Summary
We propose a novel approach to history matching finite-difference models
that combines the advantages of streamline models with the versatility of
finite-difference simulation. Current streamline models are limited in their
ability to incorporate complex physical processes and cross-streamline
mechanisms in a computationally efficient manner. A unique feature of
streamline models is their ability to analytically compute the sensitivity of
the production data with respect to reservoir parameters using a single flow
simulation. These sensitivities define the relationship between changes in
production response because of small changes in reservoir parameters and, thus,
form the basis for many history-matching algorithms. In our approach, we use
the streamline-derived sensitivities to facilitate history matching during
finite-difference simulation. First, the velocity field from the
finite-difference model is used to compute streamline trajectories, time of
flight, and parameter sensitivities. The sensitivities are then used in an
inversion algorithm to update the reservoir model during finite-difference
simulation.
The use of a finite-difference model allows us to account for detailed
process physics and compressibility effects. Although the streamline-derived
sensitivities are only approximate, they do not seem to noticeably impact the
quality of the match or the efficiency of the approach. For history matching,
we use a generalized travel-time inversion (GTTI) that is shown to be robust
because of its quasilinear properties and that converges in only a few
iterations. The approach is very fast and avoids many of the subjective
judgments and time-consuming trial-and-error steps associated with manual
history matching. We demonstrate the power and utility of our approach with a
synthetic example and two field examples. The first one is from a CO2 pilot
area in the Goldsmith San Andreas Unit (GSAU), a dolomite formation in west
Texas with more than 20 years of waterflood production history. The second
example is from a Middle Eastern reservoir and involves history matching a
multimillion-cell geologic model with 16 injectors and 70 producers. The final
model preserved all of the prior geologic constraints while matching 30 years
of production history.
Introduction
Geological models derived from static data alone often fail to reproduce the
field production history. Reconciling geologic models to the dynamic response
of the reservoir is critical to building reliable reservoir models. Classical
history-matching procedures whereby reservoir parameters are adjusted manually
by trial and error can be tedious and often yield a reservoir description that
may not be realistic or consistent with the geologic interpretation. In recent
years, several techniques have been developed for integrating production data
into reservoir models. Integration of dynamic data typically requires a
least-squares-based minimization to match the observed and calculated
production response. There are several approaches to such minimization, and
these can be classified broadly into three categories: gradient-based methods,
sensitivity-based methods, and derivative-free methods. The derivative-free
approaches, such as simulated annealing or genetic algorithms, require numerous
flow simulations and can be computationally prohibitive for field-scale
applications. Gradient-based methods have been used widely for automatic
history matching, although the convergence rates of these methods are typically
slower than the sensitivity-based methods such as the Gauss-Newton or the LSQR
method. An integral part of the sensitivity-based methods is the computation of
sensitivity coefficients. These sensitivities are simply partial derivatives
that define the change in production response because of small changes in
reservoir parameters.
There are several approaches to calculating sensitivity coefficients, and
these generally fall into one of three categories: perturbation method, direct
method, and adjoint-state methods. Conceptually, the perturbation approach is
the simplest and requires the fewest changes in an existing code. Sensitivities
are estimated simply by perturbing the model parameters one at a time by a
small amount and then computing the corresponding production response. This
approach requires (N+1) forward simulations, where N is the number of
parameters. Obviously, it can be computationally prohibitive for reservoir
models with many parameters. In the direct or sensitivity equation method, the
flow and transport equations are differentiated to obtain expressions for the
sensitivity coefficients. Because there is one equation for each parameter,
this approach requires the same amount of work. A variation of this method,
called the gradient simulator method, uses the discretized version of the flow
equations and takes advantage of the fact that the coefficient matrix remains
unchanged for all the parameters and needs to be decomposed only once. Thus,
sensitivity computation for each parameter now requires a matrix/vector
multiplication. This method can also be computationally expensive for a large
number of parameters. Finally, the adjoint-state method requires derivation and
solution of adjoint equations that can be quite cumbersome for multiphase-flow
applications. Furthermore, the number of adjoint solutions will generally
depend on the amount of production data and, thus, the length of the production
history.
© 2005. Society of Petroleum Engineers
View full textPDF
(
1,629 KB
)
History
- Original manuscript received:
12 January 2004
- Revised manuscript received:
27 July 2005
- Manuscript approved:
5 August 2005
- Version of record:
15 October 2005
View Cart
