Exploring the physics behind the interaction between acoustic waves and unsaturated porous media

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1 Exploring the physics behind the interaction between acoustic waves and unsaturated porous media Wei-Cheng Lo Hydraulic and Ocean Engineering, National Cheng Kung University, Taiwan Garrison Sposito Civil and Environmental Engineering, University of California at Berkeley Ernest L. Majer Geophysics, Earth Sciences Division, Lawrence Berkeley National Laboratory Funding: Department of Energy, USA

3 +26 kg Berkeley, 2003 Cornell, 1994

4 Compensation at University of California (2012) Berkeley Gross pay above $200K 430 people! Gross pay above $300K 89 people! Los Angles Gross pay above $200K 1471 people! Gross pay above $300K 611 people! $ 305,733.30!! => NT $25128/per day

5 Robert E. Horton Medal 2013 Soroosh Sorooshian (UC Irvine) 2012 Keith Beven (Lancaster ) 2011 Murugesu Sivapalan (UIUC) 2010 Jacob Bear (IIT) 2009 William E. Dietrich (UC Berkeley) 2008 Vijay K. Gupta (UCo, Boulder) 2007 Rafael L. Bras (GIT) 2006 Thomas Schmugge (USDA) 2005 Gedeon Dagan (Tel Aviv) 2004 Garrison Sposito (UC Berkeley) 2003 Shlomo P. Neuman (U Arizona) 2002 Jean-Yves Parlange (Cornell) 2001 Donald R. Nielsen (UC Davis) 2000 M. Gordon Wolman (Die) 1999 Wilfried H. Brutsaert (Cornell) 1998 Ignacio Rodriguez-Iturbe (Princeton 1997 John D. Bredehoeft 1996 Mark Meier 1995 Don Kirkham 1994 Mikhail I. Budyko 1992 Luna B. Leopold 1990 Paul A. Witherspoon 1988 Peter S. Eagleson 1986 Abel Wolman 1984 Charles V. Theis 1982 John R. Philip 1980 William C. Ackermann 1978 Harold A. Thomas, Jr Walter B. Langbein

6 Motivation (Field Observation) Oil Production, Barrels/Day Oil Production Oil Cut Earthquake Events ASR Stimulations Oil Cut, % Group of 26 Wells to the Northeast that responded both to the earthquake events of Sept-Oct 1999 and to ISS Oil Production Jan-99 Apr-99 Jul-99 Oct-99 Jan-00 Apr-00 Jul-00 Oct-00 Jan-01 Diatomite: Northwest Group of 26 Wells 10 0 Oil Production, Bbls/Day Oil Cut Oil cut, % Group of 5 Wells to the Southeast that did not respond either to the earthquake events or to ISS 50 0 Jan-99 Apr-99 Jul-99 Oct-99 Jan-00 Apr-00 Jul-00 Oct-00 Jan Diatomite: Souteast Group of 5 Wells

7 Field Observation Fluctuations of water level in a 52-m deep well induced by seismic waves excited by passing trains and an earthquake.

8 Field Experiments

9 Laboratory Experiment

10 Laboratory Observation Stimulation time: 360s Pore pressure gradient: 3 kpa/m Free-phase TCE observed Permeability: 1.1x10-10 m 2 (111 d) Roberts et al., Environ. Engin. Sci. 18(2):67-79 (2001)

11 Problems to be Addressed Although the potential benefits of seismic wave stimulation have been demonstrated in laboratory and field experiments, they lack a sound theoretical basis. Seismic wave stimulation will be not fully developed into a predictable and reliable field technology until more fundamental research is performed to understand better the basic science controlling enhancement phenomena.

12 Statement of Problem Porous medium containing two immiscible fluids (oil and water or air and water) Solid: porous, isotropic, homogeneous, and elastic Fluids: compressible and viscous

13 Methodology Two-phase fluid flow in porous media [continuum mechanics of mixtures] Coupled Elastic wave propagation [linear stress-strain relations]

14 Classical Hydrological Model An uncouple model - representing the intricate interaction between interstitial fluid flow and solid matrix deformation simply using a single lumped parameter, known as storage coefficient.

