1 Horizontal Infiltration using Richards Equation. The Bruce and Klute approach for horizontal infiltration

2 Richards Eq: let’s derive it again for kicks Richards Equation is easy to derive, so let's do it here for one-dimensional horizontal flow. For horizontal flow Darcy’s law says: The conservation of mass tells us: the time rate of change in stored water is equal to the negative of the change in flux with distance (i.e., an increase or decrease in flux with distance results in respective depletion or accumulation of stored water) Richards Equation is easy to derive, so let's do it here for one-dimensional horizontal flow. For horizontal flow Darcy’s law says: The conservation of mass tells us: the time rate of change in stored water is equal to the negative of the change in flux with distance (i.e., an increase or decrease in flux with distance results in respective depletion or accumulation of stored water)

3 Taking the first derivative of Darcy’s law with respect to position, and substituting the result from the conservation of mass for the left side Richards Equation for horizontal flow. Taking the first derivative of Darcy’s law with respect to position, and substituting the result from the conservation of mass for the left side Richards Equation for horizontal flow. recall and

4 Richards Eq: Using the definition of soil-water diffusivity this may be written in the form of the diffusion equation Our goal: to solve this rascal for horizontal infiltration into dry soil Using the definition of soil-water diffusivity this may be written in the form of the diffusion equation Our goal: to solve this rascal for horizontal infiltration into dry soil

5 So what are the rules here? If we find a solution to Richards Equation which satisfies the boundary and initial conditions, then we have the unique solution.If we find a solution to Richards Equation which satisfies the boundary and initial conditions, then we have the unique solution. The Green and Ampt solution had a one-to-one relationship between the square root of time and position.The Green and Ampt solution had a one-to-one relationship between the square root of time and position. With that motivation, let’s introduce a “similarity variable" referred to as either the Boltzman or Buckingham transform If we find a solution to Richards Equation which satisfies the boundary and initial conditions, then we have the unique solution.If we find a solution to Richards Equation which satisfies the boundary and initial conditions, then we have the unique solution. The Green and Ampt solution had a one-to-one relationship between the square root of time and position.The Green and Ampt solution had a one-to-one relationship between the square root of time and position. With that motivation, let’s introduce a “similarity variable" referred to as either the Boltzman or Buckingham transform

6 Now putting the equation in terms of  We are going to write Richards Eq. with  in place of t and z. We need to calculate the substitutions for the derivatives. For z: For t: We are going to write Richards Eq. with  in place of t and z. We need to calculate the substitutions for the derivatives. For z: For t:

7 We have and and Using the expressions for  z and  t, we see that the right side of Richards Eq is NOTE: The partial derivatives are now simple derivatives since there is only one variable in the similarity version of the equation. The left side can be put in terms of  : Using the expressions for  z and  t, we see that the right side of Richards Eq is NOTE: The partial derivatives are now simple derivatives since there is only one variable in the similarity version of the equation. The left side can be put in terms of  :

8 Finishing the substitution Putting this all together we find: multiply by t: Putting this all together we find: multiply by t:

9 We have in terms of  Multiplying each side by d  and integrating from  =  i to  we obtain where  ’ is the dummy variable of integration. This may be rearranged: The Bruce and Klute Eq.! Multiplying each side by d  and integrating from  =  i to  we obtain where  ’ is the dummy variable of integration. This may be rearranged: The Bruce and Klute Eq.!

10 Wait! That integral is constant! If  i is zero (initially dry soil), the integral is identified as a soil-water parameter which is referred to as the soil “sorptivity.” For infiltration into initially dry soil the B&K Eq. is Pretty Simple! Clearly solutions to will depend on the form of D(  ) and S(  ). If  i is zero (initially dry soil), the integral is identified as a soil-water parameter which is referred to as the soil “sorptivity.” For infiltration into initially dry soil the B&K Eq. is Pretty Simple! Clearly solutions to will depend on the form of D(  ) and S(  ).

11 Now what? Need D and S!  What forms of the function D(  ) allow for analytical solutions?  Philip (1960 a, b) developed a broad set of forms of D(  ) which produce exact solutions.  Brutsaert (1968) then provided an expression for diffusivity which fit well to natural soils and allow solution  What forms of the function D(  ) allow for analytical solutions?  Philip (1960 a, b) developed a broad set of forms of D(  ) which produce exact solutions.  Brutsaert (1968) then provided an expression for diffusivity which fit well to natural soils and allow solution

