Flow in porous media:
physical, mathematical and
numerical aspects
13/04/201 - Stavanger-CFD Workshop
5
Peppino Terpolilli TFE-Pau
OUTLINE
• Darcy law
• Mathematical issues
• Some models: Black-oil, Dead-oil,BuckleyLeverett………
• Numerical approach
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Darcy law
• Navier-Stokes equations:
v
1
 vv   p  v  f
t

• Darcy law:
q 
• K
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K

p
is the matrix of permeability: porous
media characteristic
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Darcy law
• Continuum mechanics:
at a REV located at x :
 ( x)
K ( x)
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porosity: ratio of void to bulk volume
permeability: Darcy law
REV
4
Darcy law
• Darcy law:
empirical law (Darcy in 1856)
• theoretical derivation:
Scheidegger, King Hubbert, Matheron
(heuristic)
Tartar (homogeneization theory)
Stokes
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G


Darcy law
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Darcy law
• Poiseuille flow in a tube:
single-phase, horizontal flow
steady and laminar
no entrance and exit effects
 R 2 p
v
8 L
v
R
L
p
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mean velocity
radius
length
pressure gradient
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Darcy law
• Poiseuille flow in a tube:
K
 R2
8
unit: darcy   m2 1012 m2
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Darcy law
• Different scale:
pore level: Stokes equations
lab: measures
numerical cell: upscaling
field: heterogeneity
Darcy law
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G


Darcy law
8
Black-oil model
• Extended Darcy law:
qp  
Kkrp
p
( p p   p gD)
• krp relative permeability of phase p
• D the depth
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Darcy law
• Continuum mechanics:
at a REV located at x :
So , w, g
kr (S )
pc (S )
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saturation: fraction of pore volume
relative permeability
capillary pressure
REV
10
Kr-pc
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Kr-pc
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Math issues
• For single-phase flows Darcy law leads to
linear equation:
 C
p
 div ( K ( x).p)  f
t
• For multi-phase flow we recover nonlinear
equtions: hyperbolic, degenerate parabolic
etc…..
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Math issues
• The mathematical model is a system of PDE
with appropriate initial and boundary
conditions
• the coefficients of the equations are poorly
known  stochastic approach
• geology + stochastic = geostatistic
K ( x,  )
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A field….
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Math issues
Data:
• wells
: core, well-logging, well test
• extension: geophysic, geology
• scale problems and uncertainty
(geostatistic)
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Uncertainty
• SPDE:
p
 div ( K ( x,  ).p )  f
t
• These problems are difficult:
experimental design approach
‘ Grand projet incertitude ’
Industrial tools
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Black-oil model
• Hypotesis:
three phases: 2 hydrocarbon phases and
water
hydrocarbon system: 2 components
a non-volatile oil
a volatile gas soluble in the oil phase
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Black-oil model
• Hypotesis:
components

oil
oil

 gas

water
oil
gas
gas
water
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phases
19
Black-oil model
phases:
water: wetting
oil
: partially wetting
gas : non wetting
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Sw
So
Sg
saturation
saturation
saturation
Black-oil model
• Validity of the hypothesis:
dry gas
depletion, immiscible water or gas injection
oil with small volatility
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Black-oil model
• PVT behaviour: formation volume factor
Bo 
Vo Vdg RC
Vo STC
;
Vg RC
Bg 
Vg STC
;
VW RC
Bw 
VW STC
• where:
volume of a fixed mass at reservoir
VRC
conditions
VSTC
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volume of a fixed mass at stock tank
conditions
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Black-oil model
• Mass transfer between oil and gas phases:
 Vdg 
RS  

V
 o  STC
Vdg : gas component in the oil phase
Vo : oil component in the oil phase
functions of the oil phase pressure
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Black-oil model
• Thermo functions for oil:
160
140
120
1,35
1,3
1,1
1,25
1
1,2
0,9
1,15
0,8
0,7
1,1
1,05
0
100
200
300
400
P
Bo
0,6
muo
0,5
1
0
100
200
P (bars)
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300
0,4
400
muo (cP)
100
80
60
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1,4
1,2
20
0
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1,4
1,3
Bo (Sm3/m3)
Rs(m3/m3)
200
180
Black-oil model
• Mass balance:
water
oil
gas
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
(S w  w )  div(  w qw )  Qw
t

(So  o )  div (  o qo )  Qo
t

(S g  g  So  dg )  div (  g q g   dg qo )  Qg  Qdg
t
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Black-oil model
• Extended Darcy law:
qp  
Kkrp
p
( p p   p gD)
• krp relative permeability of phase p
• D the depth
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Black-oil model
• Water:
• oil:
• gaz:
 Kkrw

  Sw 


div

Pw


wgZ

  0



 t  Bw 
 Bw w

 Kkro

  So 




div

Po


ogZ
0




t  Bo 
 Bo o

 Kkrg

  Sg
So 
  Rs
  Pg   ggZ  

  div 
 t  Bg
Bo 
 Bg  g

 KkroRs

 div 
  Po   ogZ    0
 Bo o

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Black-oil model
• saturation:
So  Sw Sg 1
• capillary pressures:
pw  po  pcow
p g  po  pcog
• we obtain 3 equations with 3 unknowns:
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po , Sw , Sg
if
po  pb
po , Sw , Rs
if
po  pb
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Black-oil model:boundary conditions
• Boundaries
closed: no flux at the extreme cells
aquifer: source term in corresponding cells
• wells:
Dirichlet condition: bottom pressure
imposed
Neumann condition: production rate
imposed
source terms for perforated cells (PI)
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Black-oil model: initial conditions
• capillary and gravity equilibrium
• pressure imposed in oil zone at a given depth
• oil pressure in all cells and then Pc curves
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Black-oil model: theoretical results
• Antonsev, Chavent, Gagneux:
existence results for weak solutions
• PME: porous media equation
more resuts: Barenblatt, Zeldovich,
Benedetti,…Vazquez.
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