SPE 73144 A Geostatistical Approach to Streamline-Based History Matching

History matching large reservoir models under a realistic geological continuity constraint remains an outstanding problem. We propose a new combined geostatistical and streamline-based method potentially capable of history matching large reservoir models, while accounting consistently for geological continuity as provided by a permeability histogram and variogram, and production data. While most existing history matching techniques somehow rely on the calculation of single grid-block sensitivity coefficients, the proposed method avoids such calculation entirely by jointly perturbing effective permeabilities along a set of streamlines. Such perturbation results in large changes of the permeability field that significantly reduce error in each iteration of the history matching procedure. The problem of mapping streamline effective permeability perturbations to single grid blocks is performed under a Gauss-Markov random function constraint. This novel stochastic mapping procedure accounts for the target histogram and variogram, while honoring the streamline effective permeability perturbations. Forward flow simulation is achieved by a streamline simulator. The methodology is presented on synthetic cases; it appears to be computationally efficient and robust. Introduction Historic production data can provide important information about permeability connectivity and distribution within a reservoir. The process of integrating production into reservoir models can be tedious for various reasons: (1) the flow simulator, also termed forward model, is CPU demanding, (2) other information pertaining to permeability is available that needs to be integrated jointly with dynamic data. The latter information usually consists of hard permeability (core or well-log) in wells, seismic (3D and 4D) information and information about the geological continuity of the reservoir traditionally modeled with a variogram. We propose a combined geostatistical and streamline-based method for integration of production data, accounting for geological continuity. We concentrate on waterflood applications, but in principle, the method can be extended to other scenarios. The fractional flow (i.e., water cut) response at wells is mostly dominated by preferential flow paths of high permeability or low permeability flow barriers. Moreover, the information provided by dynamic data on the permeability field relates to a global measure of flow paths existing within the reservoir. An ideal tool for detecting such paths is a streamline simulator 4 , 3 , 2 , 1 As opposed to more conventional finite difference simulators, streamline simulators move fluids along "streamlines" instead of between single grid blocks. Although it is sometimes argued that streamline simulators do not solve the "full physics" of the flow (low compressibility and injection dominated displacement are generally assumed), they have the potential to solve that part of the physics that is mostly related to permeability heterogeneity. Hence, the properties of streamlines appear especially attractive when fashioning history-matching techniques to infer heterogeneity. Streamline simulators have the advantage of being order(s) of magnitude faster than traditional methods, moreover they have the capability to filter out useful information in the production data that relates most to permeability heterogeneity. Streamline-based inversion methods are not new. 7 , 6 , 5 , 3 In fact, some of the basic ideas used here were elucidated by Brigham and AbbaszadehDehghani 8 for tracer testing. However, the current approaches do not integrate, in a consistent framework, geological information about the reservoir. Unlike current inversion methods, we propose to perturb the effective permeability along single streamlines, also termed streamline effective permeability, instead of perturbing single grid-block permeabilities. In the proposed approach, we avoid calculating single grid-block sensitivity coefficients on the global grid. Instead we perturb streamline effective permeabilities serially. Because a streamline generally passes through a number of grid-blocks, potentially large changes are made to the permeability field per iteration step. Geological information is integrated within a consistent stochastic framework using a geostatistical methodology. The proposed geostatistical algorithm spreads the streamline effective permeability perturbation to the grid-block level, accounting for any given histogram and variogram of the grid-block permeabilities. We show that for the examples presented our approach is fast, simple, and flexible. In the discussion we present a strategy to apply this methodology to 3D cases. Perturbing Streamline Effective Permeability Although any existing streamline inversion method can possibly be used, we will follow the methodology developed by Wang and Kovscek. 6 Streamline-based history matching is essentially an iterative process whereby streamline simulations are run each time a change in the permeability field is made, until the error between calculated and measured production and pressure data is minimized. Wang and Kovscek 6 propose a streamline-based history matching method that calculates the change in permeability in two steps. First, the error between measured and calculated fractional flow is related to an error in effective permeability along each streamline; this results in a proposed change or perturbation of the streamline effective permeability. Secondly, the perturbation in streamline effective permeability is mapped into a change/perturbation of individual grid cells. The algorithm is summarized as follows: • Choose an initial model for the permeability field ) k u ( at each grid cell ∈ ∀ = u u z) y, (x, . • Start iteration max ,... 1 L l = (a) Run a flow simulator to obtain the saturation and pressure in each grid-block, and water-cut curve for a well of interest. For SL N streamlines, compute streamline geometry and time of flight for each streamline, SL , through every grid cell, u , using the pressure field. (b) Propose a perturbation of effective streamline permeability, m SL k ∆ , for each streamline, following the approach of Dykstra and Parsons,10 that matches oil and water production. (c) Translate the joint perturbation for all streamlines into a perturbation for each grid block (see Appendix A). D ∆k k k (l) l ∈ ∀ + = + u u u u ( ( ) 1 ( • Until the measured production matches the simulated production. The advantages and limitations of this approach are amply discussed by Wang and Kovscek. 6 In this paper, we propose a geostatistical methodology for step (c) that updates the permeability field accounting for the proposed change in streamline effective permeability SL m SL l m SL l m SL N m k k k ..., , 1 , ) ( ) 1 ( = ∆ + = + consistent with the model variogram and histogram of the grid cell permeability u ( k . We recall briefly the results of Wang and Kovscek. 6 The streamline effective permeability is a weighted harmonic average of the single grid cell permeability ) (u k along the streamline: m SL : 10