12 Using the Brutsaert D B n and D o determined experimentally for soil. n and D o determined experimentally for soil. 1< n < 10, depending on pore size distribution 1< n < 10, depending on pore size distribution D o is the diffusivity at saturation D o is the diffusivity at saturation Using this for D(  ), B&K is generally solved by n and D o determined experimentally for soil. n and D o determined experimentally for soil. 1< n < 10, depending on pore size distribution 1< n < 10, depending on pore size distribution D o is the diffusivity at saturation D o is the diffusivity at saturation Using this for D(  ), B&K is generally solved by

13 THE SOLUTION!!! This may be easily checked by putting the solution into B&K and turning the crank.This may be easily checked by putting the solution into B&K and turning the crank. This equation gives the exact solution for the shape of the wetting front as a function of time.This equation gives the exact solution for the shape of the wetting front as a function of time. Exactly the information that we could not get out of the Green and Ampt approach.Exactly the information that we could not get out of the Green and Ampt approach. This may be easily checked by putting the solution into B&K and turning the crank.This may be easily checked by putting the solution into B&K and turning the crank. This equation gives the exact solution for the shape of the wetting front as a function of time.This equation gives the exact solution for the shape of the wetting front as a function of time. Exactly the information that we could not get out of the Green and Ampt approach.Exactly the information that we could not get out of the Green and Ampt approach.

Horizontal infiltration as a function of n for a Brutsaert soil with Do = 1, and n = 2, 5, and 10. The wetting front becomes increasingly sharp as n increases, making the pore size distribution narrower.

15 HYDRUS-2D Simulations of Horz. Infil. Plotting moisture content vs position above and moisture content vs x/t 1/2 on the lower plot They fit the Boltzman transform! Also recall that in Miller similarity time scales with the square root of macroscopic length scale…not bad! Plotting moisture content vs position above and moisture content vs x/t 1/2 on the lower plot They fit the Boltzman transform! Also recall that in Miller similarity time scales with the square root of macroscopic length scale…not bad!

16 Now about that sorptivity... Can also solve for the S. Using Brutsaert’s equation and the definition of S Let’s pull out the constant Can also solve for the S. Using Brutsaert’s equation and the definition of S Let’s pull out the constant

17 Computing the Sorptivity... So we have the result Which is easy to integrate to obtain Most often sorptivity is reported for saturated soil,  = 1 So we have the result Which is easy to integrate to obtain Most often sorptivity is reported for saturated soil,  = 1

18 Why bother (with S)? Suppose we want to calculate the infiltration Integrating the moisture content over all positions at a given time We can evaluate the same integral by switching the bounds of integration so that we integrate all positions over the moisture content or in of  Suppose we want to calculate the infiltration Integrating the moisture content over all positions at a given time We can evaluate the same integral by switching the bounds of integration so that we integrate all positions over the moisture content or in of   x

19 Computing cumulative infiltration Which is just which is exactly the form obtained by Green and Ampt! (i.e. square root of time) Can calculate the rate of infiltration Again, identical Green and Ampt! Which is just which is exactly the form obtained by Green and Ampt! (i.e. square root of time) Can calculate the rate of infiltration Again, identical Green and Ampt!

20 OK, but what is sorptivity? A parameter which expresses the macroscopic balance between the capillary forces and the hydraulic conductivity. Recall From the discussion of the Green and Ampt results that K sat goes up with 2 and  f goes with 1/ ( is the characteristic microscopic length scale, for instance d 50 ), then we can guess that sorptivity will get larger for coarser soils, but only with 1/2 A parameter which expresses the macroscopic balance between the capillary forces and the hydraulic conductivity. Recall From the discussion of the Green and Ampt results that K sat goes up with 2 and  f goes with 1/ ( is the characteristic microscopic length scale, for instance d 50 ), then we can guess that sorptivity will get larger for coarser soils, but only with 1/2

21 Miller Scaling Big Time From the definition of S(  ) we know that where S is the sorptivity, D is the diffusivity, K is the conductivity,  is the moisture content and  is the Boltzman transform variable,  = xt -1/2 From the definition of S(  ) we know that where S is the sorptivity, D is the diffusivity, K is the conductivity,  is the moisture content and  is the Boltzman transform variable,  = xt -1/2

22 Miller Scaling S To derive the scaled value of sorptivity we must replace the variables with the appropriate scaled quantities (L is macro- scopic scale) We find: To derive the scaled value of sorptivity we must replace the variables with the appropriate scaled quantities (L is macro- scopic scale) We find:

23 Wrapping this up... S scales with 1/2, (Just as we saw in the Green and Ampt Sorptivity). As a little bonus, we see what the effect of changing fluid properties would produce in the value of sorptivity S scales with 1/2, (Just as we saw in the Green and Ampt Sorptivity). As a little bonus, we see what the effect of changing fluid properties would produce in the value of sorptivity