Stochastic Modeling of Multiphase Transport in Subsurface Porous Media: Motivation and Some Formulations

Thomas F. Russell

National Science Foundation, Division of Mathematical Sciences

David Dean, Tissa Illangasekare, Kevin Barnhart

In honor of the 60th birthday of Alain BourgeatScaling Up and Modeling for Flow and Transport in Porous Media

Dubrovnik, Croatia, October 13-16, 2008

Philosophy 1 Goal: (motivated, e.g., by DNAPL)

Macro-model of complex multiphase transport – pooling, fingering, etc. – amenable to efficient computation

Pore-scale physics very important, but won’t be seen in that form at macro-scale

Homogenization can yield important insights, but is too restricted

Philosophy 2 For practical purposes, heterogeneous

multiphase effects can’t be characterized deterministically

Micro-scale phenomena show traits of randomness when viewed through a coarser lens

Thus: Consider modeling with stochastic processes

Seek: Stochastic micro-model that yields macro capillary behavior (end effect, etc.), analogous to Einstein-Fokker-Planck

Outline

2-phase stochastic transport equation for position of nonwetting fluid particle, derived from Itô calculus

Capillary barrier effect Channeling / fingering Qualitatively capture experimental

behavior Extension to bubble flow

Ito Stochastic Differential Equations (SDE’s)

Particle Trajectory SDE:

nxtxtxt

tWdXtdtXtatXd tt

),,(),(2

1),(

)( ),( ,)(

TT DDDBBD

BD

TTT

TT

T

Integral Form:

)( ))(,( ))(,()()(0 0

0 WdXdXatXtXt

t

t

t

BDTT

Conditional Probability Density PDE (Fokker-Planck Equation):

0),(),(),(,),(

txpxttxpXta

t

txpt

D

The connection between the SDE and the PDE is through the semimartingale version of Ito’s Lemma.

Sand Tank Experiments

Soil

Type

Uniformity

Index

#8 0.954 1.608 1.778 1.863

#16 0.616 0.983 1.041 1.690

#30 0.358 0.497 0.525 1.466

#70 0.116 0.187 0.201 1.736

#140 0.056 0.095 0.105 1.874

10d 50d 60d 1060 dd

2) (UniformIndex y Uniformit

Finer) (60%

Finer) (50%Diameter SizeGrain Mean

Finer) (10%Diameter SizeGrain Effective

1060

60

50

10

dd

d

d

d

Experimental Plume – Heterogeneous Tank

The picture on the right shows the development of a Soltrol plume in the heterogeneouspart of the tank. Five sands, #8, #16, #30, #30:50(2:1), #70, were used in packing the lowerportion of the tank depicted in these pictures.

Approximate Sand Domain Dimensions:70 cm Wide; 50 cm High; 5 cm Thick

Model Behavior – Heterogeneous Tank

The last simulation demonstrates the behavior of the SDE model in a highly heterogeneoussand tank. Five sands, #8, #16, #30, #30:50(2:1), #70, were used in packing the lowerportion of the tank depicted in the picture on the left.

Water Flow

Sample Interface Control Experiment

A two phase, water/NAPL, problem in which the tank is initially saturated with water. The NAPL enters the tank through a plug of high permeability, #8, sand embeddedin the tank. Pooling of the NAPL occurs where sand changes from coarse to fine.

Sample Interface Control Experiment

The points at which the NAPL breaks through along the coarse/fine sand interface aredetermined, in part, by porosity variations.

Fingering

Fingering has been linked to several factors:

In this pore scale model, we focus on pore characteristics and try to simulate instabilitiesbased on pore scale variations and the accompanying pressure variations.

• Mobility

• Gravity

• Capillary Forces

• Permeabilities

• Others

Fingering

The fingering algorithm is based on the grain size cumulative density function, derivedfrom Taylor mesh data, the grain size PDF and the grain size relationship to the equivalentpore size.

Grain

Size

Sieve

Analysis

Equivalent PoreSize

Grain Size Grain Size

By probing along the coarse/fine sand interface, the plume finds points in the fine sand where the porosity is higher than average and fingering into these areas is possible. In this simulation, one such point develops as a finger.

Fingering

Fingering And Secondary Pooling

In this example, a finger is spawned at the primary interface but later pools ata secondary interface.

Fingering And Secondary Pooling

In the following example, a finger is spawned at the primary interface but laterpools a second time at a slightly higher secondary interface.

LNAPL Spreading Due To Heterogeneities

In this animation, the LNAPL plume spreads uniformly in the coarse sand until it encounters a broken fine sand lens where the plume pools under the lens and spreadsthrough the gap.

Channel Flow

Air displaces liquid along continuous paths of least resistance

Bubble flow aroundimpermeable lenses

Computations based on experiments of Ji et al. (1993)

Air Sparging

Air bubbles pass through NAPL plume and carry off volatile contaminant

Summary

• Propose The Use Of The Ito Calculus To Develop Stochastic Differential Equation (SDE) Descriptions Of Saturation Phases

• Test The Ability Of The SDE Model To Capture The Interface Effects Of Plume Development, Such As Pooling, Channeling And Fingering, Bubble Flow

• Extend This Work To A Nonlinear Up-scaling Methodology

• Develop A Macro-scale Stochastic Theory Of Multiphase Flow And Transport Accounting For Micro-scale Heterogeneities And Interfaces