Statistical Geometry of Petroleum Reservoirs in Exploration and Exploitation
Authors: Dowds John P. 
Published: December 2007

An English clergyman, the Reverend Thomas Bayes, introduced his theorem of conditional probabilities approximately 200 years ago. It has since become the foundation for solving geologic, engineering, and economic problems by statistical approaches.

Introduction

Management needs to be advised on (1) how often a petroleum reservoir may be tapped, (2) how much oil or gas is in place, and (3) how much it is going to cost to drill a well or an ensemble of wells.

Simply stated, the problem is how often and how much.

People are fascinated by the workings of probability theory witness, for example, a dice game. probability theory witness, for example, a dice game. The classical laws of probabilities and statistics amply analyze the chances taken in such a game; but a dice game is a closed system, so there is not much uncertainty as to the outcome, provided that the game continues for a long enough time. In contrast, because of the uncertainty involved, people in the oil business must measure risks and learn to deal with all of the data within open systems, and decide what alternatives are available and what courses of action must be taken.

Within recent years mathematical theories have been developed or revived in a few academic centers in the U.S. Leaders among them are Kansas U., Pennsylvania State U., U. of Arizona, Colorado Pennsylvania State U., U. of Arizona, Colorado School of Nines, Northwestern U., Dartmouth, Harvard, Stanford and MIT.

As everyone in the industry is aware, petroleum is becoming more difficult to find and produce. There is constant talk about the advent of new tools to explore and exploit petroleum reservoirs. The techniques, models, and case histories presented here are some of those new tools. This breakthrough covers many disciplines that are evolving today and that are being applied toward better, non-descriptive numerical forecasts.

Theory and Definition

The premise is that rocks and their hydrocarbon contents exhibit patterns that can be examined statistically. Rock distributions are not entirely random and haphazard, but are a result of a continuous chain of events that can generate a geometric reservoir body within probabilistic limits. From a single grain to an enormous, complex reservoir host rock, there exists something of a memory as to where one sand grain is located and bow that sand grain influences where the next sand grain comes to rest by the energy that provides the depositional mechanics for the distribution provides the depositional mechanics for the distribution of the lithologies. This is a generate, reversible process. Using sequential analysis and iteration, we process. Using sequential analysis and iteration, we can study the geometry of the reservoir rocks by applying theories that deal with conditional probabilities, i.e., Bayes' theorem, and Markov schemes as prescribed by the Information Theory.

The prime sources of this Information Theory are Shannon and Weaver and Wiener, as well as Pask and Ashby. It is beyond the scope of this paper to deal at length with cybernetics and the statistical communication/information theory, but for a fuller understanding of their application to subsurface geology and engineering, these writers should be studied further.

Bayesian statistics requires two sources of input:

deterministic data or facts that can be observed and measured, and the statistical inference or subjective attitude (judgment) that an individual adds to the system.

JPT

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