Spe-192002-ms case Study Applied Machine Learning to Optimise pcp completion Design in a cbm field
Download 1.89 Mb. View original pdf
|
- Navigate this page:
- Likelihood and Bayesian Inference
| Covariance and Hyper-parameters The covariance function defines the covariance between the random variables (the functions, and is calculated using the input variables (Rasmussen & Williams, p14): (9) Where k is function describing the covariance between the input variables. In modelling failure time using GPs, selection of covariance function is a critical decision. Additionally, covariance functions contain parameters, called hyper-parameters, which can be tuned for improved model accuracy. Likelihood and Bayesian Inference The GP provides a distribution of functions from which a random sample is drawn Bayesian inference provides a method for selection of functions from the sample which are useful for modelling the failure time. The application of Bayes rule to joint distributions requires integration over variables and is beyond the scope of this paper (refer to Rasmussen & Williams or Murphy). Bayesian inference relies on Bayes rule: (10) Downloaded from http://onepetro.org/SPEAPOG/proceedings-pdf/18APOG/2-18APOG/D021S016R002/1220497/spe-192002-ms.pdf/1 by Vedanta Limited - Cairn Oil & Gas user on 28 June 2023 6 SPE-192002-MS Where P(A|B) and P(B|A) are probabilities of event A given event Band vice versa, and PA) and P(B) are the probabilities of events A and B independently. Intuitively, Bayes rule enables the use of prior information (the prior) on the probability of an event. The prior information is used with new information on a related event. The new information conditions the prior via calculation of likelihood i.e. the likelihood of observing the new information given the prior. The posterior probability (the posterior) is the probability of an event, given the new information and prior. (11) For a GP of functions, the prior is a distribution of functions according to m(x) and k(x,x'). For obtaining the functions in the GP which are useful for failure time modelling, the prior is conditioned on the input variables of the test data. The functions of the GP which agree with the test data are retained, and those functions are used to calculate the pointwise mean and variance of the posterior. The pointwise mean and variance are used for estimation of failure time, or this case, PCP runlife. Discussion The best model converged well to the problem with a mean absolute error of 126 days looking at failed pumps only. This model was used as the basis fora recommender system for the next down- hole configuration. The recommender system provided consistently reasonable configurations for the next production string completion to be installed in the well. The residual error of the model could be improved further if the causes of pump failure were isolated to the PCP design given the reservoir fluids and completion type. Download 1.89 Mb. Share with your friends:
|