Compressible Streamlines and Three-Phase History Matching

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.