MATHEMATICS

Oil Production Mathematics



Oil Production Mathematics

It is estimated the cost of biodiesel production based on assumptions, made by their authors, regarding production volume, feedstock, and chemical technology. There could be great value, however, in having a flexible model that allows the user to make changes in these variables and examine the impact of such changes on product cost. We can simplify this analysis by representing the production profile by three phases:

Build up: The period when you drill wells to gain enough production to fill the facilities.

Plateau: After reaching the desired production rate (plateau), the period when you continue. Production at that rate as long as the reservoir pressure is constant and until you produces a certain fraction of the reserves. In the early stages of development, you can only estimate this fraction, and production above certain rate influences plateau duration. Decline:

The period when production rates, P, decline by the same proportion in each time step, leading to an exponential function: P (t) = P (0) exp (-c*t), where t is the time since the plateau phase began and c is some constant.

With only estimates for the total Stock Tank Oil Initially in Place (STOIIP = reserve size) and percent recovery amounts, the objective is to select a production rate, a facility size, and well numbers to maximize some financial measure. In this example, the measure used is the P10 of the NPV distribution. In other words, the oil company wants to optimize an NPV value which they are 90% confident of achieving or exceeding. As described, the problem is neither trivial nor overly complex. A high plateau rate doesn't lose any reserves, but it does increase costs with extra wells and larger facilities. However, facility costs per unit decrease with a larger throughput, so choosing the largest allowed rate and selecting a facility and number of wells to match might be appropriate.

Crystal Ball enhances your Excel model by allowing you to create probability distributions that describe the uncertainty surrounding specific input variables. This model includes five probability distributions, referred to in Crystal Ball as "assumptions." These five assumptions describe the uncertainty around the STOIIP, Recovery, Well rate, Discount factor, and well cost input variables. Each assumption cell is colored green and is marked by an Excel note (mouse over the cell to view the note). To view the details of an assumption, highlight the cell and either select Define Assumption from the Define menu or click on the Define Assumption button on the Crystal Ball toolbar.

This model also includes a Crystal Ball forecast, shown in light blue. Forecasts are equations, or outputs, that you want to analyze after a simulation. During a simulation, Crystal Ball saves the values in the forecast cells and displays them in a forecast chart, which is a histogram of the simulated values. In this example, you want to analyze the Net Present value (NPV). To view a forecast with Crystal Ball, highlight the cell and either select Define Forecast from the Define menu or click ...