Preferential Flow and Extensions to Richards Equation

Transcription

1 Preferential Flow and Extensions to Richards Equation David A. DiCarlo Dept. of Petroleum and Geosystems Engineering University of Texas at Austin

2 Preferential flow Uniform infiltration Preferential flow paths Can result in large differences between actual and modeled flow behavior

3 Gravity driven fingering gk Preferential flow in a initially uniform media Result of an Saffman-Taylor instability with the heavy water displacing the light air or ( 2 ρ1 ρ2) q( µ 1 µ ) > q < K Finger width predicted well by Parlange and Hill (1976) Glass, Selker, Steenhuis (early 90s) 0 in conductivity units

4 Why are not all infiltrations unstable? Infiltrations into silts and loams are observed to be stable Soils with initial water have stable fronts Hendrickx and Yao (1996) showed that even dry sands highly susceptible to instability are stable at very low infiltration rates Look into dynamics of fingers for the answer!

5 Saturation overshoot Saturation overshoot is always associated with fingering Is overshoot a result or a cause of gravity driven fingers? Compare 1-D overshoot to 3-D fingers Key is saturation at the wetting front (Parlange)

6 Gravity driven fingering gk Preferential flow in a initially uniform media Result of an Saffman-Taylor instability with the heavy water displacing the light air or ( 2 ρ1 ρ2) q( µ 1 µ ) > q < K Finger width predicted well by Parlange and Hill (1976) Glass, Selker, Steenhuis (early 90s) 0 in conductivity units

7 Saturation overshoot Saturation overshoot is always associated with fingering Is overshoot a result or a cause of gravity driven fingers? Compare 1-D overshoot to 3-D fingers Key is saturation at the wetting front (Parlange)

8 Experiments Infiltrate into tubes smaller in diameter than fingers Measure saturation and front velocity using light transmission Compare profiles versus infiltrating flux

9 Saturation profile vs flux

10 Saturation profile vs flux No overshoot at low and high fluxes

11 Saturation profile vs flux No overshoot at low and high fluxes Overshoot at fluxes in between

12 Saturation profile vs flux No overshoot at low and high fluxes Overshoot at fluxes in between Measure tip and tail saturations vs flux

13 Tip and tail saturations vs flux Tail saturation is the unsaturated conductivity Tip is unsaturated and tip saturation decreases slowly with flux

14 Tip and tail saturations vs flux Tail saturation is the unsaturated conductivity Tip is unsaturated and tip saturation decreases slowly with flux Overshoot

15 Effect of initial water content Less overshoot and smaller range with initial water

16 1-D overshoot controls 3-D fingering Range of fluxes over which 1D overshoot occurs is the same range over which 3D fingering is observed (Hendrickx and Yao, 1996) Magnitude of 1D overshoot versus initial water content agrees with fingers in 2D slab studies (Bauters et al., 2000)

17 No overshoot - Stable flow q Displacement stays on the wetting curve Media wets up to the saturation that matches the applied flux No driving force for the instability A Cap Pressure B θ* C Saturation C B A q = K(θ*)

18 Overshoot - Unstable flow q Displacement starts on the wetting curve and finishes on the drying curve Media saturation overshoots the saturation that matches the flux Overshoot provides a driving force for the instability A Cap Pressure θ* E D C B Saturation q < K(θ max ) E D C B A

19 Fundamental physics of overshoot Tail saturation is the unsaturated conductivity Tip saturation is controlled by the porefilling processes at the initial wetting front which are velocity dependent Fast front -> collective pore filling Slow front -> snap-off ahead of front

20 Micromodel experiments (Lenormand and Zarcone, 1984) Decreasing front velocity Sharp front High tip saturation Diffuse front Low tip saturation

21 Micromodel experiments (Lenormand and Zarcone, 1984) Decreasing front velocity A Pc θ* D B Saturation C A Pc B θ* C Saturation

22 Instability, overshoot, and multiphase flow equation extensions Although there is a qualitative mechanistic understanding of overshoot and instability, one would like a continuum model of flow which accurately reproduces the instability Instability and overshoot are forbidden by Richards Equation Extensions have been proposed; need to compare model extensions to data Measure saturation overshoot as a function of Grain size Grain roughness Infiltrating fluid alkanes and alcohols Edge effects - tomography

23 Effect of grain size Increasing grain size Flux range for overshoot increases with grain size

24 Effect of grain size Increasing grain size Flux range for overshoot increases with grain size

25 Overshoot in continuum descriptions of fluid flow Richards equation strictly forbids instability and overshoot Different continuum extensions have been proposed Relaxation term (Hassanizadeh et al. or Barenblatt) S S Pc = Pcs τ H or Se = S+τ B t t Hyperbolic or hypodiffusive (Eliassi and Glass) S TS ( ) t t or [ FS ( ) S] Fourth order term (Cueto-Felgueroso and Juanes) 2 2 ( S)

26 Overshoot simplifies solving continuum equations Solving the flow equations for instability requires discretization due to their nonlinearity and multi-dimensionality (3 space, 1 time) For saturation overshoot there is only 1 spatial dimension; (x,y,z,t) -> (z,t) Infiltrations are found to be traveling waves; S(z,t) -> S(z-vt) For any continuum extension, we can convert the PDE to an ODE

27 Overshoot can be modeled exactly for a traveling wave For Richards equation, ODE is first order with an always positive diffusivity => monotonic, non-overshoot solution For Hassanizadeh or Barrenblatt extension, ODE is second order => can get saturation overshoot For Eliassi and Glass terms, ODE is first order with a range of negative diffusivities => can get saturation overshoot

28 Comparing model extensions to data Dynamic Pc Non-monotonic Pc; inertial term Both models show overshoot; both require a fitting term

29 Hyperbolic and hypodiffusive terms Transition flux does not scale correctly with grain size

30 Relaxation term Transition flux does not scale correctly with grain size Still possibly correct; results depend highly on initial saturation

31 Summary Gravity driven instability is more than ever the important question in multi-phase flow Forbidden by Richards equation Likely a result of dynamic pore scale processes Can be related to overshoot Experimental measurements of overshoot are an excellent test of multi-phase extensions Quantitatively constrain dynamic network (mechanistic) models Continuum models can be tested using traveling wave solutions Can we make a connection between the mechanistic physics and a continuum model? Verdict is still out

32 Thanks For Your Attention! Experimenters Robert Smith Don Seale Kyungmin Ham Behdad Aminzadeh Co-authors Ruben Juanes Tara LaForce Thomas Witelski Discussions Jean-Yves Parlange Tammo Steenhuis Martin Blunt Richard Hughes Clinton Willson All data available!

33 Drainage curves overshoot infiltration Pressure - saturation curves obtained match well with scanning curves Flux determines the starting point for the scanning curves