# Immiscible Front Evolution in Randomly Heterogeneous Porous Media

## Abstract

The evolution of a sharp interface between two immiscible fluids in a randomly heterogeneous porous medium is investigated analytically using a stochastic moment approach. The displacing fluid is taken to be at constant saturation and to have a much larger viscosity than does the displaced fluid, which is therefore effectively static. Capillary pressure at the interface is related to porosity and permeability via the Leverett J-function. Whereas porosity is spatially uniform, permeability forms a spatially correlated random field. Displacement is governed by stochastic integro-differential equations defined over a three-dimensional domain bounded by a random interface. The equations are expanded and averaged in probability space to yield leading order recursive equations governing the ensemble mean and variance of interface position, rate of propagation and pressure gradient within the displacing fluid. Solutions are obtained for one-dimensional head- and flux-driven displacements in statistically homogeneous media and found to compare well with numerical Monte Carlo simulations. The manner in which medium heterogeneity affects the mean pressure gradient is indicative of how it impacts the stability of the mean interface. Capillary pressure at the interface is found to have a potentially important effect on its mean dynamics and stability.

- Authors:

- Publication Date:

- Research Org.:
- Idaho National Laboratory (INL)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 912178

- Report Number(s):
- INEEL/JOU-03-00229

TRN: US200801%%631

- DOE Contract Number:
- DE-AC07-99ID-13727

- Resource Type:
- Journal Article

- Journal Name:
- Physics of Fluids

- Additional Journal Information:
- Journal Volume: 15; Journal Issue: 11

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 58 - GEOSCIENCES; INTEGRO-DIFFERENTIAL EQUATIONS; PERMEABILITY; POROSITY; PRESSURE GRADIENTS; PROBABILITY; SATURATION; STABILITY; VISCOSITY; heterogeneous; immiscible fluids; porous media

### Citation Formats

```
Tartakovsky, A M, Neuman, S P, and Lenhard, R J.
```*Immiscible Front Evolution in Randomly Heterogeneous Porous Media*. United States: N. p., 2003.
Web. doi:10.1063/1.1612944.

```
Tartakovsky, A M, Neuman, S P, & Lenhard, R J.
```*Immiscible Front Evolution in Randomly Heterogeneous Porous Media*. United States. doi:10.1063/1.1612944.

```
Tartakovsky, A M, Neuman, S P, and Lenhard, R J. Sat .
"Immiscible Front Evolution in Randomly Heterogeneous Porous Media". United States. doi:10.1063/1.1612944.
```

```
@article{osti_912178,
```

title = {Immiscible Front Evolution in Randomly Heterogeneous Porous Media},

author = {Tartakovsky, A M and Neuman, S P and Lenhard, R J},

abstractNote = {The evolution of a sharp interface between two immiscible fluids in a randomly heterogeneous porous medium is investigated analytically using a stochastic moment approach. The displacing fluid is taken to be at constant saturation and to have a much larger viscosity than does the displaced fluid, which is therefore effectively static. Capillary pressure at the interface is related to porosity and permeability via the Leverett J-function. Whereas porosity is spatially uniform, permeability forms a spatially correlated random field. Displacement is governed by stochastic integro-differential equations defined over a three-dimensional domain bounded by a random interface. The equations are expanded and averaged in probability space to yield leading order recursive equations governing the ensemble mean and variance of interface position, rate of propagation and pressure gradient within the displacing fluid. Solutions are obtained for one-dimensional head- and flux-driven displacements in statistically homogeneous media and found to compare well with numerical Monte Carlo simulations. The manner in which medium heterogeneity affects the mean pressure gradient is indicative of how it impacts the stability of the mean interface. Capillary pressure at the interface is found to have a potentially important effect on its mean dynamics and stability.},

doi = {10.1063/1.1612944},

journal = {Physics of Fluids},

number = 11,

volume = 15,

place = {United States},

year = {2003},

month = {11}

}