## 4 Decision Making in Reservoir Engineering While Facing Uncertainties  
In reservoir management problems, the most critical problem is the geological uncertainty. In today's practices of the upstream petroleum engineering technologies, reservoir engineering principles and fundamentals are instrumental in making sound decisions towards the development and production of a hydrocarbon reservoir. Most of the time these decisions need to be made in the presence of limited data. Figure 2: Due to existing uncertainties in deciding on a choice, a high-quality decision can still result in poor outcome and vice versa. Fundamentally, the basic relation for reservoir evaluation for a given process using a deterministic protocol can be expressed as [5]:  
\[q(t)[or\ p(t)]=\Omega(D,\ R) \tag{1}\]  
In Equation (1), the following terms are identified:  
\(q(t)\): Production/injection rate at point \(x,\ y,\ z\) at time \(t\)  
\(p(t)\): Pressure at point \(x,\ y,\ z\) at time t  
\(\Omega(D,\ R)\): Mathematical flow model with the relevant \"built in\" physics and thermodynamics  
\(D\): Flow domain characteristics (e.g., fluid types, spatial and directional dependencies)  
\(R\): Some predefined production/injection mechanism, and/or recovery process including the project design parameters (e.g., well geometry). Equation (1) may appear in front of a reservoir engineer in three different forms. If Equation (1) is used to solve for \(q(t)[or\ p(t)]\) using the available information on \(\Omega(D,\ R)\) as it appears on the right-hand-side, then, this solution is known as \"forward solution\". The accuracy of the forward solution depends on the degree of accuracy of the representation of the flow mechanisms in terms of physics and thermodynamics as well as accurate description of the spatial distribution of the domain characteristics, the design parameters imposed on the reservoir system in terms of boundary and initial conditions as well as the key features of the project implemented. Equation (1) can also be rearranged to solve for \(D\), which is known as \"history matching\", so that a reservoir can be characterized. This inverse form of the solution of Equation (1), as expected, inherently suffers from non-uniqueness. The second inverse solution involves the design of the project parameters. For example, in Equation (1), if a desired \(q(t)\) is specified, then, the same equation can be rearranged to solve for the project design parameters that will effectively generate the expected \(q(t)\) solution. Like any inverse solution, this second inverse solution is also prone to non-uniqueness problems. In solving Equation (1), if the right-hand-side of the equation, \(\Omega(D,\ R)\), is fully known, solving for the left-hand-side is straightforward and such a solution protocol is known as a deterministic process. The main difficulty faced here is how certain one is of the entries on the right-hand-side, namely \(\Omega,\ D\) and \(R\). If \(\Omega,\ D\), and \(R\) are partially known and several scenarios are created for the unknown entries of \(\Omega,\ D\), and \(R\) and the problem is solved for each scenario, then the solution protocol is called stochastic. The \"hybrid modeling\" that will be discussed later incorporates deterministic, stochastic, and proxy modeling approaches in a synergistic manner. There are four main sources of uncertainty in reservoir characterization including quantity and quality of the existing data, geological understanding, scaling-up, and mathematical representations. Unfortunately, out of these four sources of uncertainty, only one of them, the uncertainties originating from mathematical representation, is quantifiable.