Reduced Order Models for Production Optimization

M.A. Cardoso and L. Durlofsky

Progress in seismic and well log tools over the last decade resulted in exceptionally detailed geological models and, as a consequence, more complex reservoir flow models (with an elevated number of grid blocks, typically in the order of tens of thousands to millions).

Maximizing the hydrocarbon production or the Net Present Value (NPV) of a reservoir in a Smart Field can be accomplished by applying Optimal Control Theory. This usually requires many simulations of the reservoir model. The time needed to calculate optimized controls increases with the number of controls and with the complexity of the reservoir model.

Model order reduction (MOR) techniques can be used to overcome this issue. MOR generates simple models based on a detailed model of the system under study. A reduced order model

• uses a considerably smaller number of variables or states than the original model to describe a particular system,
• is relatively inexpensive to simulate, when compared to the original model.
  
  

Proper Orthogonal Decomposition (POD) (also known as Karhunen-Loeve (KL) expansion, Principal Component Analysis (PCA), empirical eigenfunctions, and empirical orthogonal eigenfunctions) is one of the most popular model reduction techniques for large scale models.

Figure 1. Relative energy distribution within the basis functions.

By POD the user can obtain a reduced order model tailored to the accuracy required. A POD analysis starts by taking snapshots of state variables data (pressure and saturation) for all grid blocks. These snapshots are generated by evaluating the computational solution (detailed reservoir model) at several times. Then, the main dynamics are extracted and approximated by a subset of optimal (orthonormal) basis functions that can capture these dynamics within the precision needed. In general, the number of states used in the reduced order model is very small when compared to those in the original reservoir model (see Figure 1).

For complicated dynamic systems a large number of snapshots may be required and this is translated in a prohibitive computing time needed for determining the basis functions. Therefore, to reduce this computational complexity, a clustering technique that reduces a large snapshot set to a much smaller one is applied (see Figure 2).

Figure 2: Snapshot clustering technique.

For practical nonlinear systems, a fairly accurate POD-based reduced order model is often only two/three times faster than the original model (the reduced order model is also based on a nonlinear system of partial differential equations, but the linearized system, though smaller, is not sparse anymore).

To improve the computational efficiency of the reduced order model, the method of Missing Point Estimation (MPE) is considered. With that we can further reduce the number of states that describe the system. The MPE method is based on the estimation of the POD coefficients from a selected number of grid blocks in the spatial domain. The criteria for this selection are based on algebraic considerations (conditioning of associated matrices) that involve well locations, boundary conditions and other relevant blocks.

The results so far demonstrate the applicability of the POD-based model order reduction approach. In the near future it will be implemented into the Stanford’s General Purpose Research Simulator (GPRS).

Water Flood Management and Optimization

D. Echeverría Ciaurri and M. Thiele

With 70% of oil production coming from fields that are 30 years or older, the optimal management of reservoir floods, and of water floods in particular, is a specially important topic. Despite many advances in reservoir simulation and management, reservoir engineers still struggle in setting optimal injection/production target rates for such fields.

Figure 1. Typical streamlines view of a field.

Streamline-based flow simulation (see Figure 1) is generally associated with fast simulations. Therefore, they are a good proxy (surrogate) to use in any workflow that requires many simulations. An overlooked factor of streamline-based simulation is the novel data that is generated. Examples of this are the identification of injector-producer pairs and the quantification of the reservoir volumes associated with these pairs. Armed with this information it is possible to promote the more efficient injector/producer pairs and demote the less efficient ones in a manner computationally very efficient. This can lead to a more successful field management strategy.

In this project, we propose to verify how well the streamline-based heuristic flood management approach presented in Thiele and Batycky (2006) compares to a solution obtained using a more rigorously framed optimization scheme. The project will be helpful in shedding light on:

• the quality of the flood management solution given by the streamline-based method; a direct comparison of approaches should in turn reveal a number of hybrid efficient methodologies for the management of water floods (for example, a good initial guess for a gradient-based optimization algorithm might be quickly obtained by streamlines);
• delineate situations where streamlines can/cannot be used for flood management optimization;
• investigate if streamline specific information can be directly incorporated in the more rigorous flood optimization and management problem formulation, in the hope of improving convergence and solution accuracy.

Initially, the same problem presented in Thiele and Batycky (2006) will be used. This is a small but real reservoir with nine producers and eight injectors (see Figure 2) with a grid of size 80x81x20 (129,600 grid points). The goal is to improve (optimize) sweep over a period of five years by setting target rates every three months on both producers and injectors (this yields a total of 340 optimization variables). Additionally, the problem can be constrained to total available injection rate, for example. After this first test, the reservoir considered can be larger and more realistic.

