Transcript Why Mixed Finite Elements are not used in the Petroleum Industry and what can we do about it ? Ilya D.

Why Mixed Finite Elements are not used in
the Petroleum Industry and what can we do
about it ?
Ilya D. Mishev
IMA Workshop on Compatible Discretizations
May 11-16 14, 2004
Outline
 Introduction to Petroleum Industry
 Compositional model (Black Oil model)
– Black Oil is considered as a particular case of Compositional
 General framework
 Examples
2
Introduction to Petroleum Industry
Seismic map
Geologic model
Analogs
interpretation
Reservoir model
Cores
3
Logs
Compositional Model
Phases
- liquid (l), vapor (v), and aqueous (a) - N P phases,
components - methane, ethane, propane, etc.,
- N C components
n
Conservation of mass

Wi
V


 Ui  n  Ri ,
V
V
Wi  T zi , T 
Volume V
Wi 
Ui 
Ri 
Sj 
t
overall concentration,
component flow rate,
sources and sinks,
saturation of phase j.
4

j
zi 
xij 
i  1,, NC ,
NP
 S ,
j j
j 1
NP
Ui 
x  v .
ij j j
j 1
porosity,
molar density of phase j,
mole fraction of component i,
mole fraction of component i in phase j
Compositional Model - cont.
Darcy law (generalized)
vj 
Kk rj
j
Pj   j gD ,
Pj  P  Pc , j , j  v, a, P  Pl .
vj
Pj
Pc , j
j




K  absolute permeability,
phase velocity,
k rj  relative permeability of phase j,
phase pressure,
capillary pressure,  j  mass density of phase j,
viscosity of phase j, g  gravitational acceleration,
D  depth.
5
Compositional Model - cont.
Volume balance laws - total volume balance
=
pore volume - VP
total volume of fluids - VT
VP ( P)  VT ( P, N)
 VP VT  P




t  t
 t
VT  Vl  Vv  Va ,
N  ( N1 ,, N NC ),
6
N i  moles of comp
i
NC

i 1
VT
Ri    U i 
N i
“pressure equation”
Compositional Model - cont.
Volume balance laws - liquid phase volume balance
=
liquid saturation V P
volume of liquid - V l
S lVP ( P)  Vl ( P, N),
S l  VP P 
Vl P
VP




Sl 
t  P t 
P t
NC

i 1
Vl
Ri    U i 
N i
The equations for the other phase saturations are similar.
7
“saturation of oil
equation”
Compositional Model - cont.
1
Simplify - no capillary pressure, no saturation equations.
N i
   U i  Ri , i  1,  , N C ,
t
N
K Ui 

P
ij ( N)P
j 1
 0 , i  1,  , N C
VP  VT
Linearize (typically first discretize then linearize)
N i
   U i  Ri , i  1,  , N C ,
t
K 1U i 
NC
 N
ij
j
  iPP  f i , i  1,  , N C ,
j 1
NC

i 1
8 i
iP N
  PP P  f P .
N
U  R
t
N 
1
K U   A   f
 P 
Compositional Model - cont.
K 1u  p  0
u  f
  (Kp)  f
p0

K 1u  v  p  v  0




K 1u  v  ap  v  0


K
N 
U  A   f ,
 P 
T N 
a     fP.
P


K 1u  v  p  (av)  0

x

1

K 1u  v  p  v  0,



x
ui
Wang, Yotov, Wheeler, et. al introduced
(possible problems for9 non smooth solution)
1
K A

1
N 
U     A1f
 P 
We have to discretize N i , p.
 Pi
Grids
pinchouts
10
General framework
Given general cell centered grid
– build dual grid to approximate the fluxes,
– choose approximation space for the pressure,
– define local approximation of the flux on the dual volume,
– exclude fluxes to get the finite volume method
11
General framework
Model problem written as a system
Find

(u, p) U  P
such that (primal dual MFEM)

K 1u  v  p  v  0

K 1u  p  0
 u  f
v  V ,

 u q   f q

q  Q .

