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Flow in porous media: physical, mathematical and numerical aspects

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Transcript of Flow in porous media: physical, mathematical and numerical aspects
19/04/23  StavangerCFD Workshop
Peppino Terpolilli TFEPau
Flow in porous media:
physical, mathematical and
numerical aspects
 CFD Stavanger 219/04/23
OUTLINE
• Darcy lawDarcy law• Mathematical issuesMathematical issues• Some models: Blackoil, Deadoil,BuckleySome models: Blackoil, Deadoil,Buckley
Leverett………Leverett………• Numerical approachNumerical approach
 CFD Stavanger 319/04/23
Darcy law
• NavierStokes equations:NavierStokes 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
 CFD Stavanger 419/04/23
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
 CFD Stavanger 519/04/23
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
 CFD Stavanger 619/04/23
Darcy law
• Poiseuille flow in a tube:Poiseuille flow in a tube:
singlephase, horizontal flowsinglephase, 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
 CFD Stavanger 719/04/23
Darcy law
• Poiseuille flow in a tube:Poiseuille flow in a tube:
unit: darcy unit: darcy
2
8
RK
2 12 210m m
 CFD Stavanger 819/04/23
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
 CFD Stavanger 919/04/23
Blackoil 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
 CFD Stavanger 1019/04/23
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
 CFD Stavanger 1319/04/23
Math issues
• For singlephase flows Darcy law leads to linear For singlephase flows Darcy law leads to linear equation:equation:
• For multiphase flow we recover nonlinear For multiphase flow we recover nonlinear equtions: hyperbolic, degenerate parabolic etc…..equtions: hyperbolic, degenerate parabolic etc…..
( ( ). )p
C div K x p ft
 CFD Stavanger 1419/04/23
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
 CFD Stavanger 1619/04/23
Math issues
Data:Data:
• wells : core, welllogging, well test wells : core, welllogging, well test
• extension: geophysic, geologyextension: geophysic, geology
• scale problems and uncertainty scale problems and uncertainty
(geostatistic) (geostatistic)
 CFD Stavanger 1719/04/23
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
 CFD Stavanger 1819/04/23
Blackoil model
• Hypotesis: Hypotesis:
three phases: 2 hydrocarbon phases andthree phases: 2 hydrocarbon phases and
waterwater
hydrocarbon system: 2 componentshydrocarbon system: 2 components
a nonvolatile oila nonvolatile oil
a volatile gas soluble in the oil phasea volatile gas soluble in the oil phase
 CFD Stavanger 1919/04/23
Blackoil model
• Hypotesis:Hypotesis:
components phasescomponents phases
oil oiloil oil
gas oilgas oil
gas gasgas gas
water waterwater water
 CFD Stavanger 2019/04/23
Blackoil 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
 CFD Stavanger 2119/04/23
Blackoil 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
 CFD Stavanger 2219/04/23
Blackoil 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
 CFD Stavanger 2319/04/23
Blackoil 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
 CFD Stavanger 2419/04/23
Blackoil 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
 CFD Stavanger 2519/04/23
Blackoil 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
 CFD Stavanger 2619/04/23
Blackoil 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
 CFD Stavanger 2719/04/23
Blackoil 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
 CFD Stavanger 2819/04/23
Blackoil 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
 CFD Stavanger 2919/04/23
Blackoil 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)
 CFD Stavanger 3019/04/23
Blackoil 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
 CFD Stavanger 3119/04/23
Blackoil 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.