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Soil Steady-State Evaporation

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Morteza Sadeghi
Morteza Sadeghi

In this presentation, an exact analytical solution to steady state evaporation from porous media is introduced. The solution is presented in terms of a set of infinite series. An advantage of this solution compared to previous derivations is that the infinite series can be very closely approximated using a closed-form solution (i.e., excluding integrals or series).

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An Analytical Solution to Soil Steady-State Evaporation Morteza Sadeghi Utah State University Ferdowsi University of Mashhad Nima Shokri Boston University Scott B. Jones Utah State University
Motivation
Unsaturated soil Surface water Water table Ground water
Steady State Evaporation 1 – Near surface water table (Phase one): 2- Deeper water table (Phase two):
Darcy’s law: e Drying Front Film Region z Saturated Region Suction Air-entry Liquid flow region (Dmax) Water table depth (D) Gas Region dh K ( h) 1 dz
 When D < Dmax (phase one), evaporation rate is high.  When D > Dmax (phase two), evaporation rate significantly decreases due to the hydraulic discontinuity between water table and soil surface.
Analytical solutions have been developed using: Gardner function Ks K 1 h / hb P Brooks-Corey function K K: Unsaturated conductivity Ks: Saturated Conductivity h: suction head hb : Air-entry suction head P: Shape parameter Ks K s h / hb (h P hb ) (h > hb )
Literature Review
In this research:
Mathematical Derivations Darcy: K dh K e z z: depth to water table K: Unsaturated Conductivity h: suction head e: evaporation rate hb : Air-entry suction head he : h (K=e) hDF : h at the Drying front Defining variables: T e/ K (hb <h he ) U K /e (he <h hDF ) Applying Brooks-Corey model for K(h): K Ks K s h / hb (h P hb ) (h > hb )
Mathematical Derivations 1 1 e / Ks z 1 1 e / Ks 1 1 e / Ks h (h hb ) dh hb 1 T he dh hb 1 T h h h ( hb< h he) h he Udh 1 U (he < h hDF ) Maclaurin series expansion for |x| < 1 as (1 – x)-1 = 1 + x + x2 + x3 +… 1 e / Ks z 1 e / Ks 1 e / Ks 1 1 1 h h h (h hb ) h hb he hb 1 T T 2 ... dh 1 T T 2 ... dh (hb <h he ) h he U U 2 U 3 ... dh (he <h hDF )
Mathematical Derivations Exact Solution 1 e 1 Ks z h 1 e 1 Ks z he (h hb ) hb 1 he n h / he i 0 1 he i 1 1 iP h / he 1 iP 1 he 1 iP i 1 1 iP i 0 1 he i 1 n hb / he 1 iP (hb <h he ) 1 iP i 1 1 iP (he <h hDF ) Suction head distribution above the water table as a function of hydraulic properties and
Mathematical Derivations Closed-form Solution 1 e 1 Ks h hb e e ) Ks Ks e 1 P 1 Ks hb e e ln(1 ) Ks Ks e 1 P 1 Ks (h ln(1 z 1 P 1 h 1 P e Ks Ks h ln 1 (h / hb ) e P 1/ P 1 e h ln 1 (h / hb ) P Ks ln 2 1 P hb ) (hb <h he ) 2 /12 ln 2 ln 2 1 P 1 P 1 P (he <h hDF )
Dmax hb e e ln(1 ) Ks Ks e 1 P 1 Ks 2 e Ks 1/ P Gas Region Film Region z Saturated Region Suction Dmax Drying Front ln 2 1 P ln 2 12 P 1 P ln 2 1 1 P
Results & Discussions Suction head distribution 3.0 2.5 z/hb 2.0 1.5 Warrick [1988], Brooks-Corey K(h) Warrick [1988], Gardner K(h) New solution, Exact New solution, Approximate Salvucci [1993] h=h h = he b 1.0 0.5 0.0 0.001 0.01 0.1 1 h/hb 10 100
Results & Discussions Suction head distribution 3.0 2.5 z/hb 2.0 1.5 Warrick [1988], Brooks-Corey K(h) Warrick [1988], Gardner K(h) New solution, Exact New solution, Approximate Salvucci [1993] 1.0 0.5 0.0 0.001 0.01 0.1 1 h/hb 10 100
Results & Discussions Suction head distribution 2.0 z/hb 1.5 1.0 Warrick [1988], Brooks-Corey K(h) Warrick [1988], Gardner K(h) New solution, Exact New solution, Approximate Salvucci [1993] 0.5 0.0 0.01 0.1 1 h/hb 10 100
Results & Discussions Dmax (cm), Approximate solution Liquid flow region 200 Chino Pachappa 1.02 mm 0.48 mm 0.16 mm coarse sand fine sand silt 150 100 50 0 0 50 100 150 Dmax (cm), Exact solution 200
Results & Discussions When D > Dmax, evaporation rate decreases significantly due to hydraulic discontinuity. 1.0 D = Dmax Chino Pachappa 1.02 mm 0.48 mm 0.16 mm coarse sand fine sand silt 0.8 e/e0 0.6 0.4 0.2 0.0 0 1 2 D/Dmax 3 4 5
Conclusions
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Soil Steady-State Evaporation

