We demonstrate the use of path integrals to calculate pressure statistics of Darcy flow in a porous medium. A path integral is used to generate simulations of the pressure and flow rates.
To predict flow through geologically realistic porous media, it is necessary to have a representation of the permeability. We use a stochastic model for the permeability. The engineering strategy to calculate pressure statistics is through explicit realisations of the geological model. The pressure field corresponding to each realisation is obtained through Darcy’s law.
For a reliable estimate, a large number (commonly 10^6) of independent realisations of the inverse permeability matrix is needed. In repeating the matrix inversion 10^6 times, we run into heavy computational requirements. In practice, these can be so heavy that too few realisations are used, making the results unreliable.
We propose a path integral over all pressure realisations as a computationally lighter alternative. These pressure realisations are generated through Markov Chain Monte Carlo methods, which rely exclusively on the probability distribution of the permeability. The path integral proves an elegant tool to reduce computational time.