15 Classical Hydrological Model Transient groundwater flow in confined aquifer Terzaghi theory T S 2 h h = t S = bρg( α + nβ ) 2 p α 2 z = p t K α = sat ρgm v

16 Physically-based Model (Poroelasticity) The solid and fluid constituents should be treated on an equal footing, and their displacement vectors were systemically formulated in the coupled equations of motion.

17 Mass Balance Equations ( ρ α t θ α ) + ( ρ α θ α v α ) = 0 Storage Outflow

18 Momentum Balance Equations r α D v α ur ur uur r ρθ α α tα ρθ α αg ρθ α αm α = 0 Dt Inertia Stress Gravity Interphase Exchange

19 Constraints on Constitutive Relationships Objectivity Symmetry Entropy inequality Linearity

20 Mass Balance Equations with Constitutive Relationships ( ρ θ α t α ) + ( ρ α no change! θ α v α ) = 0

21 Momentum Balance Equations with Constitutive Relationships Fluid r α D v α ur ur r r ρθ α α = θα pα + ρθ α αg+ Rαα ( vα vs ) Dt r r + A ( aβ as ) α = 1,2 β = 1,2 β αβ Solid r s Dv ur ur ur ur r r s ρθ s s = ts p1 θ1 p2 θ2 + ρθ s sg Rαα ( vα vs ) Dt α r r A ( aβ as ) α β αβ

22 Linear Stress-Strain Relations in Unsaturated Porous Media ur r ur r ur r φs1 p1 = a12 us + a22 u1+ a23 u2, ur r ur r ur r φ (1 S1 ) p2 = a13 us + a23 u1+ a33 u2, 2 ur r ur r ur r ts = 2 Ge+ [( a11 G) us + a12 u1+ a13 u2] δ 3 r where uα = displacement of the α phase a = elastic coefficients ij

23 Porous Medium with Two Fluids (Acoustic Motions) Governing Equations [Lo et al., 2005, Water Resources Researches]: e ε1 e ε2 e ε1 e ρθ s s + A 2 11( ) + A ( ) + A ( ) 2 2 t t t t t t t 2 2 ε2 e ε1 e ε2 e + A22( ) + R ( ) + R22( ) t t t t t t = a e+ a ε + a ε ε1 ε1 e ε2 e A 2 11( A R ρθ ε e ) ( ) ( ) t t t t t t t = a e+ a ε + a ε ε 2 2 ε1 ε2 ε2 2 2 A 2 21 A R ρθ e e e ( ) ( ) ( ) t t t t t t t = a e+ a ε + a ε

24 Porous Medium with One Fluid (Acoustic Motions) Governing Equations [Biot, 1956]: t = + t 2 ( ρ11e ρ12ε) b ( e ε) ( Pe Qε) t = + t 2 ( ρ12e ρ22ε) b ( e ε) ( Qe Rε) ur r ur r where e= us = dilatation of solid, ε = u f = dilatation of fluid r r us and u f are the displacement vectors of solid and fluid ( P- G) Ks + QK f = 1 φ and QKs + RK f = φ are elasticity parameters 3 b is the Biot viscous coupling parameter (inversely proportional to permeability) ρ, ρ, and ρ ( ρ < 0) are the Biot inertial coupling parameters

25 Dispersion Relations ω 3 ω 2 ω D11( ) + D 2 22( ) + D 2 33( ) + D 2 44 = 0 k k k 2 ω This equation is a cubic polynomial in and, therefore, 2 k it will in general have three complex roots. When the wave excitation frequency is stipulated, the corresponding phase speed and attenuation coefficient can be deduced. The amplitude of the bulk waves always diminishes with distance, and this condition requires k i > 0, which, in turn, implies that only three of the six solutions for the attenuation coefficient are physically possible. k = k + k where k = wave number k = attenuation r i r i

