1. Predictive Models for Saturated Hydraulic Conductivity12
Hillel, pp
Predictive Models for Saturated Hydraulic Conductivity

2. The Kozeny-Carman Predictive Model for Ksat
Efforts to predict hydraulic conductivity based on soil structural attributes such as porosity, particle size, etc. date back to early 1900.
The idea is extremely attractive because of the obvious connection between water flow and the distribution of solids and pore spaces in the soil medium.
In 1927 Kozeny introduced a model relating soil permeability to porosity.
Fluidity
Permeability
r liquid density [M L-3]
g acceleration of gravity [L t-2]
h viscosity coefficient of the liquid in [Pa s] [M L-1 t-1]
Fluidity at 20oC is: 9.81x106 [m s]-1. Note that permeability has dimensions of L2.

3. The Kozeny-Carman Predictive Model for Ksat
Kozeny’s model was based on application of Poiseuille’s law to flow through granular porous media lacking structure or consolidation.
He idealizes the soil porous system as a bundle of cylindrical capillaries with a single representative radius.
Rearranging Poiseuille’s law for laminar flow in tubes, considering the fraction of pores given by porosity n, yields Kozeny’s model in its simplest form:
RH= Hydraulic Radius (ratio pore volume to wetted surface area) n Porosity

4. The Kozeny-Carman Predictive Model for Ksat
Kozeny’s approach was extended by Carman to take the form:
c Empirical coefficient representing particle size and shape (2-3 for non-cubic particles)
t Tortuosity of the flow path (21/2 for main flow direction 45o from the vertical axis)
Av Specific surface per unit volume.
Above equation can be simplified by introducing a hydraulic radius RH:

5. The Kozeny-Carman Predictive Model for Ksat
Tortuosity: Ratio of the average roundabout path of a fluid molecule to the apparent or straight flow path.
Tortuosity is a dimensionless geometric parameter of porous materials, which, though difficult to measure precisely, is always greater than 1 and may exceed 2.

6. The Kozeny-Carman Predictive Model for Ksat
The product c2 is commonly about 5. The pore shapes, surfaces, and tortuosity are interdependent, thus the product c2 Rh2 is often lumped into a single “formation factor”, F2.
Ks is estimated as:

7. The Kozeny-Carman Predictive Model for Ksat
The Kozeny-Carman approach provides unsatisfactory estimates of Ks in many soils, due to the assumption of uniform pore radii.
Works well for sands and other materials with uniform pore size distribution.

8. Revil & Cathles Predictive Model for Ksat
Revil and Cathles [1999] developed a predictive model for permeability using concepts from electrical conductance in porous media. They use the concept of a formation factor F that is a dimensionless, scale invariant parameter that characterizes the pore space topology. F is related to the porosity n and pore geometry by the empirical Archie relationship:
m is the so-called “cementation exponent” and varies with pore geometry in a range of 1 to 4. (ratio of pure solution electrical conductivity to the bulk electrical conductivity of the saturated porous medium.

9. Revil & Cathles Predictive Model for Ksat
Cementation Exponent m
For media with perfectly spherical grains m=1.5
For sands with porosities between 0.03 and 0.3 m=1.5 to 2.0
For media where the pore space consists entirely of open interconnected fractures and cracks m=1.1 to 1.3 (most of the porosity conducts electrical current)
If large pores are connected to narrow throats m>2.5
The derivations of Revil and Cathles [1999] yielded a simple expression:
d is the mean grain diameter (from particle size analysis) of granular materials.

10. Revil & Cathles Predictive Model for Ksat
The approach yields remarkable match for granular materials, and was later extended to predict hydraulic conductivities of sand-clay mixtures.
1 mD (milidarcy) = 1x10-15 m2 ~10-8 m/s (water 200 C)

11. Characterization of Flow Conditions
Flow regimes in a porous medium are characterized according to:
Changes in flow attributes in time and space (steady vs. non steady flows)
Saturation state of the system (saturated vs. unsaturated)
Geometry of flow (one-dimensional or multi-dimensional).
Constant Head Method for Ksat: Steady-state, saturated, and one-dimensional flow conditions. (water flux and other flow attributes such as hydraulic potentials and water content do not change over time at any given location).
Falling Head Method for Ksat: Non steady, saturated, and one-dimensional flow conditions. (Flux density and hydraulic potentials vary with time.)

