volume factorPVT behaviour: formation volume factor

• where: where:

volume of a fixed mass at reservoirvolume of a fixed mass at reservoir

conditionsconditions

volume of a fixed mass at stock tankvolume of a fixed mass at stock tank

conditionsconditions

; ;o dg g WRC RC RC

g wo WgSTC STCSTC

V V V VBo B B

V VV

RCV

STCV


Black-oil model

• Mass transfer between oil and gas phases:Mass transfer between oil and gas phases:

: gas component in the oil phase: gas component in the oil phase

: oil component in the oil phase: oil component in the oil phase

functions of the oil phase pressurefunctions of the oil phase pressure

dgS

o STC

VR

V

dgV

oV


Black-oil model

• Thermo functions for oil:Thermo functions for oil:

0

20

40

60

80

100

120

140

160

180

200

0 100 200 300 400

P

Rs(m

3/m

3)

1

1,05

1,1

1,15

1,2

1,25

1,3

1,35

1,4

0 100 200 300 400

P (bars)

Bo

(Sm

3/m

3)

0,4

0,5

0,6

0,7

0,8

0,9

1

1,1

1,2

1,3

1,4

muo

(cP

)

Bo

muo


Black-oil model

• Mass balance:Mass balance:

waterwater

oiloil

gas gas

( ) ( )o o o o oS div q Qt

( ) ( )w w w w wS div q Qt

( ) ( )g g o dg g g dg o g dgS S div q q Q Qt


Black-oil model

• Extended Darcy law:Extended Darcy law:

• relative permeability of phase prelative permeability of phase p

• the depththe depth

( )rpp p p

p

Kkq p g D

rpk

D


Black-oil model

• Water:Water:

• oil:oil:

• gaz: gaz:

0t

Sw Kkrwdiv Pw wgZ

Bw Bw w

0t

ogZPooBo

Kkrodiv

Bo

So

gt

0

Sg So KkrgRs div Pg g Z

Bg Bo Bg g

KkroRsdiv Po ogZ

Bo o


Black-oil model

• saturation:saturation:

• capillary pressures: capillary pressures:

• we obtain 3 equations with 3 unknowns:we obtain 3 equations with 3 unknowns:

1o w gS S S

w o cowp p p

g o cogp p p

, ,o w g o bp S S if p p

, ,o w s o bp S R if p p


Black-oil model:boundary conditions

• BoundariesBoundaries

closed: no flux at the extreme cellsclosed: no flux at the extreme cells

aquifer: source term in corresponding cellsaquifer: source term in corresponding cells• wells:wells:

Dirichlet condition: bottom pressureDirichlet condition: bottom pressure

imposedimposed

Neumann condition: production rate Neumann condition: production rate

imposedimposed

source terms for perforated cells (PI)source terms for perforated cells (PI)


Black-oil model: initial conditions

• capillary and gravity equilibriumcapillary and gravity equilibrium

• pressure imposed in oil zone at a given depthpressure imposed in oil zone at a given depth

• oil pressure in all cells and then Pc curvesoil pressure in all cells and then Pc curves


Black-oil model: theoretical results

• Antonsev, Chavent, Gagneux:Antonsev, Chavent, Gagneux:

existence results for weak solutions existence results for weak solutions

• PME: porous media equationPME: porous media equation

more resuts: Barenblatt, Zeldovich, Benedetti,…more resuts: Barenblatt, Zeldovich, Benedetti,…Vazquez.Vazquez.

Flow in porous media: physical, mathematical and numerical aspects

Documents

Transcript of Flow in porous media: physical, mathematical and numerical aspects