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Soil Steady-State Evaporation

  • 1. An Analytical Solution to Soil Steady-State Evaporation Morteza Sadeghi Utah State University Ferdowsi University of Mashhad Nima Shokri Boston University Scott B. Jones Utah State University
  • 4. Steady State Evaporation 1 – Near surface water table (Phase one): 2- Deeper water table (Phase two):
  • 5. Darcy’s law: e Drying Front Film Region z Saturated Region Suction Air-entry Liquid flow region (Dmax) Water table depth (D) Gas Region dh K ( h) 1 dz
  • 6.  When D < Dmax (phase one), evaporation rate is high.  When D > Dmax (phase two), evaporation rate significantly decreases due to the hydraulic discontinuity between water table and soil surface.
  • 7. Analytical solutions have been developed using: Gardner function Ks K 1 h / hb P Brooks-Corey function K K: Unsaturated conductivity Ks: Saturated Conductivity h: suction head hb : Air-entry suction head P: Shape parameter Ks K s h / hb (h P hb ) (h > hb )
  • 10. Mathematical Derivations Darcy: K dh K e z z: depth to water table K: Unsaturated Conductivity h: suction head e: evaporation rate hb : Air-entry suction head he : h (K=e) hDF : h at the Drying front Defining variables: T e/ K (hb <h he ) U K /e (he <h hDF ) Applying Brooks-Corey model for K(h): K Ks K s h / hb (h P hb ) (h > hb )
  • 11. Mathematical Derivations 1 1 e / Ks z 1 1 e / Ks 1 1 e / Ks h (h hb ) dh hb 1 T he dh hb 1 T h h h ( hb< h he) h he Udh 1 U (he < h hDF ) Maclaurin series expansion for |x| < 1 as (1 – x)-1 = 1 + x + x2 + x3 +… 1 e / Ks z 1 e / Ks 1 e / Ks 1 1 1 h h h (h hb ) h hb he hb 1 T T 2 ... dh 1 T T 2 ... dh (hb <h he ) h he U U 2 U 3 ... dh (he <h hDF )
  • 12. Mathematical Derivations Exact Solution 1 e 1 Ks z h 1 e 1 Ks z he (h hb ) hb 1 he n h / he i 0 1 he i 1 1 iP h / he 1 iP 1 he 1 iP i 1 1 iP i 0 1 he i 1 n hb / he 1 iP (hb <h he ) 1 iP i 1 1 iP (he <h hDF ) Suction head distribution above the water table as a function of hydraulic properties and
  • 13. Mathematical Derivations Closed-form Solution 1 e 1 Ks h hb e e ) Ks Ks e 1 P 1 Ks hb e e ln(1 ) Ks Ks e 1 P 1 Ks (h ln(1 z 1 P 1 h 1 P e Ks Ks h ln 1 (h / hb ) e P 1/ P 1 e h ln 1 (h / hb ) P Ks ln 2 1 P hb ) (hb <h he ) 2 /12 ln 2 ln 2 1 P 1 P 1 P (he <h hDF )
  • 14. Dmax hb e e ln(1 ) Ks Ks e 1 P 1 Ks 2 e Ks 1/ P Gas Region Film Region z Saturated Region Suction Dmax Drying Front ln 2 1 P ln 2 12 P 1 P ln 2 1 1 P
  • 15. Results & Discussions Suction head distribution 3.0 2.5 z/hb 2.0 1.5 Warrick [1988], Brooks-Corey K(h) Warrick [1988], Gardner K(h) New solution, Exact New solution, Approximate Salvucci [1993] h=h h = he b 1.0 0.5 0.0 0.001 0.01 0.1 1 h/hb 10 100
  • 16. Results & Discussions Suction head distribution 3.0 2.5 z/hb 2.0 1.5 Warrick [1988], Brooks-Corey K(h) Warrick [1988], Gardner K(h) New solution, Exact New solution, Approximate Salvucci [1993] 1.0 0.5 0.0 0.001 0.01 0.1 1 h/hb 10 100
  • 17. Results & Discussions Suction head distribution 2.0 z/hb 1.5 1.0 Warrick [1988], Brooks-Corey K(h) Warrick [1988], Gardner K(h) New solution, Exact New solution, Approximate Salvucci [1993] 0.5 0.0 0.01 0.1 1 h/hb 10 100
  • 18. Results & Discussions Dmax (cm), Approximate solution Liquid flow region 200 Chino Pachappa 1.02 mm 0.48 mm 0.16 mm coarse sand fine sand silt 150 100 50 0 0 50 100 150 Dmax (cm), Exact solution 200
  • 19. Results & Discussions When D > Dmax, evaporation rate decreases significantly due to hydraulic discontinuity. 1.0 D = Dmax Chino Pachappa 1.02 mm 0.48 mm 0.16 mm coarse sand fine sand silt 0.8 e/e0 0.6 0.4 0.2 0.0 0 1 2 D/Dmax 3 4 5