Next Generation Geological Modeling for Hydrocarbon Reservoir Characterization - Advances in Data, Methods, Models and Their Applications in Geoscience - page 227

permeability, represents a crucial part of any modeling activity. To quantify and reduce the
uncertainty in the description of hydrocarbon reservoirs, the parameters of geological model
are usually adjusted and reconciled with pressure and multi-phase production data by
history-matching (HM). With an advent of computing capabilities in recent decades, the
classical ( i.e . manual) HM has evolved to so-called computer-Assisted (or Automated) HM
(AHM) technology. When we hereafter in this document refer to the history-matching
process, the AHM workflow is assumed.
As an inverse problem, HM is highly non-linear and ill-posed by its nature, which means
that, depending on the prior information, one can obtain a set of non-unique solutions that
honor both the prior constraints and conditioned data with associated uncertainty. To assess
the uncertainty in estimated reservoir parameters, one must sample from the posterior
distribution, and the Bayesian methods (Lee, 1997) provide a very efficient framework to
perform this operation. Using Bayes' formula, the posterior distribution ( i.e., the probability
of occurrence of model simulated parameter, m , given the measured data values, d ) is
represented as being proportional to the product of prior and likelihood probability
distributions of the reservoir model:

   



p d|m m p

dm | m

 

p m|d  (3)

md |

p d

d

where,

p m represent the posterior, likelihood, and prior
distribution, respectively. The normalization factor

p

| (|)

md , p

| (| )

dm and ()

md dm

p

() d represents the probability

associated with the data and usually treated as a constant.
When the distribution of prior model parameters, m ( M  m , where M represents the

number of parameters), follows a multi-Gaussian probability density function (pdf), the
()

0

p m , centered around the prior mean m , is given by:

m

1  1 

  

T



  

0  1 0

p m  exp  mm C mm   (4)

 

m M

12 /

M/ 2

2

2 π C  

M

MxM

with C as the prior covariance matrix ( C  ). The distribution of likelihood data is

M M

defined as the conditional pdf p (| ) dm of data, d , given model parameters, m :

dm |

1 1

 

 

T

 

   

 1

p dm |  exp  d gm  C d gm  (5)

 

dm | D

N /2 1/2

 

2

 

2  C

D

NxN
D  C ). The relationship between the data and
the model parameters is expressed as a non-linear function that maps the model parameters
into the data space, d (m) , where d is the data vector with N observations representing
the output of the model, m is a vector of model parameters, and g is the forward model
operator that maps the model parameters into the data domain. For history-matching
problems, g represents the reservoir simulator. Using Bayes' theorem (Eq. 3), both prior and
likelihood pdfs are combined to define the posterior pdf as:

with C as the data covariance matrix (

D

   

p m|d  exp  O m  (6)

 

md |