The project aims at a formal study of a streamline-based optimization methodology and assessment of its use in water flood management scenarios, with several applications of this methodology in cases of practical interest.

References

Thiele, M.R., and Batycky, R.P., Using Streamline-Derived Injection Efficiencies for Improved WaterFlood Management, SPE Reservoir Evaluation & Engineering (SPEREE), Vol 9, No 2, 187-196, April 2006 (SPE84080-PA).

Optimization of Carbon Dioxide Sequestration in Deep Brine Aquifers

D. Cameron and L. Durlofsky

Capturing the carbon dioxide produced in power plants and storing it in geological structures has been suggested as a way to reduce anthropogenic carbon emissions. Our research follows this approach; the injection of CO2 in deep brine aquifers is treated as an optimal control problem where some injection parameters are the controls, and cost is minimized while safety is maximized.

Figure 1. The synthetic reservoir used in the problem.

We begin by solving the simplest problem: controlling the injection rates from multiple injectors, keeping the total injected pore volume constant and maximizing the amount of CO2 that is residually captured in the reservoir (the safest of the trapping mechanisms known). The problem is solved using synthetic reservoir data, in a heterogeneous reservoir with three injectors and the structure shown in Figure 1.

Figure 2. Objective function to maximize in the problem.

Due to the non-linearity and discontinuous nature of this heterogeneous reservoir, we expect the presence of multiple minima/maxima as well as difficulties when applying gradient-based optimizers. Figure 2 is a granular plot of the objective function to maximize in our simple case with two primary control variables (there are three injectors in the problem and the total injected pore volume has to be constant). Therefore we aim to investigate global and derivative-free optimization schemes (e.g., genetic and direct search methods) as well as algorithms that hybridize these two strategies.

It is our intent to eventually provide a robust algorithm for optimizing both safety and cost in real reservoirs where the geological data is uncertain. In such an algorithm a history matching component would be included via a feedback loop.

Joint Quantification of Uncertainty on Spatial and Non-Spatial Reservoir Parameters

C. Scheidt and J. Caers

Uncertainty in reservoir performance is often very significant due to the small amount of data available to describe the reservoir. Reservoirs are modeled using a combination of spatial and non-spatial parameters. Spatial parameters describe properties that are correlated spatially, such as facies type, and are often modeled using geostatistical methods. Non-spatial parameters describe phenomena that do not vary spatially such as bubble point pressure of the oil. Uncertainty exists in both type of parameters, and due to their distinct nature, the quantification of uncertainty of these parameters is done using differing and often incompatible approaches.

The experimental design methodology is widely used to quantify uncertainty on non-spatial parameters. However, it is not well adapted for the case of geostatistical (spatial) uncertainty, due to the discrete nature of many input parameters as well as the potential nonlinear response with respect to those parameters. One way to handle this type of uncertainty is called the joint modeling method (JMM). This method incorporates both non-spatial and spatial parameters within an experimental design framework. The method consists of the construction of two proxy models, a mean model which accounts for the non-spatial parameters and a dispersion model which accounts for the spatial uncertainty. Classical Monte-Carlo simulation is then applied to obtain the probability density and quantiles of the response of interest (for example the cumulative oil production). Figure 1 describes the JMM workflow.

Figure 1. Workflow of the Joint Modeling Methodology.

In this example, the porosity and permeability of the channels are treated as constant values but uncertain. The channel geometry can have 4 configurations. A total of 36 simulations are performed to create a mean model and a dispersion model. The dispersion model is used to capture the variability in the response due to the 4 channel realizations. Monte-Carlo sampling is then performed.

Another method to quantify spatial uncertainty is the distance kernel method (DKM) proposed by Scheidt and Caers (2007), which defines a realization-based model of uncertainty. Based on a distance measure between realizations, the methodology uses kernel methods to select a small subset of representative realizations which have the same characteristics as the entire set. Flow simulations are then run on the subset, allowing for an efficient and accurate quantification of uncertainty.

Figure 2. Proposed workflow for uncertainty quantification: (a) distance between two models, (b) distance matrix, (c) models mapped in Euclidean space, (d) feature space, (e) pre-image construction, (f) P10, P50, P90 estimation.

