1. IMA Workshop on Compatible DiscretizationsMy
6/13/2001
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 , 2004

2. Outline Introduction to Petroleum IndustryMy
6/13/2001
Outline
Introduction to Petroleum Industry
Compositional model (Black Oil model)
Black Oil is considered as a particular case of Compositional
General framework
Examples

3. Introduction to Petroleum IndustryMy
6/13/2001
Introduction to Petroleum Industry
Seismic map
Geologic model
Analogs
interpretation
Reservoir model
Logs
Cores
Boat shoot the seismic survey, geologists create and interpret it and create the geological model
(millions of cells). Frequently analogs are used.
Exploration wells are drilled,
cores and logs are used to get extra information.
The properties are “upscaled” and a simulation model is created.

4. Compositional Model n Conservation of massMy
6/13/2001
Compositional Model
Phases liquid (l), vapor (v), and aqueous (a) -
components - methane, ethane, propane, etc.,
phases,
components n
Conservation of mass
Volume V
Compositional models are the work horse of the industry.
The black oil models are particular case of the compositional.
Multicomponent multiphase flow with mass transfer between the phases. Local thermodynamic equilibrium is assumed.
For simplicity consider isothermal model with no diffusion/dispersion.
Volume -representative elementary volume (big enough with respect to the pores, small compared to the field.
overall concentration,
component flow rate,
sources and sinks,
saturation of phase j.
porosity,
molar density of phase j,
mole fraction of component i,
mole fraction of component i in phase j

5. Compositional Model - cont.My
6/13/2001
Compositional Model - cont.
Darcy law (generalized)
phase velocity,
phase pressure,
capillary pressure,
viscosity of phase j,
absolute permeability,
relative permeability of phase j,
mass density of phase j,
gravitational acceleration,
depth.
W_i - overall concentration,
xi_T - total molar density,
xi_j - molar density of phase j with respect to fluid volume,
S_j - saturation of phase j,
z_i - mole fraction of component i
U_i - flow rate
x_{ij} - mole fraction of component i in phase j ,
v_j - Darcy velocity,
K - absolute permeability,
k_{rj} - relative permeability of phase j,
mu_j - viscolity of phase j,
P_j - j phase pressure,
rho_j - mass density of phase j,
g - gravitational acceleration,
D - depth

6. Compositional Model - cont.My
6/13/2001
Compositional Model - cont.
Volume balance laws - total volume balance =
pore volume -
total volume of fluids -
“pressure equation”
moles of comp i

7. Compositional Model - cont.My
6/13/2001
Compositional Model - cont.
Volume balance laws - liquid phase volume balance =
liquid saturation
volume of liquid -
“saturation of oil
equation”
The equations for the other phase saturations are similar.

8. Compositional Model - cont.
Simplify - no capillary pressure, no saturation equations.
Linearize (typically first discretize then linearize)

9. Compositional Model - cont.x x
We have to discretize
Wang, Yotov, Wheeler, et. al introduced
(possible problems for non smooth solution)

10. Grids
pinchouts

11. 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

12. General framework Model problem written as a systemMy
6/13/2001
General framework
Model problem written as a system
Find such that (primal dual MFEM)
Find such that

13. Examples Rectangular grid / full tensor (Ware, Parrott, and Rogers)
Dual grid - rectangles,
- piecewise constants,
- piecewise constant
vectors with continuous normals
Basis vectors
M., “Analysis of a new Mixed Finite Volume Method”,
Comp. Methods Appl. Math. V. 3, 2003

14. Examples (Ware, Parrott, and Rogers)

15. Examples (Ware, Parrott, and Rogers)
Theorem:
Numerical example:

16. Examples (Ware, Parrott, and Rogers)
-error of the pressure,
- error of the pressure,
-error of the flux,
- error of the flux

17. Voronoi/Donald mesh / full tensor
Examples
Voronoi/Donald mesh / full tensor

18. Unstructured (Voronoi) grids
- scalar coefficient
(M. “Finite Volume Methods on Voronoi Meshes”,Numer. Meth. PDE,
Herbin, et. al.)
What about
approximation

19. Unstructured (Voronoi) grids
For grids with extra regularity
Hypothesis: The approximation of
could be improved with post-processing.

20. Examples (Voronoi/Donald mesh / full tensor)
Dual grid - triangles,
- linear piecewise continuous functions,
- piecewise constants on Voronoi volumes,
- piecewise constants on triangles
with continuous normals
M. “A New Flexible Mixed Finite Volume Method”, submitted

21. Examples (Voronoi/Donald mesh / full tensor)
Discrete problem:
Find such that

22. Error estimates
Find such that

23. Extra

24. Black Oil Model Phases Components + + Gas + Oil + + + Water
Standard Conditions
Reservoir Conditions
Reservoir Conditions

25. Black Oil Model C/P l v a o x g x x w x
Phases liquid, vapor, aqueous
components - oil, gas, water
C/P
l v a
o x
g x x
w x

26. Black Oil Model C/P l v a o x w x Phases - liquid, vapor, aqueous
components - oil, gas, water
C/P
l v a
o x
w x
No mass transfer
between the phase
If only 2 phases exist
total velocity, global pressure
(Chavent, Jaffre)

27. Black Oil Model C/P l v a o x x g x x x w x
Phases liquid, vapor, aqueous
components - oil, gas, water
C/P
l v a
o x x
g x x x
w x

28. pressure to be eliminatedMy
6/13/2001
Examples
Quadrilateral mesh / full tensor (M. Edwards et. al.)
pressure
Dual grid - cell-centers connected with the middles of the edges/faces
- piecewise linears
(nonconforming space)
- piecewise constants on
the cell
-piecewise constants with
continuous normals (4 dof),
- piecewise constants (8 dof)
pressure
pressure to be eliminated
velocity

29. Examples (quads)

30. Example (quads)