**J.A. MacKay**

Rose & Associates

Houston

**Ian Lerche**

Martin Luther University

Halle-Wittenberg, Germany

This article is concerned with showing that the dependence of probable success between two reservoirs has far-reaching consequences for the expected value, volatility, and cumulative probability of anticipated hydrocarbon reserves.

In particular, varying the allowed probability for the dependent reservoir is shown to not influence either the expected project value or the standard deviation around the expected value. The result is that the cumulative probability of obtaining a hydrocarbon reserve in excess of a fixed amount is also not influenced by the dependence probability chosen and behaves as though the two reservoirs were totally independent of each other.

This result indicates that a corporation could spend an inordinate amount of time and money attempting to resolve an irresolvable (or at least noncritical) problem.

#### Introduction

Making quantitative assessments, ahead of drilling, of the likely amount of hydrocarbons one expects to find in a reservoir is one of the mainstays of economic projections for oil exploration, and one that colors to a large extent precisely what a corporation should do to ensure financial health.

Indeed, precisely that sort of effort has been the dominant theme of two volumes^{1 2} treating with such problems. The basic concerns of risk and economic worth have been one of the mainstays of hydrocarbon exploration for over half a century as is ably recorded in many works.^{3 11}

And yet it is to be doubted that all possible configurations of exploration assessment have been investigated to date. Perhaps one of the lacunae in such treatments has been the unresolved issue of what happens with assessments when there is dependence between likely reservoirs in terms of their probabilities of being successful in estimating hydrocarbon reserves.

Note that dependence is different than correlation; when dealing with a correlated assessment of reservoir success chances one is usually in the situation where if the probability is high or low of one reservoir being successful then the high (low) probability of a second reservoir being hydrocarbon bearing is related to the probability of the first reservoir.

In the case of dependence the argument is different. Here one has to deal with some form of constraint so that there is a relationship between the two reservoirs either in terms of a fixed relationship, a so-called holonomic constraint (for instance so that a given number of barrels of oil is divided between the two reservoirs, representing a closed system, or when a finite number of barrels can be lost to the reservoirs) or in terms of a range of values, a so-called nonholonomic constraint (for instance, the probability of the second reservoir being successful (or unsuccessful) given that the first was successful (or unsuccessful) should not exceed a predetermined value).

Note that a nonholonomic probability constraint differs from an holonomic constraint because one refers to the probability of success and the other refers to the amount likely to be found. Thus if one reservoir is likely to contain a large hydrocarbon accumulation then one can either take it that a second reservoir will likely also contain a large supply or that there is a high chance of success with the second reservoir irrespective of the amount to be found.

Such dependence arguments have also had a long history in the hydrocarbon exploration arena and modern interpretations, with many references to earlier works, can be found in Rose,^{10} Bickel and Smith,^{12} Lerche, and Mackay,^{2} and Bickel, Smith, and Meyer.^{13}

In any event the quantitative evaluation of a dependency has far-reaching repercussions for attempts to assess the exploration potential. It is this fact that the present article will illustrate with a simple example.

More complex dependency situations than that to be discussed in the next section of the article will, presumably, just enhance the unexpected patterns of behavior uncovered with the simple illustration.

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