Development of Low-Order Controllers for High-Order Reservoir Models and Smart Wells

Gildin, Eduardo; Klíe, Héctor; Rodríguez, Adrián; Wheeler, Mary F.

doi:10.2118/102214-ms

Cited by 13 publications

(8 citation statements)

References 14 publications

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“…2, decrease very rapidly. This is in line with earlier results from [15,19,28] and means that the 441 th order reservoir model behaves like a model of much lower order.…”

Section: Example 1: Homogeneous Permeabilitysupporting

confidence: 92%

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Controllability, observability and identifiability in single-phase porous media flow

et al. 2008

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Over the past few years, more and more systems and control concepts have been applied in reservoir engineering, such as optimal control, Kalman filtering, and model reduction. The success of these applications is determined by the controllability, observability, and identifiability properties of the reservoir at hand. The first contribution of this paper is to analyze and interpret the controllability and observability of single-phase flow reservoir models and to investigate how these are affected by well locations, heterogeneity, and fluid properties. The second contribution of this paper is to show how to compute an upper bound on the number of identifiable parameters when history matching production data and to present a new method to regularize the history matching problem using a reservoir's controllability and observability properties. Keywords

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“…2, decrease very rapidly. This is in line with earlier results from [15,19,28] and means that the 441 th order reservoir model behaves like a model of much lower order.…”

Section: Example 1: Homogeneous Permeabilitysupporting

confidence: 92%

“…In the following sections, this type of model reduction is not actually applied to reservoir models (as done in [15,19,28]). We merely point out that we can analyze when a high-order model in fact behaves like a loworder one.…”

Section: Balancing and Truncationmentioning

confidence: 99%

Controllability, observability and identifiability in single-phase porous media flow

et al. 2008

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“…[2,33]. For recent applications of system theory to reservoir modeling see [6,14,19,24,25,34,37,38,43].…”

Section: System-theoretical Conceptsmentioning

confidence: 99%

A system-theoretical approach to selective grid coarsening of reservoir models

Vakili-Ghahani

2011

Comput Geosci

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From a system-theoretical point of view and for a given configuration of wells, there are only a limited number of degrees of freedom in the inputoutput dynamics of a reservoir system. This means that a large number of combinations of the state variables (pressure and saturation values) are not actually controllable and observable from the wells, and accordingly, they are not affecting the input-output behavior of the system. In an earlier publication, we therefore proposed a control-relevant upscaling methodology that uniformly coarsens the reservoir. Here, we present a control-relevant selective (i.e. non-uniform) coarsening (CRSC) method, in which the criterion for grid size adaptation is based on ranking the grid block contributions to the controllability and observability of the reservoir system. This multi-level CRSC method is attractive for use in iterative procedures such as computer-assisted flooding optimization for a given configuration of wells. In contrast to conventional flowbased coarsening techniques our method is independent of the specific flow rates or pressures imposed at the wells. Moreover the system-theoretical norms employed in our method provide tight upper bounds to the 'input-output energy' of the fine and coarse systems. These can be used as an a priori error-estimate of the performance of the coarse model. We applied our algorithm to two numerical examples and found that it can accurately reproduce results from the corresponding fine-scale simulations, while significantly speeding up the simulation.

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“…System-theoretical model reduction techniques such as proper orthogonal decomposition (POD) appear to provide another helpful tool to reduce the complexity of a large-scale model (Heijn et al, 2004;Antoulas 2005;Gildin 2006;Markovinović 2002 and. In these methods, we use the spatial correlation in the states (pressures and saturations) to compute a limited number of spatial patterns (directions) in the state-space coordinates, which can be used to characterize the dominant dynamical variations of the system.…”

Section: Approachmentioning

confidence: 99%

Control-Relevant Upscaling

Ghahani

2008

Europec/Eage Conference and Exhibition

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The conventional reason for upscaling in reservoir simulation is the computational limit of the simulator. However, we argue that, from a system-theoretical point of view, a more fundamental reason is that there is only a limited amount of information (output) that can be observed from production data, while there is also a limited amount of control (input) that can be exercised by adjusting the well parameters; in other words, the input-output behavior is usually of much lower dynamical order than the number of grid blocks in the model. Therefore, we propose an upscaling approach to find a coarse model that optimally describes the input-output behavior of a reservoir system. In this control-relevant method, the coarse-scale model parameters are calculated as the solution of an optimization problem that minimizes the distance between the input-output behavior of the fine- and coarse-scale models. This distance is measured with the aid of the Hankel- or energy norms, where Hankel singular values are used as a measure of the combined controllability and observability of the system. The method is particularly attractive to scale up simulation models in flooding optimization and/or history matching studies for a given configuration of wells. An advantage of our upscaling method is that it intrinsically relies most heavily on those parameter values that directly influence the input-output behavior. It is a global method, in the sense that it relies on the system properties of the entire reservoir. It does not, however, require any forward simulation, neither of the full nor of the upscaled model. it also does not depends on a particular control strategy, but instead uses the dynamical system equations directly. A drawback, however, is its dependency on well positions, which implies that it should be (partially) repeated when those positions are changed. We tested the method on several examples and for nearly all cases obtained coarse scale models with a superior input-output behavior compared to common upscaling algorithms. Introduction Simulation of flow in porous media is an important tool to predict and optimize reservoir performance. As a precursor to the simulation, reservoir data are collected from different sources with various temporal and spatial scales, which are usually integrated into detailed geological models with a large number of cells (107 to 108). The large size of geo-models makes it often impossible to undertake reasonable multiphase flow simulations, sensitivity analysis, uncertainty assessment of a large number of realizations, computer-assisted history matching or life-cycle optimization. In order to perform these processes within a practical time frame and/or a reasonable cost, an upscaling procedure is required to transfer the flow and transport processes from the detailed fine-scale model to a lower-order, coarser representation. The approximated coarse-scale model (typically up to 106 cells) should reduce the complexity, while retaining the main properties of the reservoir system. Upscaling background Various upscaling techniques have been developed in both hydrology and petroleum engineering. These methods vary from simple averaging methods on Cartesian cells to sophisticated flow-based techniques on unstructured grids. Extensive reviews on upscaling were written by e.g. Wen et al. (1996), Renard and Marsily (1997) and Durlofsky (2005). Upscaling methods are classified based on the type of the parameters that are scaled up. In single-phase upscaling, we only consider the pressure (flow) equation and calculate equivalent permeability and porosity values, whereas in two-phase upscaling both pressure and transport equations are considered. Therefore, in addition to the single-phase parameters, equivalent relative permeability and capillary pressure curves are calculated. In principle, upscaling includes both parameters and equations. However, in most single-phase upscaling procedures, coarse-scale equations are chosen to have the same mathematical form as the fine-scale equations.

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Development of Low-Order Controllers for High-Order Reservoir Models and Smart Wells

Cited by 13 publications

References 14 publications

Controllability, observability and identifiability in single-phase porous media flow

Controllability, observability and identifiability in single-phase porous media flow

A system-theoretical approach to selective grid coarsening of reservoir models

Control-Relevant Upscaling

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