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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 60 th birthday of Alain Bourgeat - PowerPoint PPT Presentation
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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