Find (u h , ph )  U h  Ph such that


K 1u h  v h 
  p
h
 vh  0
v h  Vh ,
K i Fh K i
 u
K i K i
12
h
 n qh 
fq

h
qh  Qh .
Examples
Rectangular grid / full tensor (Ware, Parrott, and Rogers)
Dual grid - rectangles,
k
l
ekl
Ph ,Qh - piecewise constants,
Uh ,Vh - piecewise constant
vectors with continuous normals
Basis vectors eij , eik , e jl , ekl
v  vijeij  vikeik  v jle jl  vklekl
i
j
vij   v  n ij , vik   v  n ik ,
lij
lik
v jl   v  n jl , vkl   v  n kl ,
M., “Analysis of a new Mixed Finite Volume Method”,
Comp. Methods Appl. Math. V. 3, 2003
13
l jl
lkl
Examples
1
K
 uh  vh 

 v
E h E
h
(Ware, Parrott, and Rogers)
 n E [ ph ]  0 v h  eij , eik , e jl , e kl ,
uijtij,ij  uik tik ,ij  u jlt jl ,ij  uklt kl ,ij  p j  pi  0,
uijtij,ik  uik tik ,ik  u jlt jl ,ik  uklt kl,ik  pk  pi  0,
uijtij, jl  uik tik , jl  u jlt jl , jl  uklt kl, jl  pl  p j  0,
uijtij,kl  uik tik ,kl  u jlt jl ,kl  uklt kl,kl  pl  pk  0,
where t mn , pq   K 1e mn  e pq , or


u h  Tp h , p h  pi  p j , pi  pk , p j  pl , pk  pl ,
u
V
h
T
n   f
V
[ p]E  lim  p( x  tn E )  p( x  tn E ) 
t 0, t  0
14
Examples
(Ware, Parrott, and Rogers)
Theorem:
|| u  u h ||0,  || p  ph ||1,h  Ch || u ||1,  | p |2, ,
where || q ||1,2 h ~
2
(
q

q
)
 i j
neighbors
Numerical example:
K ( x, y )  RD( x, y ) RT ,
d1 0 
cos( )  sin( )

D ( x, y )  
,
R

,


,

 sin( ) cos( ) 
0
d
4


2

d1  1  2 x 2  y 2  y 5 , d 2  1  x 2  2 y 2  x 3 ,
p( x, y )  sin(x) sin(y ).
15
Examples
(Ware, Parrott, and Rogers)
Errors
Error
L2 -error of the pressure,
H 1 - error of the pressure,
L2 -error of the flux,
H div - error of the flux
8
16
32
64
Mesh points
16
128
256
Examples
Voronoi/Donald mesh / full tensor
17
Unstructured (Voronoi) grids
I )x ( k  K - scalar coefficient
E
p  ph
1,
 Ch p 2,
(M. “Finite Volume Methods on Voronoi Meshes”,Numer. Meth. PDE,
Herbin, et. al.)
hu
What about


K 1u h  v h 
 v
approximation
h
 n E [ ph ]  0
h
 n qh 
E h E
 u
K i K i
fq

v h  U h ,
h
qh  Ph .
Ph  piecewise constants,
U h  piecewise constants to approximate
the normal component of the f lux on DE .
18
Unstructured (Voronoi) grids
E
u  uh
0,
 p  ph
1,
 Ch p
2, 
,
u
2
0,

  u  n
.
2
E
E edge DE
For grids with extra regularity
u  uh
 Ch
1/ 2
h ( div, )
p
2, 
,
u
2
h ( div, )
Hypothesis: The approximation of h u
could be improved with post-processing.
19

  u  n
E edge E

2
E
Examples
(Voronoi/Donald mesh / full tensor)
Dual grid - triangles,
Ph - linear piecewise continuous functions,
Qh- piecewise constants on Voronoi volumes,
U h , Vh - piecewise constants on triangles
with continuous normals
e
ij
 n ij  1,
lij
e
vij   v  nij , vik   v  nik , v jk   v  n jl ,
lij
lik
e
ij
 n ik  0,
lik
ik
 n ij  0,
lij
v  vijeij  vikeik  v jke jk
e
 n ij  0,
lij
l jl
M. “A New Flexible Mixed Finite Volume Method”, submitted
20
ij
 n jk  0,
l jl
e
ik
 n ik  1,
lik
jk
e
e
lik
e
ik
 n jk  0,
l jl
jk
 n ik  0,
e
l jl
jk
 n jk  1,
Examples
(Voronoi/Donald mesh / full tensor)
Discrete problem:
Find
(uh , ph ) Uh  Ph
1
K
 u h  v h   p h  v h  0

such that
v h  U h ,

 u
K i K i
h
 n qh   f qh qh  Ph .

