This work is based on Zengyi Dou's MS thesis.
The oil well placement problem
The oil well placement problem is vital part of secondary oil production. Since the calculation of the net present value (NPV) of an investment depends on the solution of expensive partial differential equations that require tremendous computational resources, traditional methods are doomed to fail. The problem becomes exceedingly more difficult when we take into account the uncertainties in the oil price as well as in the ground permeability. In this study, we formulate the oil well placement problem as a global optimization problem that depends on the output of a finite volume solver for the twophase immiscible flow (wateroil). Then, we employ the machinery of Bayesian global optimization (BGO) to solve it using a limited simulation budget. BGO uses Gaussian process regression (GPR) to represent our state of knowledge about the objective as captured by a finite number of simulations and adaptively selects novel simulations via the expected improvement (EI) criterion. Finally, we develop an extension of the EI criterion to the case of noisy objectives enabling us to solve the oil well placement problem while taking into account uncertainties in the oil price and the ground permeability. We demonstrate numerically the efficacy of the proposed methods and find valuable computational savings.

Problem specification
The objective of the oil well placement problem is to select the well locations (injection and production) that maximize the NPV of the investment. The NPV of the investment is:
\begin{equation}
f_T(\mathbf{x}) = \int_0^T \left\{\sum_{\text{prod. wells}\;i}\left[c_o(t)q_{o,j}^(\mathbf{x}, t)  c_{w,\text{disp}}q_{w,j}^(\mathbf{x},t)\right]
 \sum_{\text{inj. wells}\;j}c_{w,\text{inj}}q_{w,j}^+(\mathbf{x},t)
\right\}(1 + r)^{t/365}dt,
\label{eq:npv}
\end{equation}
where
\begin{equation}
\mathbf{x}^* = \arg\max_{\mathbf{x}}\mathbb{E}\left[ f_T(\mathbf{x})\right],
\end{equation}
where the expectation is taken over the uncertainty in oil price as well as the uncertainty in the permeability field.
\begin{equation}
f_T(\mathbf{x}) = \int_0^T \left\{\sum_{\text{prod. wells}\;i}\left[c_o(t)q_{o,j}^(\mathbf{x}, t)  c_{w,\text{disp}}q_{w,j}^(\mathbf{x},t)\right]
 \sum_{\text{inj. wells}\;j}c_{w,\text{inj}}q_{w,j}^+(\mathbf{x},t)
\right\}(1 + r)^{t/365}dt,
\label{eq:npv}
\end{equation}
where
 \(\mathbf{x}\) is a design vector representing the location of all wells, e.g., if we consider one injection and one production well, then \(\mathbf{x}\) is fourdimensional.
 \(T\) is the time horizon of the investment.
 The first sum is over all production wells and the second sum over all injection wells.
 \(c_o(t)\) is the uncertain oil price time series, \(c_{w,\text{dis}}\) is the price for disposing contaminated water, and \(c_{w,\text{inj}}\) is the price of injecting water.
 \(q_{o,i}^(\mathbf{x},t)\), \(q_{w,i}^(\mathbf{x},t)\), and \(q_{w,j}^+(\mathbf{x},t)\) are the oil production rate, the water disposal rate, and the water injection rate, respectively, for a choice of design variables \(\mathbf{x}\) at time \(t\). Note that these rates depend on the solution of a coupled systems of partial differential equations that model twophase flow through the ground. They also depend on the uncertain ground permeability.
 \(r\) is a discount factor.
\begin{equation}
\mathbf{x}^* = \arg\max_{\mathbf{x}}\mathbb{E}\left[ f_T(\mathbf{x})\right],
\end{equation}
where the expectation is taken over the uncertainty in oil price as well as the uncertainty in the permeability field.
Quantifying uncertainties
For the oil price we use a lognormal random walk with a constant drift. That is, we assume that:
$$
\log c_o(t+1) = \log c_o(t) + \mu t + \sigma z_t,
$$
where \(\mu\) is the growth rate, \(\sigma\) the volatility, and \(z_t \sim \mathcal{N}(0, 1)\).
For the permeability we use a logGaussian process model. That is, we assume that:
$$
\log K(\mathbf{x}_s) = \log K_0(\mathbf{x}_s) + G(\mathbf{x}_s),
$$
where \(\mathbf{x}_s\) stands for the spatial location, \(K_0(\mathbf{x}_s)\) for a mean permeability profile, and \(G(\cdot)\) is a Gaussian process defined over the reservoir with zero mean and exponential covariance function.
$$
\log c_o(t+1) = \log c_o(t) + \mu t + \sigma z_t,
$$
where \(\mu\) is the growth rate, \(\sigma\) the volatility, and \(z_t \sim \mathcal{N}(0, 1)\).
For the permeability we use a logGaussian process model. That is, we assume that:
$$
\log K(\mathbf{x}_s) = \log K_0(\mathbf{x}_s) + G(\mathbf{x}_s),
$$
where \(\mathbf{x}_s\) stands for the spatial location, \(K_0(\mathbf{x}_s)\) for a mean permeability profile, and \(G(\cdot)\) is a Gaussian process defined over the reservoir with zero mean and exponential covariance function.
Solution strategy 
When these quantities become uncertainty, this uncertainty is propagated to the NPV. The figure on the left demonstrates the broadening of the uncertainty in the NPV that is induced by this fact. Notice that including the permeability uncertainty more than doubles the risk associated with the investment.

We solved the oil well placement problem by employing the Bayesian global optimization (BGO) framework, albeit using a novel extension of the expected improvement that is robust under uncertainty.

The following video shows simulations that BGO picks at each iteration.
The final video shows the evolution of water saturation (water is blue) as a function of time for the best well locations found by our approach.