Both methods are applied to a synthetic channelized case which has spatial uncertainty on the channel representation of the facies, and non-spatial uncertainties on the channel permeability, porosity, and connate water saturation. In this case, 2025 realizations are generated to describe the uncertainty. The results show that the DKM provides for a more accurate quantification of uncertainty compared to JMM, reducing clearly the number of simulations required to have an accurate estimation of the densities and quantiles of the production. Performance of the JMM suffers due to the lack of an efficient method to sample to spatial uncertainty. We propose an improvement to the JMM which combines aspects of both approaches. The DKM selects a good sample of realizations to employ in the JMM, instead of choosing them arbitrarily. Results show that the accuracy of the JMM can be significantly improved.

The efficiency of the DKM is dependent upon an accurate measure of distance which must be correlated with the production response(s) of interest. In the case presented here, the DKM method requires flow-based distance measure to capture uncertainty in changes in permeability, porosity and connate water saturation. Employing static measures may not be as accurate.

References

Scheidt, C. and Caers, J, A workflow for Spatial Uncertainty Quantification using Distances and Kernels, SCRF report 20, Stanford University.

Production Optimization Using Gradient-Free Methods

O. Isebor and L. Durlofsky

The determination of optimum well controls represents a challenging optimization problem which is an integral component of the more general closed-loop approach to reservoir management. A very efficient technique that has been used to address this problem is the adjoint method, which provides gradients of the objective function (cumulative oil production, net present value, etc.) with respect to the well controls. Having these gradients, an appropriate gradient-based optimization algorithm can be used to maximize the objective function. A limitation of this technique is that it is difficult to implement and requires linkage to the simulator at the level of source code for the calculation of the gradients. This can be very restrictive if using commercial simulators, or if an entire workflow is involved in the computations and one needs to obtain gradients from multiple applications. In addition, the adjoint method converges to local minima. For these reasons, it is useful to investigate alternative methods that treat the simulator as a black box and are also easy to implement.

Figure 1. Synthetic 2D reservoir used to assess performance of methods.

This work involves a study of some gradient-free alternatives to the adjoint method for the purpose of solving the production optimization problem. The optimization methods considered thus far are genetic algorithms (GAs), general pattern search (GPS), and mesh adaptive direct search (MADS). The performance of these methods, together with the adjoint method, is compared for a synthetic reservoir under waterflood (See Figures 1 and 2).

Figure 2. Comparison of evolution of NPV for the different methods.

As expected, the adjoint method significantly outperforms the gradient-free procedures, typically achieving better optimum solutions in hundreds of simulations as compared to thousands of simulations for the gradient-free methods. However, it must be kept in mind that the gradient-free methods can be easily parallelized and require much less effort to implement.

In addition to comparing different algorithms, we also investigate using proxies to improve the performance of the GA, as well as hybridization methodologies that combine aspects of the GA and adjoint procedures. It has been shown that the use of a neural network proxy with the GA substantially improves its performance, and that the hybridization of the GA and adjoint methods represents a viable means for providing different initial guesses for the adjoint procedure. Current work is geared towards improving the performance of the above mentioned gradient-free methods, as well as looking into other methods.

History Matching Using Integrated Data With an Application to a North Sea Field

D. Echeverria, E.T.F. Santos, M. Dadashpour, J. Kleppe, M. Landrø and T. Mukerji

This research aims at making optimal updates of geological models by jointly using flow simulation and tomographic inversion, while honoring the geologic spatial continuity, in closed-loop reservoir management.

Initial reservoir models are generally based on 3D seismic characterization and log data. To improve these original estimations we integrate different types of instrumented field data: time-lapse seismic and production data. Production data provides an integrated response of the reservoir to fluid flow, while time-lapse seismic data yields a spatially distributed characterization of the changes in elastic velocities due to saturation and pressure variations.

Tomographic inversion uses recorded seismograms and traveltimes to estimate elastic velocities, which are linked through rock physics to porosity and saturations. The high-dimensional spatial random field is represented by a principal component expansion, which allows us to apply optimization techniques while maintaining the geostatistical structure of the reservoir.

Figure 1. a) Continuous facies map view of one layer of the Stanford VI synthetic reservoir. Top left: true model; top right: initial guess of the history matching; bottom left: history matching solution after only production data match; bottom right: history matching solution after optimizing both production data and seismic data match in an alternating scheme.

The approach combines information of different nature: spatially localized and of high periodicity (production data) and spatially distributed and of low periodicity (seismic data). The calibrated models, optimally constrained to both integrated and distributed responses, will give a better description of the reservoir, and consequently, more reliable forecasts.

Figure 2. A discretization of a North Sea Field.