26 Dispersion Relations for the Free Vibration Problem Input elasticity and hydraulic data e= β exp i( kz ωt) s ε = β exp ikz ( ωt) 1 1 ε = β exp ikz ( ωt) 2 2 Viscous and inertial coupling parameters Water retention curve Hydraulic conductivity function Select vibrational frequency Three roots for wave number k i = v = ω k r (P hasev elocity) Im ( k ) (A ttenuation C oefficient)

27 Acoustic Wave Propagation in Unconsolidated Fine Sandy Loam Phase Velocity (P1 Wave) air-water oil-water 1 50 Hz 100 Hz 150 Hz 200 Hz 1200 Phase Speed (m/s) Dimensionless Speed Water Saturation 0.2 H Vc = ρ 4 where H = a11 + a22 + a33 + 2( a12 + a13 + a23) + G 3 ρ = ραθα Water Saturation

28 Acoustic Wave Propagation in Unconsolidated Fine Sandy Loam Attenuation Coefficient (P1 Wave) air-water system oil-water system 1.2 x Hz 100 Hz 150 Hz 200 Hz 1.2 x Hz 100 Hz 150 Hz 200 Hz Attenuation Coefficient (1/m) Attenuation Coefficient (1/m) Water Saturation Water Saturation

29 Physical Mechanism First term is proportional to the square of the difference in material densities of the two pore fluids, multiplied by the product of their relative mobilities. A second term in the model expression is inversely proportional to the square of an average kinematic shear viscosity weighted by relative permeability. The first term should be large for an air-water mixture, but small for an oil-water mixture, whereas the reverse should be true for the second term.

30 Physical Mechanism air-water system oil-water system 14 x first term second term 10 x first term second term (s 2 /m 4 ) 6 (s 2 /m 4 ) Water Saturation Water Saturation

31 Acoustic Wave Propagation in Unconsolidated Fine Sandy Loam Phase Velocity (P2 Wave) air-water system oil-water system Hz 100 Hz 150 Hz 200 Hz Hz 100 Hz 150 Hz 200 Hz Phase Speed (m/s) Phase Speed (m/s) Water Saturation Water Saturation

32 Acoustic Wave Propagation in Unconsolidated Fine Sandy Loam Attenuation Coefficient (P2 Wave) air-water system oil-water system Hz 100 Hz 150 Hz 200 Hz Hz 100 Hz 150 Hz 200 Hz Attenuation Coefficient (1/m) Attenuation Coefficient (1/m) Water Saturation Water Saturation

33 Physical Mechanism Effective dynamic shear viscosity parameter for a two-fluid system defined in terms of relative mobilities η eff 1 η1η 2 = = b + b ( η k + η k ) r1 1 r2

34 Physical Mechanism air-water system oil-water system 1.2 x x η eff 0.6 η eff Water Saturation Water Saturation

35 Acoustic Wave Propagation in Unconsolidated Fine Sandy Loam Attenuation Coefficient (P2 Wave) 1 50 Hz 100 Hz 150 Hz 200 Hz 0.8 Dimensionless Attenuation k = 2Q k i r Water Saturation

36 Acoustic Wave Propagation in Unconsolidated Fine Sandy Loam Phase Velocity (P3 Wave) air-water system oil-water system Hz 100 Hz 150 Hz 200 Hz Hz 100 Hz 150 Hz 200 Hz Phase Speed (m/s) Phase Speed (m/s) Water Saturation Water Saturation

37 Acoustic Wave Propagation in Unconsolidated Fine Sandy Loam Attenuation Coefficient (P3 Wave) air-water system oil-water system 10 x Hz 100 Hz 150 Hz 200 Hz 10 x Hz 100 Hz 150 Hz 200 Hz Attenuation Coefficient (1/m) Attenuation Coefficient (1/m) Water Saturation Water Saturation

