TPWL

MULTI-PHYSICS, OPTIMIZATION, AND UNCERTAINTY PROPAGATION
Leonardo Guimarães, Bernardo Horowitz, Ramiro Willmersdorf, Ezio Araújo, Silvana Afonso, Paulo Lyra
The Federal University of Pernambuco
Email: [email protected]
Website: http://lmcg0.ctg.ufpe.br/chair/
Geomechanics of Fractured Reservoirs
Coupled Hydro-Mechanical simulation is an important tool for
understanding the
hydrocarbon production of unconventional
reservoirs, since faults and fractures are critical to the behavior of
naturally fractured rocks.
Unconventional Reservoirs, such as
naturally fractured carbonates and shales
Streamline-Geomechanics Coupling
Reduced Order Methods - TPWL
Coupled geomechanics and multi-phase flow simulation is essential
to predict the behavior of stress sensitive reservoirs. However,
coupled hydro-mechanical simulation involves a high complex
physics associated with a large number of degrees of freedom. The
simulation domain includes reservoir and its surroundings. For
these reasons, simulations have a high computational cost.
Residual Equation – Terms: A – Accumulation; F – Flow; Q – Well Controls
Sequential Coupling scheme
Trajectory Piecewise Linearization + Proper Orthogonal Decomposition
TPWL
Linearization of Residual Equation around
stored previously converged states; good
approximation near training trajectory.
POD – Projects the state vector into a
reduced space using basis 
x  Φy
Linear Reduced Equation to be solved NO ITERATIONS REQUIRED
Much more complex behavior of fractures...
Modeling Discontinuities
Stochastic 3D model of fractures based on observed dips and azimuths
(Carmod Network Project. Funding: Petrobras)
Continuum Strong Discontinuity Approach can introduce
discontinuities (fractures or faults) in a finite element mesh, avoiding
excessive discretization or remeshing algorithms. Two different
techniques may be adopted:
1) Embedded Discontinuity
y
 y
i 1
 (J r )
Local post-processing of velocity - LPPV
 Velocity field continuous between
element-element interface;
1
i 1
   A i 1 

 Q

n
i
n 1
i 1
u
 u 

 y  y   
i
i 1  
 u
r
 x r

Where:
 Velocity field satisfy
at all point;
2) Interface Elements
Without LPPV
With LPPV
Large heterogeneous case
Discontinuity placed inside the finite
element
n 1
i 1
Reduced order model is a sequence
of linear system solutions. In order to
solve it, it is necessary to find the
closest state among the stored ones.
The procedure linearize new solution
around this state. Finally, the actual
state is reconstructed by the inverse
projection.
 Velocity must be consistent with
pressure equation.
Fluid injection in faulted reservoir
Interface elements are positioned along
the faces of regular finite elements
TPWL Accuracy Study
SPE10-Modified
Case
(a)Incompressible/
Same Densities
(b)Compressible/
Diff Dens
(c) B + Retraining
(a)
Saturation
field
(b)
(c)
Multifidelity Optimization – MSAO Algorithm
Fluid injection in naturally fractured media:
Results for SPE10-Modified Case
Surrogate Model
k
k
k
fˆ ( x )  f T P W L  x    K r ig in g
x
TPWL Retraining Criterion
streamline
Speed-up:400
E TPW L 
Speed-up:40
N
1
T 
vz

n 1
y z  y z   tn
n
i

Algorithm Tren
d
Emax
min
Iterat. HF
LF
Retrain NPV (108$)
.
SQP (HF)
-
-
-
5
88
-
-
6.92
SQP (LF)
-
-
-
11
1
187
-
6.48
MSAO
0
10-2
0.25
7
165*
777
1
6.53
MSAO
1
10-2
0.25
5
120*
715
0
7.20
MSAO
1
10-2
-
11
254* 1446
2
7.22
Retraining Criteria – Paper to be published at RSS 2015