# SECTION II.III: Mathematical Approaches

INTRODUCTION: SESSION ll.lll

Chapters 11 and 12 share one common identity. These two chapters are classified as mathematical tools to evaluate economic profitability of oil projects. They involve mathematical manipulations in order to arrive to an economic decision, that is why are grouped as “mathematical approach” as shown in the graph.

Fundamentals of economic engineering economic analysis is presented next

## Risk and Decision Analysis in Oil Operations

11.1 INTRODUCTION

The ability to evaluate critical risk factors for oil and gas projects is crucial to optimizing outcomes and planning for effective and cost-efficient risk mitigation programs. Substantial investments are required for gas and oil exploration and production projects.

This chapter highlights methods, tools, and techniques used to study risk, uncertainty, and decision analysis for oil operations. It does not give details to any of these methods.

Some types of risks are briefly described in Figure 11.1.

Why oil projects? The oil and gas industry invests significant money and other resources in projects with highly uncertain outcomes. Risks and uncertainties are everywhere in many oil projects. Risk and uncertainty impact decision-making by the projects chosen. In the first place how these projects are developed, and what is their economic performance. In simple words, a risk assessment is a careful examination of anything that may cause harm during the course of a given work. Once this is done, you will then be able to decide upon the most appropriate action to take to minimize the likelihood of expected hurt by someone.

On the other hand, decision-making is defined as the selection of a course of action from among alternatives or the selection of a logical choice from available options. Risk is a prevalent issue in investment decisions because it could not be avoided, but can be managed. Investment without risk element might not be a worth-while investment because overcoming risk could launch the business into unprecedented success.

Some important applications involving risk and economic analysis in oil operations may include: [1]

FIGURE 11.1 Representation of risks encountered by the oil industry.

11.2 THEORY AND METHODS

In the study of risk analysis and uncertainties in this field, probabilistic concepts and tools are commonly used to describe projects under risk and uncertainty, the following methods are used:

• • Monte Carlo Simulation
• • Decision Trees
• • Commercial Software
• • Engineering Economy
• • Economic Indicators
• • Database

Risk and decision analysis software is as diverse as the analysis methods themselves. There are programs to do Monte Carlo simulation and decision tree analysis. Analytic models to do economics can be linked to both Monte Carlo simulation and decision trees. Monte Carlo simulation (also known as the Monte Carlo method) lets you see all the possible outcomes of your decisions and assess the impact of risk, allowing for better decision-making under uncertainty.

A decision situation is called a decision under risk when the decision maker considers several states of nature, and the probabilities of their occurrence are explicitly stated. A decision situation where several states are possible and sufficient information is not available to assign probability values to their occurrence is termed a decision under uncertainty.

Decision situations can be classified as follows:

• • Decision-making under certainty, where complete information is assumed or available
• • Decision-making under risk, where partial information is known
• • Decision-making under uncertainty, where limited information is available

Because of space limitation in this volume, detailed description of the methods is not included.

11.3 APPLICATIONS

CASE 1

Suppose an oil company w'ould like to assign three drilling rigs to drill oil wells at three different stratigraphic locations in a manner that will minimize total drilling time. The drilling times in days are presented in Table 11.1

TABLE 11.1

Drilling Times in Days

 Drilling Times in Days for Three Different Oil Wells Well Number 1 2 3 Rig Number A 30 70 40 В 40 60 60 C 30 80 50|

This example illustrates the concept of complete enumeration. This means examining every payoff, one at a time, comparing the payoffs to each other, and discarding inferior solutions. The process continues until all payoffs are examined. By complete enumeration as shown in the solution given in Table 11.2, all the alternatives are listed. It is clear that alternative number 5 is the best choice since the total drilling time is the minimum.

TABLE 11.2

Complete Enumeration Solution

 Alternative Assignment Total Drilling Time 1 A-l, B-2, C-3 30 + 60 + 50 = 140 2 A-l, B-3, C-2 30 + 60 + 80 = 170 3 A-2, B-l, C-3 70 + 40 + 50 = 160 4 A-2, B-3, C-l 70 + 60 + 30 = 160 5 A-3, B-2, C-l 40 + 60 + 30 = 130 <- 6 A-3, B-l, C-2 40 + 40 + 80 = 160

CASE 2

Consider an investment of a company, engaged in oil field services, of \$10,000 over a 4-year period that returns Rt at the end of year t, with Rt being a statistically independent random variable. The following probability distribution is assumed for Rt.

 R, Probability \$2,000 0.10 \$3,000 0.20 \$4,000 0.30 \$5,000 0.40

SOLUTION

The expected value of the return in a given year is given by: The variance of an annual return is determined as follows:

CASE 3

An oil firm has four alternatives from which one is to be selected. The probability distributions describing the likelihood of occurrence of the present worth of cash flow amounts, expected values, and variance for each alternative are given in Table 11.3.

TABLE 11.3

Comparison of Four Alternatives of an Oil Firm

 Alternatives Present Worth of Cash Flow (\$1,000) EV Var -\$40 10 60 110 160 A1 0.2 0.2 0.2 0.2 0.2 60 5 * 109 A2 0.1 0.2 0.4 0.2 0.1 60 3*109 A3 0.0 0.4 0.3 0.2 0.1 60 2.5 * 10° <- A4 0.1 0.2 0.3 0.3 0.1 65 3.85 * 10е

SOLUTION

For any given alternative, the decision maker wishes to maximize the expected value and at the same time to minimize the variance of the present worth of the cash flow. If equal weights to the expected value and variance are given, then the values of the expected value-variance criterion will be as computed in the last column of Table 11.3.

Based on the expected value-variance criterion, alternative A3 should be selected.

CASE 4

An oil-drilling company is considering bidding on a \$110 million contract for drilling oil wells. The company estimates that it has a 60% chance of winning the contract at this bid. If the company wins the contract, it will have three alternatives: (1) to drill the oil wells using the company’s existing facilities, (2) to drill the oil wells using new' facilities, and (3) to subcontract the drilling to a number of smaller companies. The results from these alternatives are given as follows:

 Outcomes Probability Profit (\$ million) 1. Using existing facilities: Success 0.30 600 Moderate 0.60 300 Failure 0.10 -100 2. Using new facilities: Success 0.S0 300 Moderate 0.30 200 Failure 0.20 -40 3. Subcontract: Moderate 1.00 250

The cost of preparing the contract proposal is \$2 million. If the company does not make a bid, it will invest in an alternative venture with a guaranteed profit of \$30 million. Construct a sequential decision tree for this decision situation and determine if the company should make a bid.

FIGURE 11.2 Starting the TreePlan program and select a tree.

FIGURE 11.3 TreePlan dialogue boxes.

SOLUTION

The oil company should make the bid because this will result in an expected payoff value of \$143.2 million. The problem is solved using an academic version of Microsoft Excel Add-in. TreePlan Software. To construct a decision tree with TreePlan, go to Tools menu and choose Decision Tree, which brings up the TreePlan as shown in Figure 11.2.

The dialogue boxes used by TreePlan for constructing a decision tree are shown in Figure 11.3. The dialogue boxes enable us to adding decision nodes, state of nature nodes, decision alternative branches, state of nature branches, probabilities, payoffs, and all other tree parameters.

• [1] Reserve Estimate • Exploration and Production (E&P) • Recovery Factors • Expected Production Profile