For v h  e ij , e ik , e kl
Tu h  Mp h  0, u h  T 1Mp h .
T heorem:
|| u  u h ||0,  || p  ph ||1,  Ch (|| u ||1,  | p |2, )
21
Error estimates
Find
(uh , ph ) Uh  Ph
such that
ah (u h , v h )  bh ( v h , ph )  g , v h 
ch (u h , ph )  f , qh 
v h  Vh
qh  Qh ,
ch (.,.) continuous and
a h (u h , v h )
i ) inf sup
  , LBB (inf-sup) conditions
u h U 0 h v V || u h||U || v h||V
h
h
h
h
bh ( v h , p h )
ii) inf sup
  , U 0 h  u h  U h , ch (u h , qh )  0, q h  Qh ,
ph Ph v V || v h||V || p h|| P
h
1h
h
h
ch (u h , qh )
iii) inf sup
  , V1h  v h  Vh , a h (u h , v h )  0, u h  U 0 h .
qh Qh u U || v h||V || q h||Q
h
h
h
h
a h (.,.),
bh (.,.),
dim(Uh )  dim(Ph )  dim(V h )  dim( Q h ) .


|| u  u h ||Vh  || p  ph ||Ph  C infvhVh || u  v h ||Vh  infqhPh || p  qh ||Ph ... .
22
Extra
23
Black Oil Model
Phases
Components
+
Gas
+
+
Reservoir
Reservoir
Conditions
Conditions
+
Oil
+
+
Water
Standard Conditions
24
Black Oil Model
Phases
- liquid, vapor, aqueous
components - oil,
gas, water
C/P
o
g
w
l
x
x
v
a
x
x
25
Uo  l vl ,
U g  x g ,l l v l  x g ,v v v v ,
Uw  a va ,
Black Oil Model
Phases
- liquid, vapor, aqueous
components - oil,
gas, water
C/P
o
l
x
v
a
U o  l vl ,
Uw  a va ,
w
No mass transfer
between the phase
x
If only 2 phases exist
total velocity, global pressure
(Chavent, Jaffre)
26
v  K (Pg  G ), v  v l  v a

v    f
t
Black Oil Model
Phases
- liquid, vapor, aqueous
components - oil,
gas, water
C/P
o
g
w
l
x
x
v
x
x
27
a
x
x
U o  xo,l l v l  xo,v v v v ,
U g  x g ,l l v l  x g ,v v v v  x g ,a a v a ,
Uw  a va ,
Examples
Quadrilateral mesh / full tensor (M. Edwards et. al.)
Dual grid - cell-centers connected with the middles of the edges/faces
Ph - P
piecewise linears
h
(nonconforming space)
Qh - piecewise constants on
the cell
pressure
U h -piecewise constants with
continuous normals (4 dof),
Vh - piecewise constants (8 dof)
pressure
pressure to be eliminated
velocity
28
Examples (quads)
 u  v   Kp  v  0

v  V

  u q   f q

u

h
 vh 
q  Q

4
   Kp
K i Fh j 1 K i
j
 u
K i K i
29
h
h
 vh  0
 n qh   f q h

v h  Vh ,
qh  Q h .
Example (quads)
u

h
 vh 
4
   K p
K i Fh j 1 K i
j
h
 vh  0
v h  Vh ,
u h  uijeij  uik eik  u jl e jl  ukle kl .
For v h  fij ,i , fik ,i , f ij, j , f jl , j , f kl ,k , fik ,k , f kl ,l , f jl ,l
solve the linear system
 ph 
Nuh  M    0,
p1,h 

N  8  4, M  8  8 matrices,



for u h  uij , uik , u jl , ukl , p1,h  pij , pik , p jl , pkl , i.e.,
T


u h  Tp h , p h  pi , p j , pl , pk .
30
T