1. ATM 301 Lecture #7 (sections 7.3-7.4) Soil Water Movements – Darcy’s Law and Richards Equation

2. infiltration (~76% of land precip.) redistribution unsaturated soil saturated groundwater Overland flow The law governing water flow through a porous medium such as soil was discovered in 1856 by Henri Darcy, a French engineer. Henri Darcy (6/10/1803–1/03/1858)

3. Hydrologic Soil Horizons (or layers): Phreatic Zone (saturated zone) Vadose Zone (unsaturated zone) p=  w g (z’-z’ o )

4. Darcy’s Law: for saturated subsurface flow Specific discharge is proportional to the pressure gradient: where q x = specific discharge (discharge per unit area, in m/s), also called Darcy velocity Q x = volume discharge in m 3 /s K hx = saturated hydraulic conductivity in m/s dh/dx = the gradient of total hydraulic head, h

5. Hydraulic Head (h) in ground and soil water: Hydraulic head (or simply head), h, is the fluid potential, or the mechanic energy per unit weight of the fluid (unit: in meters) h = z +  = z + p/  = z + p/(  g) z = elevation (m, gravitational head) p = total pressure (including air pressure, N/m 2 )  (psi)= p/(  g) is the pressure head (in m) The total head h = z +  in a saturated flow can be measured as the height to which water rises in a piezometer, a tube connecting the point to the air. Hydrostatic Downward flow

6. If there is no pressure gradient over a distance, no flow occurs (these are hydrostatic conditions); If there is a pressure gradient, flow will occur from high pressure towards low pressure (opposite the direction of increasing gradient – hence the negative sign in Darcy's law); The greater the pressure gradient (through the same formation material), the greater the discharge rate; and The discharge rate of fluid will often be different — through different formation materials (or even through the same material, in a different direction) — even if the same pressure gradient exists in both cases. Simple applications of the Darcy’s Law:

7. Porosity and actual flow velocity: Porosity ( , phi ) is the ratio of the void space (filled with air or water) between grains or particles within a porous medium divided by the volume of the medium.  ranges from ~40% for sandy soil with large particles to close to 50% for clay soil with fine particles. Flow velocity (U x, m/s) of a fluid within a medium is related to the Darcy velocity q x U x  Q x /(  A x ) = q x /  UxUx AxAx qxqx QxQx

8. Darcy’s Law: Limitations The specific discharge or Darcy velocity (q x ) represents the averaged flow speed over a small local volume of the order of several grain/particle diameters (representative elementary volume or REV). Darcy’s law does not hold on scales smaller than this REV scale. Darcy’s law only applies to parallel or laminar flows. As the flow velocity increases, nonlinear relation arises between the flow rate and head gradient. This is usually quantified using the Reynolds Number, Re: where d = average grain diameter in m v = kinematic viscosity of the fluid (  1.7x10 -6 m 2 /s) v =  / ,  = (dynamic) viscosity discussed earlier. Think of viscosity as “resistance to flow”! Darcy’s law is valid when Re <  1, which is true for most flows in the ground and soil.

9. Permeability and Hydraulic Conductivity: The hydraulic conductivity (K h ) depends on the nature of the medium (e.g., fine vs. coarse grain) and type of the fluid (e.g., water vs. oil): and k I = C d 2 is the intrinsic permeability that depends only on the nature of the medium. g = gravitational acceleration (  9.8m/s 2 ) d = the average grain diameter C = parameter that depends on grain shape, size distribution and packing. v = kinematic viscosity (  1.7x10 -6 m 2 /s) See p. 325 on how K h is measured.

10. General Saturated-Flow Equation: where S s is the specific storage (in m): S s depends on the compressibility of the fluid and of the medium. Derive it using mass conservation and the Darcy’s law.

11. 1.Steady-state flow, 2. Isotropic and homogeneous medium, K hx =K hy =K hz =K h (diffusion eq.): 3. Steady-state, isotropic and homogeneous medium (Laplace equation): Simplified Forms of the Saturated Flow Equation:

12. Hydrologic Soil Horizons (or layers): Phreatic Zone (saturated zone) Vadose Zone ( unsaturated zone ) p=  w g (z’-z’ o )

13. Darcy’s Law: for unsaturated subsurface flow where  = local volumetric water content Both K hx and pressure head  (psi) increase with  The K h vs.  and  vs.  relationships are crucial For the vertical z direction:

14. Soil-water Pressure Head  in unsaturated (or vadose) zone: Capillary Rise (or Capillary Action) is the ability of a liquid to flow in narrow spaces without the assistance of, and in opposition to, external forces like gravity. If the diameter of the narrow space (e.g., within a tube or between soil particles) is sufficiently small, then the combination of surface tension and adhesive forces between the liquid and container (or particles) act to lift the liquid. Inside the unsaturated soil layer, the capillary rise causes tension or negative (i.e., less than atmospheric, or inward) pressure on the water surface. This pressure is measured using tensiometers is called tension head. pmpm

15. Fig. 1 Jet Fill Tensiometer from Soil Moisture Equipment Corp. (www.soilmoisture.com)

16. Moisture-characteristic curve Moisture-conductivity curve Approximated as:  (  )=  ae (θ/  ) b  = porosity or void fraction in a medium. b = parameter that depends on pore size distribution K h = saturated hydraulic conductivity Effect of water content (  ) on pressure (tension head) and conductivity Both K hx and pressure head  increase with  Tension head decreases with 

17. Effect of water content on pressure (tension head) for different soils tension head decreases with grain size and with water content

18. Effect of water content on hydraulic conductivity for different soils Maidement (1993) KhKh Note: Higher conductivity at higher water contents Higher conductivities for coarse textures

19. Effect of water content on hydraulic conductivity for different soils S Higher conductivity at higher water contents Higher conductivities for coarse textures

20. q z =-K h [1+d  (  )/dz] Saturated value  =  ae (θ/ϕ) b

21. General Unsaturated-Flow Equation: Richards Equation for isotropic K h and the z direction is vertical. The specific storage (in m): In unsaturated soil, S s depends on changes in water content relative to hydraulic head, not the compressibility of the fluid and of the medium. And Darcy’s law becomes: Derive this Eq. from mass conservation and the Darcy’s law following Box 7.3 on p. 326

22. Solutions to the General Flow Equation: Given the K h (  ) and  (  ) relations and boundary conditions, the Richards equation can be solved using numerical methods for  (x,y,z) and  (x,y,z)=  (  ), and the head as h(x,y,z)=  (x,y,z) + z Vertical downward flow (percolation): no horizontal flow, most common application (Also referred to as the Richards Eq.): or where z’=-z pointing downward, i.e., increasing downward. This is also solved numerically on computers. American Lorenzo Adolph Richards (1904–1993) was one of the 20th century’s most influential minds in the field of soil physics. He derived the Richard Eq. in 1931.

23. Home work #3 (on soil moisture; due Oct. 6): 1.Ex. 1 on p. 342 of Dingman (2015) (see section 7.1.1.1, Box 7.1, use spreadsheet) (25%) 2.Ex. 2 on p. 342 (see Section 7.1.2-7.1.4, Box 7.1) (15%) 3.Ex. 5 (Experiment A only) on p. 387 (see Section 8..4.3, Need spreadsheet, use text-disk program) (30%) 4.Briefly describe the Darcy’s law using plain language and an equation (15%) 5.Briefly describe the Richards equation using plain language and an equation (for vertical direction only, i.e., no horizontal flow case). (15%)