1. Bayesian inverse theory
for subsurface characterization and
data assimilation problems
Dario Grana
Department of Geology and Geophysics and School of Energy Resources
University of Wyoming
Quebec City, Canada – 18 March 2015
2. Introduction to reservoir modeling
A reservoir model has
• A structural component (geometrical grid)
• Properties filling the structure
• Data constraining the model
• Predictions
– 2
3. Introduction to reservoir modeling
– 3
Main rock and fluid properties in a static reservoir model
(before production)
• Porosity
• Permeability
• Fluid saturation
• Lithology
• Fluid Pressure
4. Introduction to reservoir modeling
• Static reservoir models are built using measured data at the
well location and surface geophysical measurements
(seismic data).
• Geophysical data are low resolution, hence static reservoir
models are uncertain.
• Uncertainty is generally represented by ensemble of
multiple models (for example 100 realizations of porosity
with different spatial correlations)
– 4
6. Introduction to reservoir modeling
– 6
• In reservoir modeling we aim to model rock
properties: porosity, lithology, and fluid
saturations.
7. Introduction to reservoir modeling
– 7
• When we create a model of the subsurface,
we have measurements of the properties we
are interested in and measurements of other
properties.
Reservoir modeling: porosity and oil saturation
(seismic data)
Aquifer modeling: water saturation
(electromagnetic data)
Mining: ore grades
(seismic data)
d = f (m)
• The model is the solution of an inverse problem
8. Introduction to reservoir modeling
– 8
• In reservoir modeling we aim to build models of
rock properties.
• Rock properties cannot be directly measured away
from the wells. The main source of information are
seismic data.
Inverse problem
Seismic data Porosity
9. – 9
Introduction: geophysical inverse problems
• There are various approaches for quantitative
estimation of reservoir properties from seismic
data:
Linear or non linear regression
Bayesian methods
Stochastic optimization methods
• Spatial variations in reservoir properties and inter-
dependence between different properties are
complex to model.
• The probabilistic framework is ideally suited to
model the uncertainty.
11. – 11
Introduction: geophysical inverse problems
• Bayesian approach:
• Prior distribution: prior knowledge of the model (e.g.
geological information, nearby fields)
• Likelihood function: probabilistic information of the
physical model linking data and model
• Posterior distribution: probabilistic information
combining the prior and the likelihood
Prior
Likelihood
Posterior
12. – 12
Introduction: geophysical inverse problems
• Many inverse problems are solved by using a
Bayesian approach and assuming a linear (or
linearized) physical model and a Gaussian
distribution of the model.
13. – 13
Introduction: geophysical inverse problems
• We present a Bayesian inversion method based
on Gaussian mixture distributions.
• Many inverse problems are solved by using a
Bayesian approach and assuming a linear (or
linearized) physical model and a Gaussian
distribution of the model.
14. Example: multimodal behavior
Well data
P-wavevelocity(m/s)
Porosity (v/v)
Sand content
• The overall distribution of
porosity is bimodal.
• Porosity is approximately
Gaussian in each facies
(i.e. in sand and in shale)
but not overall
Shale
Sand
15. – 15
Velocity(m/s)
Porosity
• Measured data at the well location
Depth(m)
Velocity
(m/s)
Mineralogical
fractions
Porosity &
saturation
Lithology
SummaryExample: multimodal behavior
17. – 17
• A random vector m is distributed according to a
Gaussian Mixture Model (GMM) with L components
when the probability density is given by:
where each single component is Gaussian:
and the additional conditions
Gaussian mixture models
L
1k
kk )(f)(f mm
),()( )()( k
m
k
mk Nf Σμm
0,1 k
L
1k
k
Example of 1D mixture with L=2
components (PDF and 500 random
sample histogram)
18. – 18
Gaussian mixture models
Gaussian Mixture distribution ),;(~ )()(
1
k
m
k
m
L
k
k N Σμmm
Weights, means and covariance matrices estimated by EM method
(Hastie, Tibshirani, Friedman, The Elements of Statistical Learning, 2009)
19. – 19
• Linear inverse problem
Linear inverse problems (Gaussian)
),(~ mmN Σμm
εGmd
),(~ dmdm
N Σμdm
),(~ Σ0ε N
NNMMN
RRRRR εGmd and:,,
)()( 1
| m
T
m
T
mmdm GμdΣGGΣGΣμμ
m
T
m
T
mmdm GμΣGGΣGΣΣΣ 1
| )(
• If
then
Tarantola, Linear inverse problems, 2005.
20. – 20
Linear inverse problems (Gaussian)
• This result is based on two well known
properties of Gaussian distributions:
A. The linear transform of a Gaussian
distribution is again Gaussian;
B. If the joint distribution (m,d) is Gaussian,
then the conditional distribution m|d is
again Gaussian.
• These two properties can be extended to the
Gaussian Mixture case.
21. – 21
• Linear inverse problem
Linear inverse problems (GM)
εGmd
),;()(~ )()(
1
k
dm
k
dm
L
k
k N Σμmddm
),(~ Σ0ε N
NNMMN
RRRRR εGmd and:,,
)()( )(1)()()()(
|
k
m
Tk
m
Tk
m
k
m
k
dm GμdΣGGΣGΣμμ
)(1)()()()(
| )( k
m
Tk
m
Tk
m
k
m
k
dm GμΣGGΣGΣΣΣ
• If
then
),;(~ )()(
1
k
m
k
m
L
k
k N Σμmm
),;()(,
)(
)(
)(
1
k
d
k
dkL
j jj
kk
k Nf
f
f
Σμdd
d
d
d
Analytical
expression
22. – 22
Introductory example
Reference Model: Trivariate distribution with 2 Tri-Normal
distributions (Gaussian mixture with 2 components)
)(f)(f)(f 2211 mmm
),()( )1()1(
1 mmNf Σμm
3
2
1
m
m
m
m
),()( )2()2(
2 mmNf Σμm
29. – 29
Inverse problem
Problem 1: We know the seismic amplitudes of
waves (measured data) and we want to
estimate elastic attributes.