12. Unsaturated Flow

13. Flow of Water in Unsaturated Soils
Most flow processes in the field occur under unsaturated conditions, or in the presence of an air-phase.
This drastically modifies water flow pathways, with decreasing water content flow through water filled pores become constrained to narrower and tortuous channels.
Darcy's law, originally conceived for saturated flow only, was extended by Edgar Buckingham (1907) to unsaturated water flow. Buckingham's main assumption was that unsaturated hydraulic conductivity is a function of water content or matric potential, i.e. the degree of wetness.
Unsaturated Hydraulic Conductivity expressed as a function of Matric Potential h

14. Saturated versus Unsaturated Flow
The driving force for flow in a saturated soil under constant gravitational potential is the gradient of positive pressures, whereas under unsaturated conditions the gradient is of negative matric potentials.
When the soil is completely water-saturated, the pores conduct at a maximal rate Ks. When the soil desaturates, some of the water-filled pores become partially or completely air-filled, and the cross-sectional area available for mass flow decreases K(h).
Equations for saturated flow have a constant saturated hydraulic conductivity (Ks)
Unsaturated flow equations have an unsaturated hydraulic conductivity function, K(h).

15. Unsaturated Hydraulic Conductivity Function
The unsaturated hydraulic conductivity is a highly nonlinear function of water content, K(), or matric potential, K(h)- often spanning more than 8 orders of magnitude!
At saturation a sandy soil has higher conductivity than clay due to larger water-filled pore spaces.
At lower suctions large pores in sandy soil drain resulting in a steep decrease in conductivity.
Loamy and clayey soils, having smaller (on average) pores, retain greater number of water-filled pores resulting in higher hydraulic conductivity than sandy soils at low potentials.
crossover

16. Unsaturated Hydraulic Conductivity Function
Estimates for unsaturated hydraulic conductivity for various soil textural classes can be obtained from the UNSODA and NRCS Soil Survey databases.

17. Parametric Models for K(q)
Some researchers have extended the concepts discussed in the previous section and derived closed-form analytical predictions of the unsaturated hydraulic conductivity. The basic step is replacing the summations with integrals of SWC analytical (parametric) expressions
The van Genuchten and Brooks and Corey SWC Models are amenable to such integration. Van Genuchten (1980) derived an expression for the unsaturated hydraulic conductivity based on the same parameters he used for the SWC curve and on a modification of Jackson’s model proposed by Mualem (1976).
The resulting water content-dependent unsaturated hydraulic conductivity K(q) is given by:

18. Parametric Models for K(h)
van Genuchten-Mualem (VGM) model:
The unsaturated hydraulic conductivity can also be expressed in terms of matric head (h):

19. Parametric Models for K(h)
Brooks and Corey (BC) model:
Similarly expressions can be derived for the Brook's and Corey model.
where hb is the matric head at air-entry value or "bubbling pressure", and l is the pore distribution index parameter.
Expressed as K(q):

20. Parametric Models for K(q) & K(h)
As input information for the VG and BC models for unsaturated hydraulic conductivity we need to measure a few h-q pairs (SWC) and saturated hydraulic conductivity Ks:
We fit the parametric expressions for the SWC to measured data to obtain free model parameters a, n, qr (VG) or hb, l (BC)
We use measured Ks and the free model parameters from the SWC to predict unsaturated hydraulic conductivity

21. Parametric Models for K(h)
Another widely used parametric model for unsaturated hydraulic conductivity was proposed by Gardner (1958):
Where b (dimensions of L-1) is a parameter related to typical pore size of the medium.
The primary advantage of Gardner’s model is that it allows analytical solutions to certain flow problems.
Why do we need parametric models?
For obtaining analytical solutions to flow and transport problems
For comparison between different types of soils or other porous media.

22. Steady State Unsaturated Flow in Soils

23. Steady State Unsaturated Flow
Steady-state unsaturated flow may develop under a variety of conditions.
When a constant flux at the soil surface, less than the soil's Ks is maintained for extended periods of time by use of a sprinkler or other means.
The evaporation rate from a shallow water table may also attain (approximately) a steady state.
For vertical steady-state unsaturated flow we may write the Buckingham-Darcy equation:
This ordinary differential equation may be integrated to yield the integral form of the Buckingham-Darcy equation (next slide).

