Meta-analysis of hydraulic fracture conductivity data

Abstract

Previous empirical models of propped fracture conductivity are based either on data sourced from single investigations or on data not in the public domain. In this work, statistically rigorous models of propped fracture conductivity are developed using a database of fracture conductivity experiments reported in technical literature over the last 40 years. The database contains the results from about 2700 experimental runs. Propped fracture conductivity is the dependent variable and proppant types, mesh size, proppant concentration, formation hardness, closure stress, formation temperature, and polymer concentration are the independent variables. The mother database is partitioned into subsets; that is different databases with each daughter database having complete information in relation to the dependent and independent variables. As a result, the number of independent variables included in the daughter databases varied from three to six. Seventy percent of the data was used to develop the models while 30% of the data was used to validate them. First, fixed effect models were developed using regression analysis. Afterwards, three, four and five factor models were compared for two types of proppant: sand and ceramic proppant. The five factor model appeared to be the most prominent one. The analysis was further carried out using five factors of these two types of proppant. Mixed effect modeling was employed because of the disparate sources of the data and also the temporal diversity of the dataset. The mixed effect model appeared to be the better than the fixed effect model while compared the error terms. Also, because the mother database contained some missing values, two statistical imputation approaches were employed to predict the missing values which are categorical imputation and multiple imputation using chained equations. Imputations are employed because it is speculated that a model developed using a large number of data points should provide better predictions. Generally, the mean squared error (MSE) is less in the mixed effect model for sand and in the categorical imputation model for ceramic proppant. But, to be more precise on the performance of the models, model predictions were compared with an existing propped fracture conductivity model and different case histories published in literature. Subsequently, the models of this research can be arranged in order of predictive performance: multiple imputation model, mixed effect model, fixed effect/categorical imputation model. The results also indicate that mesh size, closure stress, formation hardness, and proppant concentration significantly affect fracture conductivity from a statistical point of view. Formation temperature and polymer concentration affect conductivity negatively but they were not statistically significant. Engineers will have access to a propped fracture conductivity database based on experiments reported over the past 40 years in technical literature. Engineers can use the models developed based on this database to generate statistical distributions of propped fracture conductivity for a variety of proppant characteristics and formation conditions. The models presented here are based on data from experimental investigations in different laboratories thereby reducing the bias that may be present in single laboratory investigations.