We apply the methodology and analyze the associated results on a sector of the Stanford VI synthetic reservoir (see Figure 1) and, in collaboration with the Integrated Operations (IO) center at the Norwegian University of Science and Technology (NTNU), on a section of a real North Sea field (see Figure 2).

Reduced-order Models for Nonlinear Optimization

M. Rousset and L. Durlofsky

Production of unconventional oil resources, such as oil sands in the province of Alberta, Canada, may contribute significantly to the future energy supply of the planet. It is indeed estimated that several trillions of oil barrels remain under the ground along the Athabasca River only. This extremely viscous oil however, cannot be recovered using traditional production techniques. One approach that has been developed consists in using in-situ electrical heaters in order to upgrade the oil sands directly within the subsurface. Such a technique shall however be carefully designed to remain economically viable and environmentally safe. It appears that building numerical models and applying powerful optimization procedures may be necessary to achieve these goals.

Numerical simulation of the oil sands in-situ upgrading process is however a difficult challenge. Indeed many physical and chemical phenomena must be represented in the model. In addition to the traditional transport of fluids through porous media and to the change of properties with varying pressure, in-situ upgrading (IUP) involves transport of heat and several chemical reactions. The changes versus pressure and temperature are very large. It is also necessary to account for a large number of components in the fluid. These and other complications lead to a large number of variables in the model, together with important nonlinearities. For this reason, the computation of such models is extremely CPU intensive. And although the potential benefits can be important, performing optimization with such models can lead to extremely long calculation times. A more efficient simulation capability is therefore very much needed.

Figure 1. Synthetic 2D reservoir used for thermal simulation.

Motivated by this need, this work proposes to investigate the use of reduced-order models for nonlinear thermal applications. As a first step, the proposed approach is to eliminate most of the complexity of IUP simulation. A single-phase system is used where viscosity and compressibility depend strongly on pressure and temperature, in order to reproduce some of the nonlinearities of the initial problem. A reduced order model is then created using the trajectory piecewise linearization procedure. This relatively new reduced-order modeling technique is based on a piecewise linearization combined with a proper orthogonal decomposition. It has previously been applied successfully on two-phase oil-water problems leading to runtime speed-up factors of two or three orders of magnitude.

The trajectory piecewise linearization (TPWL) procedure requires running a number of full-order training simulations (typically 1 to 4 runs) and saving some data during these runs (states and Jacobian of the system). This data is then used to compute the solution of any new set of well controls in a much faster way. Stanford’s General Purpose Research Simulator (GPRS) is used to run the full-order simulations.

Figure 2. Performance of a TPWL reduced-order model.

Using the bi-dimensional model pictured in Figure 1 with three producer wells (black) and four in-situ heaters (red), the performance of a TPWL reduced model using four training runs is shown in Figure 2.
A comparison with the solution of the full order model clearly shows that oil production rates were reproduced accurately by the reduced-order model. The runtime speed-up factor achieved here is 33. Although significant, the full acceleration potential is certainly not reached with this reservoir model. Indeed, larger speed-up factors are expected with larger and more complex reservoir models.

A Well Pattern Optimization Algorithm for Large-Scale Field Development

J. Onwunalu and L. Durlofsky

To be updated.

Uncertainties in Rock Pore Compressibility and Effects on Seismic History Matching

A. Suman and T. Mukerji

To be updated.

Retrospective Optimization of Well Location under Uncertainty

H. Wang, D. Echeverría Ciaurri and L. Durlofsky

To be updated.

Application of Particle Swarm Methods for Joint Inversion of Production and Time-Lapse Seismic Data

J.L. Fernandez Martinez, D. Echeverría Ciaurri, T. Mukerji, A. Suman and E. Garcia Gonzalo

To be updated.

Data Mining Applied to the Construction of an Empirical Model for Permanent Downhole Gauges using Flowrate and Pressure

Y. Liu and R. Horne

To be updated.

Bayesian Decision Networks for Real-time Drilling Decision Support

M. Rajaieyamchee. D. Echeverria and R. Bratvold

To be updated.

Reservoir Management Optimization based on Approximate Dynamic Programming

Z. Wen, K. Aziz, L. Durlofsky and B. Van Roy

To be updated.

A Streamline-based Network Approach to Production Optimization

D. Shamsi, D. Echeverría, M. Thiele and Y. Ye

To be updated.

Quasi-Continuous Monitoring of Sequestered CO2 in Geologic Reservoirs

Y. Arogunmati and J.M. Harris

To be updated.

Seismic Reservoir Monitoring by Target-Oriented Inversion

G. Ayeni and B. Biondi

To be updated.