Problem 2: We know the velocity of waves traveling
in the subsurface (from a model) and we
want to estimate rock properties;
Seismic data Porosity
SummaryInverse modeling in petroleum geophysics
30. – 30
Bayesian Gaussian mixture inversion
• Goal: - Estimate reservoir properties R
from seismic data S
-Evaluate the model uncertainty
)](),(),([
],,[
)|(
321
SSS
swc
P
S
R
SR
Porosity
Clay content
Water saturation
Partial-stack
seismic data
We estimate the posterior probability:
31. Bayesian Gaussian mixture inversion
• Seismic data S depend on reservoir
properties R through elastic properties m
• We can split the inverse problem into
two sub-problems:
•
• ff
gg
)(
)(
Rm
mS
))(( RS gf
seismic linearized modeling
rock physics model
36. – 36
Bayesian Gaussian mixture inversion
Isoprobability surface of 70% probability of hydrocarbon sand
Probability
37. – 37
Time-lapse studies
• In time-lapse reservoir modeling we aim to
model reservoir property changes from
repeated seismic surveys.
Inverse problem
Time-lapse seismic data
Reservoir property changes
(saturation and pressure)
43. – 43
Sequential simulations
• Sequential Gaussian Simulation (SGSim) is a specific case
of linear inverse problem with sequential approach
(the linear operator is the identity)
• The sequential approach to linear inverse problems
(Gaussian case) was proposed by Hansen et al. (2006)
• We extended this approach to Gaussian Mixture models
45. – 45
Sequential inversion (Gaussian)
mi
εGmd
NNM
RRR εG and:
ms is the subvector of direct observations of m
46. Sequential inversion (Gaussian)
Hansen et al., Linear inverse Gaussian theory and geostatistics: Geophysics, 2006.
mi
εGmd
NNM
RRR εG and:
),;(~),( ),(),( dmdm
Σμmdm sisi mmsi Nm
),(~ Σ0ε NIf
then
),;(~ mmN Σμmm
ms is the subvector of direct observations of m
47. – 47
Sequential inversion (GM)
mi
εGmd
NNM
RRR εG and:
ms is the subvector of direct observations of m
48. – 48
Sequential inversion (GM)
Analytical formulation form means, covariance matrices, and weights.
mi
εGmd
NNM
RRR εG and:
),(~ Σ0ε NIf
then
ms is the subvector of direct observations of m
),;(~),( )(
),(
)(
),(
1
k
m
k
m
L
k
ksi sisi
Nm dmdm
Σμmdm
),;(~ )()(
1
k
m
k
m
L
k
k N Σμmm
52. Introduction to reservoir modeling
– 52– 52
The static reservoir model provides a ‘snapshot’ of the
reservoir before production starts.
When production starts, for example by water injection
or depletion, fluid saturation and fluid pressure change in
time.
Dynamic reservoir modeling (i.e. fluid flow simulation)
predicts hydrocarbon displacement and pressure changes
(by solving equations of fluid flow through porous media
based on finite volumes).
53. Introduction to reservoir modeling
– 53– 53
In dynamic reservoir model, we run fluid flow
simulations and obtain:
• Production forecast at the well locations
• Snapshot of saturation and pressure at different
time steps.
54. Introduction to reservoir modeling
– 54
Due to uncertainty in the data and approximations of the models,
reservoir model predictions are uncertain. After N years of
production, we can compare production data with predictions.
History matching is a data assimilation technique that allows
updating the model until it closely reproduces the past behavior
of a reservoir.
55. History matching
– 55
Problem: Find the most likely model of initial porosity and
permeability to match the measured production history of the
first N years of production.
Method: Bayesian updating
56. History matching
– 56
Method: Ensemble Kalman Filter (EnKF)
Injector
Producer
N porosity models
N production forecasts
Data
Simulations
58. Re-parameterization
– 58
POD-DEIM response computed from 60 ‘snapshot’ of saturations
(i.e. saturation fields at 60 different times steps, every 60 days)
Example 1: 5 eigenvalues retained
DEIM point locations
59. Re-parameterization
– 59
Example 2: Similar example
with 20 eigenvalues.
The 20 DEIM points are
located along the channel. It
seems that the DEIM points
dynamically follow the water
front displacement.
Using the POD-DEIM
reduced model we can
reconstruct the true
production forecast
60. – 60
SummaryCurrent research projects
• Joint seismic-EM inversion • Seismic history matching (re-
parameterization of the water
front)
61. Conclusions
– 61– 61
Bayesian inverse methods are a powerful tool in reservoir
modeling for property estimation and uncertainty
quantification
The Gaussian mixture approach can be used for
multimodal models and preserve the analytical solution
The Bayesian approach can be extended to history
matching problems upon a re-parameterization of the
data assimilation problem
62. – 62
Acknowledgements
• Thanks to Erwan Gloaguen and INRS for the
invitation
• Thanks to IAMG to support UWyo and INRS
student chapters
• Thanks for your attention
