# A Bayesian Estimation for the Fractional Order of the Differential Equation that Models Transport in Unconventional Hydrocarbon Reservoirs

## TL;DR

This paper introduces a Bayesian method to estimate the fractional order in differential equations modeling fluid transport in unconventional hydrocarbon reservoirs, addressing the challenge of parameter estimation where traditional methods fail.

## Contribution

The paper develops a novel Bayesian estimation technique for fractional differential equations, enabling uncertainty quantification in modeling transport in reservoirs.

## Key findings

- Effective Bayesian estimation of fractional order.
- Quantification of uncertainty in model predictions.
- Simulation results demonstrate method's utility.

## Abstract

The extraction of natural gas from the earth has been shown to be governed by differential equations concerning flow through a porous material. Recently, models such as fractional differential equations have been developed to model this phenomenon. One key issue with these models is estimating the fraction of the differential equation. Traditional methods such as maximum likelihood, least squares and even method of moments are not available to estimate this parameter as traditional calculus methods do not apply. We develop a Bayesian approach to estimate the fraction of the order of the differential equation that models transport in unconventional hydrocarbon reservoirs. In this paper, we use this approach to adequately quantify the uncertainties associated with the error and predictions. A simulation study is presented as well to assess the utility of the modeling approach.