2nd International Conference Mechanics of Unsaturated Soils 7 th 9 th March 2007

Transcription

1 2nd International Conference Determination of the Soil Water Retention Curve and the Unsaturated Hydraulic Conductivity from the Particle Size Distribution Alexander Scheuermann & Andreas Bieberstein

2 Motivation Pedotransfer functions basing on the particle size distribution Arya & Paris (1981) Haverkamp & Parlange (1986) Fredlund et al. (2002) Schick (2002) Aubertin et al. (2003)

3 Motivation Pedotransfer functions basing on the particle size distribution Arya & Paris (1981) Haverkamp & Parlange (1986) Fredlund et al. (2002) Schick (2002) Aubertin et al. (2003) Is it possible to calculate the SWRC from the particle size distribution without empirical input data?

4 Pore Constriction Distribution Schulze (1992) and Schuler (1997)

5 Pore Constriction Distribution Schulze (1992) and Schuler (1997)

6 Pore Constriction Distribution 20 % 20 % 20 % 20 % 20 % Schulze (1992) and Schuler (1997)

7 Pore Constriction Distribution Consideration of the relative density D

8 Pore Constriction Distribution Consideration of the relative density D

9 Pore Constriction Distribution Consideration of the relative density D D = n n max max n n min = F F PE,max PE,max F F PE PE,min D = 0 D = 1

10 Pore Constriction Distribution Consideration of the relative density D D = n n max max n n min = F F PE,max PE,max F F PE PE,min D = 0 D = 1

11 Pore Constriction Distribution Consideration of the relative density D D = 0 D = 0,2 D = 0,4 D = 0,6 D = 0,8 D = 1

12 Pore Constriction Distribution Difference in sizes pore constrictions Determination of a minimal and a maximal pore constriction Calculation of a mean pore constriction on the logarithmic scale total number Z k of calculations for k particle classes: Z k = k 4

13 Pore Constriction Distribution Schulze (1992) and Schuler (1997)

14 Pore Constriction Distribution Experimental investigations on the pore constriction distribution (Witt 1986)

15 Pore Constriction Distribution Schulze (1992) and Schuler (1997)

16 Soil Water Retention Curve porosity limits n max = 0,428 n min = 0,314 hydraulic conductivity k f = m/s

17 Soil Water Retention Curve Laplace s equation for the definition of the matric potential ψ m = matric potential [N/m 2 ] σ wa = surface tension [N/m] r = radius of the pore constriction [m] δ = contact angle ψ m = σ 2 wa cos r ( δ ) r max r min

18 Soil Water Retention Curve Laplace s equation for the definition of the matric potential ψ m = matric potential [N/m 2 ] σ wa = surface tension [N/m] r = radius of the pore constriction [m] δ = contact angle ψ m = σ 2 wa cos r ( δ ) r min r max

19 Soil Water Retention Curve Laplace s equation for the definition of the matric potential ψ m = matric potential [N/m 2 ] σ wa = surface tension [N/m] r = radius of the pore constriction [m] δ = contact angle ψ m = σ 2 wa cos r ( δ ) r min r max

20 Soil Water Retention Curve Laplace s equation for the definition of the matric potential ψ m = matric potential [N/m 2 ] σ wa = surface tension [N/m] r = radius of the pore constriction [m] δ = contact angle ψ m = σ 2 wa cos r ( δ ) r min

21 Soil Water Retention Curve Definitions for the residual water content water will be stored in the contact zones of the particles after drainage of the pore with the smaller pore constriction radius of a particle constellation (upper boundary) with the smallest pore constriction radius for all particle constellations (lower boundary)

22 Soil Water Retention Curve Definitions for the residual water content water will be stored in the contact zones of the particles after drainage of the pore with the smaller pore constriction radius of a particle constellation (upper boundary) with the smallest pore constriction radius for all particle constellations (lower boundary) assumption of a matric potential at which residual water content will be reached (e.g. basing on the definition of the field capacity) assumption of a residual water content (e.g. by measurement)

23 Soil Water Retention Curve Comparison with experimental result (axis translation technique with hanging water column)

24 Hydraulic conductivity Basic assumptions for the determination of the hydraulic conductivity on the micro-scale (Childs & Collis George 1950) pore size distribution is calculable from the SWRC for individual pore fragments the flow law of Hagen-Poiseuille applies since the pore radius is included in Hagen-Poiseuille s law with the power of four the resistance against water flow for a sequence of two pores is dominated by the smaller pore the total hydraulic conductivity is defined by sequences of pores in series

