1. 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 Scaling Up and Modeling for Flow and Transport in Porous Media Dubrovnik, Croatia, October 13-16, 2008

2. 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

3. 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

4. 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

8. Ito Stochastic Differential Equations (SDE’s) Particle Trajectory SDE: Integral Form: Conditional Probability Density PDE (Fokker-Planck Equation): The connection between the SDE and the PDE is through the semimartingale version of Ito’s Lemma.

13. Sand Tank Experiments Soil Type Uniformity Index #80.9541.6081.7781.863 #160.6160.9831.0411.690 #300.3580.4970.5251.466 #700.1160.1870.2011.736 #1400.0560.0950.1051.874

15. Experimental Plume – Heterogeneous Tank The picture on the right shows the development of a Soltrol plume in the heterogeneous part of the tank. Five sands, #8, #16, #30, #30:50(2:1), #70, were used in packing the lower portion of the tank depicted in these pictures. Approximate Sand Domain Dimensions: 70 cm Wide; 50 cm High; 5 cm Thick

16. Model Behavior – Heterogeneous Tank The last simulation demonstrates the behavior of the SDE model in a highly heterogeneous sand tank. Five sands, #8, #16, #30, #30:50(2:1), #70, were used in packing the lower portion of the tank depicted in the picture on the left. Water Flow

17. 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 embedded in the tank. Pooling of the NAPL occurs where sand changes from coarse to fine.

18. Sample Interface Control Experiment The points at which the NAPL breaks through along the coarse/fine sand interface are determined, in part, by porosity variations.

21. Fingering Fingering has been linked to several factors: In this pore scale model, we focus on pore characteristics and try to simulate instabilities based on pore scale variations and the accompanying pressure variations. Mobility Gravity Capillary Forces Permeabilities Others

22. Fingering The fingering algorithm is based on the grain size cumulative density function, derived from Taylor mesh data, the grain size PDF and the grain size relationship to the equivalent pore size. Grain Size Sieve Analysis Equivalent Pore Size Grain Size

23. 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

24. Fingering And Secondary Pooling In this example, a finger is spawned at the primary interface but later pools at a secondary interface.

25. Fingering And Secondary Pooling In the following example, a finger is spawned at the primary interface but later pools a second time at a slightly higher secondary interface.

26. 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 spreads through the gap.

27. Channel Flow  Air displaces liquid along continuous paths of least resistance

28. Bubble flow around impermeable lenses  Computations based on experiments of Ji et al. (1993)

29. Air Sparging  Air bubbles pass through NAPL plume and carry off volatile contaminant

30. 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