### Abstract

We consider free surface flow in random porous media by treating hydraulic conductivity of a medium as a random field with known statistics. We start by recasting the boundary-value problem in the form of an integral equation where the parameters and domain of integration are random. Our analysis of this equation consists of expanding the random integrals in Taylor's series about the mean position of the free boundary and taking the ensemble mean. To quantify the uncertainty associated with such predictions, we also develop a set of integro-differential equations satisfied by the corresponding second ensemble moments. The resulting moment equations require closure approximations to be workable. We derive such closures by means of perturbation expansions in powers of the variance of the logarithm of hydraulic conductivity. Though this formally limits our solutions to mildly heterogeneous porous media, our analytical solutions for one-dimensional flows demonstrate that such perturbation expansions may remain robust for relatively large values of the variance of the logarithm of hydraulic conductivity.

Original language | English (US) |
---|---|

Pages (from-to) | 1857-1876 |

Number of pages | 20 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 61 |

Issue number | 6 |

DOIs | |

State | Published - 2001 |

Externally published | Yes |

### Fingerprint

### Keywords

- Moment equations
- Moving boundary
- Porous media
- Random
- Stochastic

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*SIAM Journal on Applied Mathematics*,

*61*(6), 1857-1876. https://doi.org/10.1137/S0036139999358180

**Dynamics of free surfaces in random porous media.** / Tartakovsky, Daniel M.; Winter, C Larrabee.

Research output: Contribution to journal › Article

*SIAM Journal on Applied Mathematics*, vol. 61, no. 6, pp. 1857-1876. https://doi.org/10.1137/S0036139999358180

}

TY - JOUR

T1 - Dynamics of free surfaces in random porous media

AU - Tartakovsky, Daniel M.

AU - Winter, C Larrabee

PY - 2001

Y1 - 2001

N2 - We consider free surface flow in random porous media by treating hydraulic conductivity of a medium as a random field with known statistics. We start by recasting the boundary-value problem in the form of an integral equation where the parameters and domain of integration are random. Our analysis of this equation consists of expanding the random integrals in Taylor's series about the mean position of the free boundary and taking the ensemble mean. To quantify the uncertainty associated with such predictions, we also develop a set of integro-differential equations satisfied by the corresponding second ensemble moments. The resulting moment equations require closure approximations to be workable. We derive such closures by means of perturbation expansions in powers of the variance of the logarithm of hydraulic conductivity. Though this formally limits our solutions to mildly heterogeneous porous media, our analytical solutions for one-dimensional flows demonstrate that such perturbation expansions may remain robust for relatively large values of the variance of the logarithm of hydraulic conductivity.

AB - We consider free surface flow in random porous media by treating hydraulic conductivity of a medium as a random field with known statistics. We start by recasting the boundary-value problem in the form of an integral equation where the parameters and domain of integration are random. Our analysis of this equation consists of expanding the random integrals in Taylor's series about the mean position of the free boundary and taking the ensemble mean. To quantify the uncertainty associated with such predictions, we also develop a set of integro-differential equations satisfied by the corresponding second ensemble moments. The resulting moment equations require closure approximations to be workable. We derive such closures by means of perturbation expansions in powers of the variance of the logarithm of hydraulic conductivity. Though this formally limits our solutions to mildly heterogeneous porous media, our analytical solutions for one-dimensional flows demonstrate that such perturbation expansions may remain robust for relatively large values of the variance of the logarithm of hydraulic conductivity.

KW - Moment equations

KW - Moving boundary

KW - Porous media

KW - Random

KW - Stochastic

UR - http://www.scopus.com/inward/record.url?scp=0035704479&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035704479&partnerID=8YFLogxK

U2 - 10.1137/S0036139999358180

DO - 10.1137/S0036139999358180

M3 - Article

AN - SCOPUS:0035704479

VL - 61

SP - 1857

EP - 1876

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 6

ER -