Summary
Reconciling high-resolution geologic models to field production history is
still by far the most time-consuming aspect of the workflow for both
geoscientists and engineers. Recently, streamline-based assisted and automatic
history-matching techniques have shown great potential in this regard, and
several field applications have demonstrated the feasibility of the approach.
However, most of these applications have been limited to two-phase water/oil
flow under incompressible or slightly compressible conditions.
We propose an approach to history matching three-phase flow using a novel
compressible streamline formulation and streamline-derived analytic
sensitivities. First, we use a generalized streamline model to account for
compressible flow by introducing an “effective density” of total fluids along
streamlines. This density term rigorously captures changes in fluid volumes
with pressure and is easily traced along streamlines. A density-dependent
source term in the saturation equation further accounts for the pressure
effects during saturation calculations along streamlines. Our approach
preserves the 1D nature of the saturation equation and all the associated
advantages of the streamline approach with only minor modifications to existing
streamline models. Second, we analytically compute parameter sensitivities that
define the relationship between the reservoir properties and the production
response, viz. water-cut and gas/oil ratio (GOR). These sensitivities are an
integral part of history matching, and streamline models permit efficient
computation of these sensitivities through a single flow simulation. Finally,
for history matching, we use a generalized travel-time inversion that has been
shown to be robust because of its quasilinear properties and converges in only
a few iterations. The approach is very fast and avoids much of the subjective
judgment and time-consuming trial-and-error inherent in manual history
matching.
We demonstrate the power and utility of our approach using both synthetic
and field-scale examples. The synthetic case is used to validate our method. It
entails the joint integration of water cut and gas/oil ratios (GORs) from a
nine-spot pattern in reconstructing a reference permeability field. The
field-scale example is a modified version of the ninth SPE comparative study
and consists of 25 producers, 1 injector, and aquifer influx. Starting with a
prior geologic model, we integrate water-cut and GOR history by the generalized
travel-time inversion. Our approach is very fast and preserves the geologic
continuity.
Introduction
Integration of production data typically requires the minimization of a
predefined data misfit and penalty terms to match the observed and calculated
production response (Oliver 1994; Vasco et al. 1999; Datta-Gupta et al. 2001;
Reis et al. 2000; Landa et al. 1996; Anterion et al. 1989; Wu et al. 1999; Wang
and Kovscek 2000; Sahni and Horne 2005). There are several approaches to such
minimization, and these can be broadly classified into three categories:
gradient-based methods, sensitivity-based methods, and derivative-free methods
(Oliver 1994). The derivative-free approaches such as simulated annealing and
genetic algorithm require numerous flow simulations and can be computationally
prohibitive for field-scale applications with very large numbers of parameters.
Gradient-based methods have been widely used for automatic history matching,
although the rate of convergence of these methods is typically slower than that
of the sensitivity-based methods, such as the Gauss-Newton or the LSQR method
(Vega et al. 2004). An integral part of the sensitivity-based methods is the
computation of sensitivity coefficients. There are several approaches to
calculating sensitivity coefficients, and these generally fall into one of the
three following categories: perturbation method, direct method, and adjoint
state methods. The perturbation approach is the simplest and requires the
fewest changes to an existing code. This approach requires (N+1) forward
simulations, where N is the number of parameters. Obviously, this 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 (Vasco et
al. 1999). Because there is one equation for each parameter, this approach can
require the same amount of work. A variation of this method, called the
gradient simulator method, utilizes the discretized version of the flow
equations and takes advantage of the fact that the coefficient matrix remains
unchanged for all parameters and needs to be decomposed only once (Anterion et
al. 1989). Thus, sensitivity computation for each parameter now requires a
matrix-vector multiplication. This method obviously represents a significant
improvement, but still can be computationally demanding for large number of
parameters. Finally, the adjoint-state method requires derivation and solution
of adjoint equations that can be significantly smaller in number compared to
the sensitivity equations. The adjoint equations are obtained by minimizing the
production data misfit with flow equations as constraint, and the
implementation of the method can be quite complex and cumbersome for multiphase
flow applications (Wu et al. 1999). Furthermore, the number of adjoint
solutions will generally depend on the amount of production data and thus can
be restrictive for field-scale applications.
© 2007. Society of Petroleum Engineers
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History
- Original manuscript received:
16 February 2006
- Meeting paper published:
22 April 2006
- Revised manuscript received:
18 January 2007
- Manuscript approved:
18 January 2007
- Version of record:
20 December 2007
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