25 Hydraulic conductivity Equation of Hagen-Poiseuille for laminar flow in capillaries p 1 Q = f HP l η Q = discharge p = pressure difference at the ends of the capillary l = length of the capillary η = dynamic viscosity f HP = form factor (for circular cross section r 4 π / ) ( ) 8

26 Hydraulic conductivity Equation of Hagen-Poiseuille for laminar flow in capillaries p 1 Q = f HP l η Q = discharge p = pressure difference at the ends of the capillary l = length of the capillary η = dynamic viscosity f HP = form factor (for circular cross section r 4 π / ) ( ) 8 hydraulic gradient i

27 Hydraulic conductivity Equation of Hagen-Poiseuille for laminar flow in capillaries p 1 Q = f HP l η Q = discharge p = pressure difference at the ends of the capillary l = length of the capillary η = dynamic viscosity f HP = form factor (for circular cross section r 4 π / ) ( ) 8 hydraulic gradient i Equitation with Darcy s law Q = k f i A

28 Hydraulic conductivity Equation of Hagen-Poiseuille for laminar flow in capillaries p 1 Q = f HP l η Q = discharge p = pressure difference at the ends of the capillary l = length of the capillary η = dynamic viscosity f HP = form factor (for circular cross section r 4 π / ) ( ) 8 hydraulic gradient i Equitation with Darcy s law Q = k f i A Saturated hydraulic conductivity A B m k f 1 = η 1 A = total surface area of all pore constellations = number of all pore constellations B m 1 f HP,m

29 Hydraulic conductivity Consideration of Tortuosity T 0 according to Bear (1972) it is feasible to use T 0 = 2/3 as a constant value for saturated condition for unsaturated condition the reduction of saturation has to be taken into account (T 0,unsat = T 0 S)

30 Hydraulic conductivity Consideration of Tortuosity T 0 according to Bear (1972) it is feasible to use T 0 = 2/3 as a constant value for saturated condition for unsaturated condition the reduction of saturation has to be taken into account (T 0,unsat = T 0 S) Connectivity K according to Vasconcelos (1998) the connectivity can be considered in context with the relationship between specific surface and its pore volume using a simplified consideration of the relationship between the surface of a pore channel formed by a pore constriction and the pore surface leads to a constant connectivity of K = 2,6 (for constant volume) connectivity is also dependent on the saturation (K unsat = K S)

31 Hydraulic conductivity Consideration of Tortuosity T 0 according to Bear (1972) it is feasible to use T 0 = 2/3 as a constant value for saturated condition for unsaturated condition the reduction of saturation has to be taken into account (T 0,unsat = T 0 S) Connectivity K according to Vasconcelos (1998) the connectivity can be considered in context with the relationship between specific surface and its pore volume using a simplified consideration of the relationship between the surface of a pore channel formed by a pore constriction and the pore surface leads to a constant connectivity of K = 2,6 (for constant volume) connectivity is also dependent on the saturation (K unsat = K S) f TK = T 0 K S 2 = 1,73 S 2

32 Hydraulic conductivity

33 Dependency on the relative density porosity limits n max = 0,428 n min = 0,314 hydraulic conductivity k f = m/s

34 Dependency on the relative density

35 Dependency on the relative density

36 Comparison with experimental results porosity limits n max = 0,483 n min = 0,347 hydraulic conductivity k f = m/s

37 Comparison with experimental results

38 Comparison with experimental results

39 Comparison with experimental results hydraulic Conductivity k f = m/s (calculated using the equation acc. to Kozeny/Carman)

40 Comparison with experimental results

41 Comparison with experimental results

42 Summary and Conclusion The pore constriction distribution of a soil can be used for the determination of the SWRC for the case of drainage. The definition for the residual water content has to be improved for a determination of the SWRC without empirical input data. Using the flow law of Hagen-Poiseuille it is possible to calculate the hydraulic conductivity for saturated and unsaturated conditions. The procedure offers the possibility for the determination of the pore range distribution forming the base for the determination of the SWRC for imbibition.

43 Dependency on the relative density Overview of the results relative density D 0 0,2 0,4 0,6 0,8 1,0 air entry value ψ AEV [kpa] 0,85 0,95 1,15 1,4 1,55 1,7 saturated hydraulic conductivity k f,calc. 7, , , , , ,2 10-4