We propose a generalized travel-time inversion method for production data integration into reservoir models using finite difference- based reservoir simulators. Our approach is motivated by seismic waveform imaging and is particularly well-suited for large-scale field applications because the computation cost depends only on the number of wells regardless of the number of parameters or the amount of observed data. Instead of matching the production data directly, we minimize a "travel-time shift" at each well derived by maximizing the cross-correlation between the observed and calculated production response. An optimal control method is used to compute the sensitivity of the travel time with respect to reservoir parameters. Finally, data integration is carried out via a modified Gauss-Newton method.
There are several advantages associated with the proposed travel-time inversion method. First, it is robust and computationally efficient. The travel-time misfit function is quasilinear with respect to changes in reservoir properties. As a result, the minimization proceeds rapidly even if the prior model is not close to the solution. Second, the computational cost associated with the sensitivity computation depends only on the number of wells, which can be orders of magnitude lower than the number of parameters or the amount of observed data. This offers a tremendous advantage over the commonly used gradient simulator method or the conventional adjoint methods that attempt to minimize the production data directly. Furthermore, the travel time approach also offers computational advantage during minimization of the misfit function using the Gauss-Newton algorithm. We have presented several examples to demonstrate the power, generality, and practical feasibility of our proposed approach for large-scale field applications.
In recent years, several techniques have been developed for integrating production data into reservoir models.1–10 These techniques allow engineers to build reservoir models that honor field production history and static information such as well logs, core, and seismic data. The theoretical basis of these techniques is generally rooted in the least-squares inversion theory. It is well known that inverse problems are typically ill-posed and can result in nonunique and unstable solutions. Proper incorporation of static data in the form of a prior model can partially alleviate the problem. However, there are additional outstanding challenges that have deterred the routine integration of production data into reservoir models. First, the computational cost is still extremely high, particularly when conventional finite-difference reservoir simulators are used for "forward" modeling. Under such conditions, most of the current methods become computationally prohibitive when a large number of parameters and observed data are involved. Second, the relationship between the production response and reservoir properties can be highly nonlinear. This often causes the solution to converge to a local minimum. All these factors make it difficult to obtain an adequate match to the production data and a meaningful estimate of the parameter field, particularly if the initial model is far from the solution.
Recently, streamline-based methods have shown significant potential for incorporating dynamic data into high-resolution reservoir models.8–12 A unique feature of the streamline-based production data integration has been the concept of a "travel-time match" that is analogous to seismic tomography. Instead of matching the production data directly, the observed data and model predictions are first lined up at the breakthrough time. This is typically followed by a conventional "amplitude match" whereby the difference between the observed and calculated production response is minimized. A major part of the production data misfit reduction occurs during the travel-time inversion, and most large-scale features of heterogeneity are resolved at this stage.8–10
There are several advantages associated with a travel-time inversion of production data. First, it is robust and computationally efficient. Unlike conventional amplitude matching, which can be highly nonlinear, it has been shown that the travel-time misfit function is quasilinear with respect to changes in reservoir properties. 13–15 As a result, the minimization proceeds rapidly even if the initial model is not close to the solution. Second, the travel time sensitivities are typically more uniform between wells compared to amplitude sensitivities that tend to localize near the wells. This prevents over-correction in the near-well regions.15 Finally, during practical field applications, the production data are often characterized by multiple peaks (for example, tracer response). Under such conditions, the travel-time inversion can prevent the solution from converging to secondary peaks in the production response.10,15
Our goal in this paper is to generalize the travel-time inversion approach to finite-difference-based reservoir models. Although the streamline models have been extremely successful in bridging the gap between geologic modeling and flow simulation, they are currently limited in their ability to incorporate complex physical processes and cross-streamline mechanisms in a computationally efficient manner.11,12 Thus, a rapid and robust approach to production data integration using finite-difference models will be particularly useful in characterizing reservoirs dominated by mechanisms such as gravity effects, transverse dispersion, and other complex compositional phenomena.
There are two important issues that need to be addressed for developing a travel-time inversion method using finite difference reservoir models. First, we need a consistent and general definition of travel time at the producing wells. This must recognize the fact that in field situations, breakthrough times are very often not well defined. Second, we need an efficient approach for computing the sensitivities of the travel time with respect to reservoir properties. These sensitivities simply define the changes in travel time resulting from small perturbations in reservoir parameters and typically constitute a critical aspect of any inversion algorithm.
In this paper, we will utilize concepts from the wave equation travel-time inversion to propose a general, robust, and efficient method for production data integration into reservoir models. Our approach is motivated by the work of Luo and Schuster in the context of waveform inversion in seismology.13 We generalize the travel-time inversion method for production data integration and actually reduce the two-step inversion (travel time and amplitude) into a single-step procedure while retaining most of the desirable features of the travel-time inversion. Furthermore, our approach becomes analogous to an amplitude inversion near the solution.15 For computing the sensitivity coefficients of the production data with respect to reservoir parameters, we resort to an optimal control method as discussed by various authors.6,16-19