A Model for Events Occurring at Random Points in Time and an Example Application to Casing Failures in Cedar Creek Anticline Wells
- Dwayne A. Chesnut (Shell Oil Co.) | Bernard Goldberg (Shell Development Co.)
- Document ID
- Society of Petroleum Engineers
- Society of Petroleum Engineers Journal
- Publication Date
- October 1974
- Document Type
- Journal Paper
- 482 - 490
- 1974. Society of Petroleum Engineers
- 4.2.3 Materials and Corrosion, 5.7.5 Economic Evaluations, 2.4.3 Sand/Solids Control, 1.6 Drilling Operations, 4.1.2 Separation and Treating
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A familiar example of the "Poisson Pure Death Process" is the phenomenon of radioactive decay. Process" is the phenomenon of radioactive decay. It contains a single parameter, lambda which assumes specific values for each distinct unstable nucleus. Rather surprisingly, data on casing failures caused by salt-flow loading in Cedar Creek Anticline wells are also consistent with this model. This analysis is presented in the hope that other complex failure processes, such as breakdown of sand consolidation processes, such as breakdown of sand consolidation treatments might also exhibit the same statistical behavior. In that case, this model could provide a useful tool for evaluating the economic consequences of such failures. Data for one homogeneous group of wells are used to test the model and to determine k for the group. The lit is quite good, and the data analysis techniques illustrated should prove useful to anyone attempting to apply the model under the usual condition of small sample size. An analysis and example show bow the model provides a basis for deciding when to exercise preventive maintenance and which remedial alternative is most economical.
Many types of complex physical phenomena are described quite accurately by a probability model known to mathematical statisticians as the "Poisson Pure Death Process." A familiar example is the Pure Death Process." A familiar example is the phenomenon of radioactive decay. One cannot phenomenon of radioactive decay. One cannot predict the precise moment that a given atom of a predict the precise moment that a given atom of a radioactive element will disintegrate. However, the Poisson model does allow one to calculate the probability that disintegration will occur within probability that disintegration will occur within any time period of arbitrary duration. This model contains a single parameter, lambda, which may be regarded as the average number of disintegrations per unit time when a large number of individual per unit time when a large number of individual atoms are observed. This parameter has a different value for each radioactive element. Thus, its value depends upon physical derails such as the chemical species involved, but the mathematical form of the probability law is the same for all radioactive probability law is the same for all radioactive elements. The model also applies to other problems in which events occur at random times, especially when the physical phenomenon depends on many variables. In this paper, the model is applied to part of the data available on casing failures in part of the data available on casing failures in Cedar Creek Anticline wells. The failures show up as complete or partial collapse of casing within salt beds at depths of 5,000 to 6,000 ft in most of Shell Oil Co.'s Cedar Creek Anticline fields. Although the model has been tested for only this particular problem, there is a good possibility that particular problem, there is a good possibility that it will apply to sand consolidation failures, casing failures caused by corrosion, etc. This paper, then, proposes to (1) briefly describe the model and proposes to (1) briefly describe the model and summarize the mathematical derivation; (2) show how to test the model and determine the parameter lambda from available data; and (3) illustrate how the model can be used in making economic decisions.
GENERAL DESCRIPTION OF THE "PURE DEATH" MODEL
Chapter XVII in Ref. 1 contains an introduction to time-dependent random processes and includes a detailed description of the general Poisson process as well as some specific examples. The case we are interested in here is known as the Pure Death Process, which is not discussed completely in Process, which is not discussed completely in Ref. 1. However, the mathematical methods presented there are easily modified to derive the necessary equations for the Pure Death model. Consider a population containing N members at time t = 0. As time passes, the number of members can either decrease, as individuals die, or remain constant; it cannot increase because no births occur and no new members are added.
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