(740f) Reservoir Optimization with Model Parameter Uncertainty Updates

As energy resources become more and more precious, it is important to employ optimal reservoir management practices. The unusual geologic structure of each reservoir not only leads to the complexity of developing a first-principle physics-based reservoir model but also results in the existence of uncertain parameters (e.g. porosity and permeability). Reservoir simulation software offers the promise of a complete and robust set of numerical solutions that predicts reservoir behaviors. Additionally, the output from this simulation can be used to identify a range (macro and micro) of models that can be used in a variety of applications including online monitoring, state estimation, and optimal feedback control. The framework of a real-time or online closed-loop management approach that combines optimal control with effective model parameter updates, first introduced by [1] (see Figure 1), will be further investigated to maximize cumulative oil production based upon a particular low-order model of these complex reservoirs. The Eclipse© software (Schlumberger, Houston, Texas) version 2009.1 is used in this study to provide simulated data of different types of reservoir models. The data produced by Eclipse will be used to identify a low-order model.

Figure 1: Schematic of the closed-loop optimal management framework. D(n): disturbance, u(n): system input, y_m(n): system output, and x(n): state variable.

In water flooding optimization studies, the injected water rates of different wells are regulated in order to maximize oil production. By studying the water injection rates to obtain the oil production rates during the production period, a low-order model of the production process can be identified from the data provided by the Eclipse model as shown in Figure 1,

where y is a vector of oil and water production rates; u is a vector of water injection rates; A and B are coefficients for y and u, respectively; n is the control step index and k is the time delay. Disturbances are not considered at the present time.

Water production is not a goal, thus the optimal controller should maximize oil production and minimize water production. The outputs of the optimal predictive controller are injected water flow rates. In the case of the closed-loop optimal management framework, model parameter updating is critical to make sure that the model's prediction is accurate. A method to carry out this updating is to use a Markov chain Monte Carlo method (MCMC) [2]. This method is based on the error (between the actual water saturation (S_w) data and the simulated data) to calculate the likelihood function and the acceptance probability (α) to determine if the uncertain parameter values generated by the Monte Carlo sampling is accepted. From those accepted uncertain parameters we can determine their distributions. Uncertain parameters such as porosity, ϕ, and permeability, K, for example, can be quantified and updated.

The MCMC method generates parameter values from a constructed Markov chain which converges to a stationary distribution. A type of MCMC method is an adaptive Metropolis (AM) algorithm that generates the uncertain parameters in a single iteration. The AM algorithm is based on a proposed distribution of the parameters. Start at iteration i = 0 with an arbitrarily chosen initial vector of parameters Θ = Θ⁰. Consider a vector Θ that has two parameters: ϕ and K. A candidate value Θ^* is generated from a proposed density based on the current value Θⁱ of Θ. The acceptance probability, α, is computed as a function of Θⁱ, Θ^* and the model. If Θ^* is accepted with acceptance probability

it then follows that Θⁱ⁺¹ = Θ^*, otherwise Θⁱ⁺¹ = Θⁱ. Here, P(X|Θ) is the likelihood function of the observed data X and P(Θ) is the prior distribution of Θ.

The likelihood function is a multi-normal joint probability density function of the data (model state variable values and observed data),

where N_X: the number of data points, X(z): observed datum at location z. M(G, z; Θ): water saturation values provided by the model at location z, G:  set of model inputs describing the domain geometry and boundary conditions and Θ: uncertain model parameters to be estimated from the data. The error term is given by ε(z) = X(z) - M(G, z; Θ) and σ_ε² is its variance.

In each MCMC iteration, the generated Θ (ϕ and K values) is used in the model to estimate S_w. The model results are compared to observed data (X in Equation (3)). The error statistics for the likelihood function are then computed. In order to reach a stationary posterior distribution of the uncertain parameters, the MCMC method usually requires the generation of tens of thousands of values of Θ with probability α. This implies that the model must be executed tens of thousands of times to generate corresponding values of S_w for each ϕ and K combination.

An approach to reducing the computational burden is to apply partial least squares (PLS) regression to find the relation between the uncertain parameters and a variable of interest. If this relation is known then execution of the Eclipse model is supplanted because the predicted variable can be estimated from the values of the uncertain parameters.

Partial least square regression finds a low-order projection space from the input data χ that is able to predict the output data Y. Without loss of generality, consider uncertain parameters ϕ, K and S_w. To establish the relation between the parameters and the outputs, different combinations of ϕ and K are inputs to the model to generate S_w.

The correlation between χ and Y is to be established. If the true range of χ is wide but the range of the data values is narrow, then the identified correlation will have limited predictability. Thus, we need combinations of ϕ and K that cover the actual operating ranges. The Latin hypercube Hammersley sampling technique (LHHS) can satisfy this condition for more than one parameter with less sampling points as demonstrated in [3]. Usually, the reservoir is divided into many grids and each grid has an S_w value. Let the χ matrix be composed of different combinations of ϕ and K and S_w the variable to be predicted. It should be obvious that to predict the value of S_w for a large number of grids from two-parameter (ϕ and K) data is not a trivial exercise. To circumvent this issue, we apply the Karhunen-Loève expansion to represent the dominant characteristic of a large data set with a small number of empirical eigenfunctions (EEFs) extracted from the data themselves,

where M(z, t): element of the matrix M at location z and time t, M̅: mean of M, {ψ} and {λ}: the empirical eigenfunctions and eigenvalues of the covariance of M, respectively and {ς}: coefficients  of the projections of M onto {ψ}. The values of S_w then can be expressed by their EEFs and their corresponding coefficients. Given the set of EEFs, the coefficients can be chosen as the variables for Y matrix [3].

The independent χ and dependent Y variables can be decomposed as

where T: score matrix of χ; P' and Q': loading matrices; and E and F are residual errors for χ and Y, respectively. In this work, χ_new is generated from the MCMC method and ͡Y is predicted from χ_new,

Given the EEFs, the values of S_w can be obtained from Equation (4).

The combination of PLS and the KL method avoids the computational burden of the Eclipse model for MCMC to reach a stationary distribution of uncertain parameters. Thus, update of uncertain parameters in the Eclipse model is efficient.

This presentation will employ a two-dimension reservoir with two-phase flow in a porous media to demonstrate the real-time optimization framework with uncertain parameter updating using a Markov chain Monte Carlo method that combines partial least square regression and Karhunen-Loève expansion method.

References

[1] D. R. Brouwer and J. D. Jansen. Dynamic optimization of water flooding with smart wells using optimal control theory. In Proc. 13th European Petroleum Conference. Society of Petoleum Engineers, 2002. Aberdeen, Scotland, U.K., SPE 78278.

[2] C. Andrieu, N. D. Freitas, A. Doucet, and M. I. Jordan. An introduction to MCMC for machine learning. Machine Learning,, 50:5?43, 2003.

[3] Yingying Chen and K. A. Hoo. Uncertainty propagation for effective reduced-order model generation. Computers and Chemical Engineering(2010), 2010. doi:10.1016/j.compchemeng.2010.02.034.