Abstract

Today different methods of calculation and software packages to design well activities properly are at the disposal of petroleum engineers. This does not mean that decision-making in the oil and gas industry has turned into a routine. The most common problem petroleum engineers face is that reservoir parameters used in these calculations are unknown or unreliable. Thus, very often petroleum engineers have to make decisions under uncertainty.

In the past this problem was ignored, but today petroleum engineers increasingly consider uncertainty and risk analysis. From the viewpoint of data quality there are three criteria for making decisions [1]:

• Under fully deterministic conditions, when all data are known and complete;

• Under risky conditions, when random data can be described in terms of probability theory; the main criterion is the expectation value of the parameter that determines the quality of the decision;

• Under conditions of uncertainty, when it is difficult or impossible to classify data by their degree of significance and when it is impossible to apply probability theory methods, as distribution functions or other statistical parameters of these values are unknown.

This specification of terms is relevant as confusion of terms often causes ambiguity. Sometimes uncertainty analysis is conducted by the Monte-Carlo method applying series of random values of some parameters in compliance with given functions of probability distribution. It is obvious that the problem is reduced to risky conditions. In reality, as a rule, distribution functions are unknown, so we have the third type of conditions (under uncertainty) and the Monte Carlo method is not applicable.

In this case the decision-making problems are usually defined in terms of game theory, representing them as a "game with Nature". [1,2]. This paper considers different game criteria, which can be rather useful tools to increase the efficiency of decisions when managing oil and gas production processes under conditions of uncertainty. To make it more illustrative, a specific example of hydraulic fracturing design is used in the subsequent text.

Payoff Matrix

In game approach, the analysis of available possibilities is connected with the help of the so-called payoff matrix A, the columns of which (j=1,2,. . ., n) correspond to the possible states of Nature, and lines (i=1,2,. . .,m) correspond to the possible strategies of the Decision Maker (DM). The matrix cell Aij, which is located at the intersection of i-th line and j-th column, defines the profit, resulting from realization of strategy i, when Nature is in state j.

For example, let us assume that DM has a task to define an optimum amount of proppant M necessary to make a frac job. The amount of proppant (for fixed proppant and formation properties) defines optimal fracture geometry [3]. But the efficiency of fractured well depends greatly upon reservoir permeability k. The exact permeability value usually is unknown; DM knows only the range of permeability variation (from 5 mD to 14 mD). In this case the state of Nature is interpreted as different permeability values; player's (DM) strategies are interpreted as different values of proppant amount pumped during treatment. The rest hydraulic fracturing parameters (type of proppant, types of hydraulic fracturing liquids etc.) remain almost the same after performing hydraulic fracturing in first wells in the current region.

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