Streamline-Based History Matching for Multicomponent Compositional Systems
In this paper, the authors introduce a novel semianalytic approach to compute the sensitivity of the bottomhole pressure (BHP) data with respect to gridblock properties.
Streamline-based history-matching techniques have provided significant capabilities in integrating field-scale water-cut and tracer data into high-resolution geologic models. However, application of the streamline-based approach for simultaneous integration of water cut and bottomhole pressure (BHP) has been rather limited. In this paper, the authors introduce a novel semianalytic approach to compute the sensitivity of the BHP data with respect to gridblock properties.
The streamline-based method has many advantages in terms of computational efficiency and applicability. The main advantage of the streamline-based method is that it is able to calculate parameter sensitivity with a single streamline simulation or post-processing of the grid-based finite-difference simulation results. The calculated sensitivity is comparable with the sensitivities computed from numerical perturbation or with adjoint-based sensitivities. It is possible to calculate the parameter sensitivity from a commercial finite-difference simulator by use of streamlines traced using the flux field. This allows accounting for detailed physics by means of finite-difference simulation while taking advantage of the power of the streamlines for sensitivity computations.
Streamline and Parameter Sensitivity
Streamline-based history matching starts with the tracing of streamlines from a given set of static and dynamic conditions. This process is outlined in detail in the complete paper.
Amplitude, Travel-Time, and Generalized Travel-Time Inversion (GTTI) Methods
The approach to minimizing the objective function through all the observed points and simulation results is defined as amplitude inversion. The travel-time inversion, instead, attempts to match single reference times such as water-breakthrough point or peak tracer response. The amplitude matching is general in terms of reduction of the objective function, because while the travel-time inversion reduces the objective function at only a single point, the amplitude matching can cover all the data points. However, the travel-time-based approach is often used because amplitude inversion is a highly nonlinear problem and presents difficulties in computational cost and in reducing the objective function when there is a large amount of wells and data points.
An alternative approach to combine the advantages of amplitude inversion and travel-time inversion is the GTTI method. The objective is to find an optimal time shift that provides good agreement with an arbitrary number of observed and calculated data points. In other words, the objective of the GTTI method is to maximize the coefficient of determination of observed and calculated data points. The GTTI method retains the desirable properties of the travel-time inversion and the amplitude matching of the production data. It is important to note that the computation of the optimal travel-time shift is carried out as post-processing of the data at each well after the production response is computed.
The sensitivity of the GTTI method is also found by travel-time inversion. The GTTI method is desirable when observed data show monotonic trends and a large number of data points are available. Amplitude matching is appropriate when data are nonmonotonic with few sampling points. In other words, GTTI is often used to integrate water cut or gas/oil ratio (GOR), while composition information is integrated by amplitude inversion.
The authors propose a new approach to integrating pressure data by calculating analytical sensitivity of the pressure or pressure drop along streamlines with respect to reservoir parameters.
The streamlines are traced with a pressure gradient generated from an injector or a producer. Normally, pressure is solved on a 3D grid by an implicit-pressure/explicit-saturation solver or is obtained from a commercial simulator. However, it is also possible to construct the pressure equation along a streamline, assuming that there is no interaction outside of the 1D coordinate. The overall idea of the proposed approach is to construct the pressure equation along a streamline with the given boundary conditions and take a derivative with respect to reservoir static properties. This concept is described in detail in the complete paper with respect to continuous space, discretized space, and BHP sensitivity.
The objective function of this study is the production-data and BHP misfit of the rate-constrained wells. Once a matrix is constructed, an iterative sparse matrix solver is used for solving the system of equations. Because the original minimization equations often lead to an unstable solution, the additional regularization term is added to improve both convergence and final solution.
It is possible to smooth the solution by adding a symmetric stencil, or to give some prior information as a covariance type of matrix. Once the change of the static variable is calculated, then the parameters are updated and the flow simulation is conducted again. The history-matching process continues until the residual reaches a certain tolerance or a given maximum iteration number.
Verification of Pressure and Arrival-Time Sensitivity
The model is tested by a synthetic case to verify the proposed pressure-sensitivity equation (results are discussed in detail in the complete paper). Because the developed formulation is applicable for multiphase corner-point geometry, all the cases are tested on the basis of the three-phase water- or gas-injection problem with capillarity and gravity. Here, the tested geometry for sensitivity verification is 1D corner point and a 2D areal model.
The first test of sensitivity verification is conducted with 1D corner-point geometry. The geometry has a zigzag shape with an uphill trend in the vertical direction, as shown in Fig. 1 (above). Three cases of different boundary conditions are tested: single producer at center, single injector at center, and rate constraint of both injector and producer located at the edge of the model.
The permeability is given as heterogeneous, ranging from 10 to 1,000 md and distributed randomly. The analytical sensitivity calculated by the proposed method is verified by the adjoint-based method implemented in a commercial simulator and by the numerical-perturbation method. Among these three methods, the perturbation method is sensitive to the magnitude of the perturbation of the static value and is not reliable in finding accurate parameter sensitivity. Because of this inaccuracy of the numerical-perturbation method, the adjoint-based sensitivity is displayed for the verification of the pressure sensitivity.
Application to History Matching
As streamline sensitivity of pressure and arrival time is verified, a developed algorithm is applied to the history-matching problem. All of the cases include matching of BHP in addition to the breakthrough of the injection fluid, which is water cut for the water-injection problem and GOR and carbon dioxide composition for the gas-injection scenario. The results of the application are discussed in detail in the complete paper.
A novel streamline-based data-integration method is proposed, and its effectiveness is demonstrated by comparisons of parameter-sensitivity calculations and history-matching scenarios. Although the approach relies on streamline-based sensitivity calculations to relate pressure and saturation responses to the reservoir parameters, it can be applied with either streamline simulators or conventional finite-difference simulators. The conclusions from the work can be summarized as follows.
- We have proposed a novel methodology for streamline-based analytic approaches to compute BHP sensitivity with respect to the permeability. Although several assumptions are made during the process of the derivation, the results of the parameter-sensitivity calculations show good agreement with results from the adjoint-based method.
- The proposed pressure-sensitivity equation can be applied with a single injector or producer, thus generalizing the streamline-based data integration, such as matching of pressure data during the primary drainage process.
- Previous work on streamline-based history matching required well-breakthrough information. The results here show the applicability of the streamline-based approach without breakthrough information by including pressure information.
- The effectiveness of the streamline-based data-integration approach to multimillion-cell problems was demonstrated. The proposed approach also showed the applicability of matching modular-formation-dynamics-tester pressure data, which provides change of permeability in fine scale, particularly with respect to depth.
- The GTTI method to integrate GOR and water cut and the use of amplitude-matching techniques to integrate pressure and production mole fraction have been demonstrated.
This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 174750, “Streamline-Based History Matching of Arrival Times and Bottomhole-Pressure Data for Multicomponent Compositional Systems,” by Shusei Tanaka, Dongjae Kam, Akhil Datta-Gupta, and Michael J. King, Texas A&M University, prepared for the 2015 SPE Annual Technical Conference and Exhibition, Houston, 28–30 September. The paper has not been peer reviewed.