computational continuum mechanics in m&m smr a. v. myasnikov d.sc., senior research scientist,...
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Computational Continuum Mechanics in M&M SMR
A. V. Myasnikov
D.Sc., Senior Research Scientist, SMRStavanger, 28th April, 2006
From Seismics to Simulations and back
1. Streamline Reservoir Simulations;
2. Finite-difference/volume simulations of poroelastic waves propagation in partially saturated layers;
3. Rheological monitoring for pay-zones
High Road Steps:
Theorem #1: If Reservoir simulations, then Streamline technologies.
Theorem #2: If Streamline technologies, then FrontSim.
From Seismics to Simulations:
FrontSim:
I. Development of an effective 1D simulators for multi-component two-phase flows
II. Development of 3 Phase Compressible Dual Porocity Models
III. Extanding of Front Tracking Technology beyond two phase black oil model
IV. Effective parallelizing of pressure solverV. … and so on ….
a) To increase the performance by development of high
resolution modern techniques (TVD, ENO, AMR, Front
Tracking)
b) To take the PVT-flash procedure off the hydrodynamic
“body” of the code
c) To search for improved algorithms for representation of
phase equilibrium in terms of alternative thermodynamic
variables.
Development of an effective 1D simulators for multi-component two-phase flows
Results: a) I order vs. II order high compressive TVD schemes
I order schemeII order scheme “exact” solution
The same C1-CO2-C4-C10 mixture
Grid =
200 pts
200 pts instead of 800!
( ) ( ),
and are known by
approximation of binodal
k k kC A C B γ γ
A B
1 2 1 1 1 2 2 1
Standard Alternative
, ,..., , , ,..., ,C CN NC C C C C γ
Alternative Set of Independent Variables
Bubble pointsDew points
Tie lines
Plait point
Two-phase domain
Tie-line extension
C1, C2
C1, γ
OilGas
1C C
2 1,...,C
m mNC C γ
Tie-line equation:
C1-CO2-C4-C10 mixture Red – 1D ECLIPSE
Black – -parametrisation
Results: b) PVT-flash procedure is taken off:
20 times CPU advantage!
One Remarkable Feature of Alternative Variables
THEOREM: The image of a Riemann problem solution for the
auxiliary system
coincides with the projection (red circles) of the C- image (red line)
of a Riemann problem solution for the general system
γ
γ γ
(blue line) 0k kA B
(not proved yet…)
“direct” I order scheme
“projective” scheme
Results: c) “projective” scheme – 3 effective components instead of 100
Some problems
in the vicinity of
the root corner
points!
II. Implementation of DPSP 3-Phase Compressible Model into FrontSim
Porous Matrix continuum
Fracture continuumFractured Porous medium
DPSP Results: 3 Phase compressible flowOil production rate vs time FrontSim
Eclipse
Water saturation
matrix
fracture
1. Streamline Reservoir Simulations;
2. Finite-difference/volume simulations of poroelastic waves propagation in partially saturated layers;
3. Rheological monitoring for pay-zones
High Road Steps:
Finite volume and finite difference
schemes for acoustic/elasticity equations:
1. Godunov’ s finite volume scheme of 1st order
accuracy in space and time and its “2nd order” TVD
extension;
2. Two step finite different Virieux’s-like staggered
grid scheme
3. >>>
+ non-reflecting boundary conditions
15 Initials
Wavelet Propagation by Virieux or Godunov (rough grid)
Virieux
SOG
FOG
1.7t 1.3t 0.9t 0.5t 0.1t
16 Initials
FLAC
SOG
FOG
1.7t 1.3t 0.9t 0.5t 0.1t
Wavelet Propagation by Virieux or Godunov (finer grid)
17 Initials
N-R conditions in 2D Elasticity (SOG)“exact”
N-R BC
18 Initials
Artificial attenuation in buffer zones (PML)Top and bottom– rigid walls
u
v
0vA 2vA 5vA
1. Streamline Reservoir Simulations;
2. Finite-difference/volume simulations of poroelastic waves propagation in partially saturated layers;
3. Rheological monitoring for pay-zones
High Road Steps:
20 Initials
Experimental (?) data, no fit by viscoelastic and classical Biot models
Uniform elastic media
Layered media, no pay-zone
Layered media, pay-zone
21 Initials
0p u
t xu p
ut x
1D for visco-X-ticity:
Bottom boundary
1 1 1 1, , ,E
3 3 3 3, , ,E
2 2 2 2, , ,E
u(x,t) p(x,t)
lx
lh
Top boundary
Top boundary:
/u x f
Bottom boundary:
nonreflecting
(Korneev, Goloshubin, et al. Geophysics, 2004, 69, 522)
One phase models: conservation laws
0t
V
d
dt
vF P
:de
divdt
qJ P V
TP P
1. Continuity equation:
2. Equations of motion:
3. Dynamic momentum equations:
4. Energy equation:
Linear and nonlinear liquids
(1/ )ds de dT p
dt dt dt
(1/ , )e e s
2
1 1: , 0
dsdiv T div
dt T T T q
q s
JJ Π v J
002
1 1 1:
ST div
T T T qJ v Π v
- definition of “fluid”
- Gibbs identity
Generelized Onsager relations for nonlinear processes:
0
32 00, ,
S
S
T div
T div
1
q
Φ vΦ vJ Π
v v
2
2
s Newtonian liquid
s s Hard plastic liquid
s s Viscoplastic liquid
The simplified diagram of the process of displacement oil by gas. Their movement paths are shown with arrows.
One phase liquids: conservation laws
0t
V
32
0S
dp
dt div
ΦvF
v v
032
0 :S
S
deT p div
dt div
Φv v
v vSome special cases:
Linear and nonlinear solids
:de ds e d
Tdt dt dt
ε
ε
( , )e e s ε
2
1 1: : 0
e dT
T T T dt
q
εJ P v
ε
Definition of “solid”
- Gibbs identity
1
2 0ε g - g - Strain tensor
Elastic solids
2
1 1: : 0
e dT
T T T dt
q
εJ P v
ε
By definition: elastic is the medium where all isothermal process are reversible
0: : 0
S e d
dt
εP v
ε
So that: e
P
ε
0Sd d
ifdt dt
0ε g
v e
Non-elastic models
Let:
: : 0e
pP e e eε
As before,
: 0e
pP P eε
0,d d
dt dt
0p pg ε
e e e
( ),
pp
e Pe
/
/
d dt e
d dt
e ε
e ε ε- System of ODE
2 /e
d dt viscoelastisity
p pe e e ε
ε
elastic plasticity p pe e
1. Comprehensive study of acoustic/seismic feature of rhelogicaly complex multiphase single-layered pay-zone
2. The same for multiple-layered zones
3. Through inverse problem to 4D seismic
Rheological monitoring: what is that?
Biot M.A. Nonlinear and semilinear rheology of porous solids. J. Geophys. Res. 1973. 78. 4924
Biot M.A. Variational irreversible thermodynamics of heat and mass transfer in porous solids: new concepts and methods. Q. Appl. Math. 1978. 36. 19
Biot M.A. New variational-lagrangian irreversible thermodynamics with applications to viscous flow, reaction-diffusion and solid mechanics. Advances in Applied Mechanics. 1984. 24. 1-91.
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