Project Title: Path Integral Method for Flow through Random Porous Media

Supervisors: Prof. Peter King and Prof. Dimitri Vvedensky

Project description:

One of the key problems in modelling flow in oil reservoirs is our lack of precise knowledge of the variations in flow properties (e.g. permeability) across the field. At best we can infer the statistics of these variations from field observations or analogue outcrops. The challenge is then to determine the statistics of the flow itself (flow rates, pressures etc.) from the statistics of the permeability variations. Hitherto this has been done by Monte Carlo simulations, which can be computationally very expensive, unless smart sampling techniques or surrogate models are used. We aim to demonstrate the use of a path integral formulation for this problem. Originally developed by R. P. Feynman as an alternative formulation of quantum mechanics, it has become a tool for use with stochastic differential equations (see, for example, Ref. [1]). To demonstrate how this methods works, we start with the one dimensional Darcy flow problem: q(x)=-K(x)dp(x)/dx where p(x) is the pressure, q(x) is the flow rate and K(x) is the rock permeability. The randomness of the porous medium is modelled by regarding K as a stochastic quantity which is assumed to follow Gaussian statistics [2]. Because of the randomly varying rock structure, there is a variety of conceivable pressure realisations p(x). The path integral Z is an integral over all realisations with an appropriate probability measure. Once Z is evaluated, either analytically, or by standard Monte Carlo methods [3], any observable of interest, including pressure correlations can be easily obtained. We investigate the numerical agreement between both approaches in one dimension. Extension of the path integral to three dimensions is straightforward. Numerical implementation is planned for the near future.

[1] C. de Dominicis and L. Peliti, Field theory renormalization and critical dynamics above Tc, Phys. Rev. B 18, no. 1 (1987).

[2] J. Law, A statistical approach to the interstitial heterogeneity of sand reservoirs, A.I.M.E. Los Angeles Meeting (1943).

M. Creutz and M. Freedman, A statistical approach to quantum mechanics, Ann. Phys. 132, 427-462 (1981).