38 Acoustic Wave Propagation in Unconsolidated Fine Sandy Loam Attenuation Coefficient (P3 Wave) Hz 100 Hz 150 Hz 200 Hz 0.8 Dimensionless Attenuation k = 2Q k i r Water Saturation

39 Insights from Numerical Results The P1 wave is a sound wave, whereas the P2 and P3 waves are related to dissipative behavior. Waves of higher frequency have higher attenuation. The P3 wave has the highest attenuation coefficient and the lowest phase velocity. The P1 and P2 waves in a two-fluid system are analogous to the fast and slow compressional waves in Biot theory.

40 Motional Modes [Lo et al., 2010; Advances in Water Resources]

41 Motional Modes (One-Fluid System) Analysis based on Normal Coordinates!! P1 Wave S F P2 Wave S F

42 Motional Modes (Two-Fluid System) P1 Wave S NF WF P2 Wave S NF WF S NF WF P3 Wave S NF WF

43 Boundary Value Problem [Lo et al., 2012, Journal of Applied Geophysics]

44 Boundary Value Problem Unconsolidated Sand saturated by TCE Hz Excess TCE concentration (ppm) Hz 50 Hz 100 Hz ml/min 10 ml/min Time (Hrs)

45 Laboratory Observation Stimulation time: 360s Pore pressure gradient: 3 kpa/m Free-phase TCE observed Permeability: 1.1x10-10 m 2 (111 d) Roberts et al., Environ. Engin. Sci. 18(2):67-79 (2001)

46 Connecting Acoustic Waves Attributes to Subsurface Hydrological and Geological Parameters [Lo et al., 2008, 2010; Journal of Hydrology]

47 Quantitative Connection between Porosity and Phase Speed water-saturated Phase speed (m/s) y = x R 2 = Porosity

48 Quantitative Connection between Permeability and Attenuation Coefficient water-saturated 9.000E-03 y = 1E+09x - 4E E-03 R 2 = 1 Attenuation coefficient (1/m) 7.000E E E E E E-03 y = 9E+08x - 3E-06 R 2 = 1 y = 5E+08x - 2E-06 R 2 = 1 y = 2E+08x - 8E-07 R 2 = Hz 200 Hz 300 Hz 400 Hz 500 Hz 1.000E-03 y = 5E+07x - 2E-07 R 2 = E E E E E E E E E-12 Permeability (m^2)

49 Consolidation Theory in Unsaturated Porous Media [Lo et al., 2014; Vadose Zone Journal]

50 Consolidation Governing Equations [Lo et al., 2013]: ε1 e ε2 e R11( ) + R22( ) = a11 e+ a12 ε1 + a13 ε2 t t t t ε1 e R11( ) = a21 e+ a22 ε1 + a23 ε2 t t ε 2 e R22( ) = a31 e+ a32 ε1 + a33 ε2 t t

51 Initial Condition Saturated porous media Terzaghi - uncoupled p(,0) z = p Biot coupled * p(,0) z * = γ p γ : loading efficiency Unsaturated porous media p (,0) z = γ p 1 1 * p (,0) z = γ p 2 2 *

52 Sand Loam Silt Clay 0.5 γ Water saturation

53 Total Settlement - Clay 14 x Total Settlement (m) S = 0.7 S = 0.8 S = 0.9 S = Time (day)

54 Future Works Dynamic Boundary conditions Layered Media Experimental verification: - Laboratory - Field

Juan E. Santos a,b,c, Gabriela B. Savioli a and Robiel Martínez Corredor c a

Juan E. Santos a,b,c, Gabriela B. Savioli a and Robiel Martínez Corredor c a Universidad de Buenos Aires, Fac. Ing., IGPUBA, ARGENTINA b Department of Mathematics, Purdue University, USA c Universidad