24. Steady State Unsaturated Flow
Integral form of the Buckingham-Darcy equation:
E.g., Gardner’s Model
When K(h) has a known functional form, the integral on the left side may be evaluated exactly to produce an analytical solution for the water flux or for the distribution of the matric potential in the soil profile, h(z).

25. Steady State Unsaturated Flow
Analytical Solution for Unsaturated Steady State Flow to a Water Table:
Given a constant flux (Jw < Ks) from the soil surface to a water table at a depth L. We assume that the unsaturated hydraulic conductivity, K(h), is given by Gardner's model K(h)=Ks ebh
We want to find the distribution of soil matric potential in the soil profile from the soil surface to the water table.
The first step is to introduce the parametric expression of K(h) into the integral form of the Buckingham-Darcy law

26. Steady State Unsaturated Flow
We set z=0 at the soil surface, and z=-L at the water table (where h=0). From a table of integrals we find the following solution:
Applying this solution to our problem with the appropriate boundary conditions yields:
Rearranging and log-transforming yields the desired form in terms of h(z):
One can verify that at z=-L, h(z=-L)=0. The term in the square brackets is limited to values between 0 and 1. This is important for analyses of maximal upward flow from a water table.

27. Steady State Unsaturated Flow
Graph showing matric potential distributions within the soil profile for various potentials (previous equation).

28. Steady State Unsaturated Flow - Example
Problem Statement:
Compute and plot the vertical distributions of the soil matric potential under steady flow into a water table at a depth of 3 m from the surface for the flow rates of Jw/Ks= , -0.01, -0.1, and Note that the sign of the downward flux is negative. The unsaturated hydraulic conductivity of the soil is based on Gardner's (1958) model, and is characterized by b=5 m-1.
Estimate the hydraulic gradient (dH/dz) from 0 to 1m below the soil surface.
Solution:
The depth dependent matric potential may be calculated according to the following relationship:
Results are listed on the next slide.

29. Steady State Unsaturated Flow - Example
Tabulated Results:

30. Steady State Unsaturated Flow - Example
Solution – Graph:

31. Steady State Unsaturated Flow - Example
Solution - Continued:
To estimate the hydraulic gradient we have to solve:

32. Prediction of Unsaturated Hydraulic Conductivity
The relationships between the hydraulic conductivity and the degree of saturation or matric potential arouse interest in methods for predicting the soil hydraulic conductivity based on SWC.
Most models are based on simplifying assumptions:
The complicated soil pore geometry is represented by a bundle of continuous cylindrical capillaries.
The flow is assumed to occur only in these capillaries (no bypass flow)
The length of the bundle of capillaries (Lc), is assumed to be larger than the length (L) of the soil column it represents; the ratio Lc/L is the tortuosity factor (t).

33. Predictive Models for Soil Hydraulic Conductivity
Poiseuille's law for laminar flow in cylindrical tubes is applied to the bundle of capillaries with a gradient DH across its length. This enables computation of the total volume of flow through the column by:
Poiseuille's Law
Nj is the number of capillaries of radius Rj in the bundle
M is the number of different (arbitrarily assigned) capillary size classes

34. Predictive Models for Soil Hydraulic Conductivity
The flux (Jw) through the column is:
nj is the number of tubes with radius Rj per unit area (nj =Nj/A)
The next step is to convert the soil water characteristic curve into a soil water content vs. equivalent pore radii curve. This is done by using the capillary-rise equation to convert a matric potential value to an equivalent capillary radius (see next slide).

35. Predictive Models for Soil Hydraulic Conductivity
The total saturation water contend qs is divided into a certain number M of equal increments Dqv where the matric potential hj corresponds to a water content of qs- jxDqv
We assume that all tubes with radii greater than Rj have drained at matric potential hj

36. Predictive Models for Soil Hydraulic Conductivity
SWC-Curve
Water Content- Equivalent Capillary Radius Curve
Empty
Full
Capillary Rise Equation

37. Predictive Models for Soil Hydraulic Conductivity
The number of capillaries having a radius Rj per unit area in each water content interval is: nj=Dqv/pRj2 (where Dqv is interpreted as the fraction of the water-filled cross sectional area that is reduced when all capillaries having a radius of Rj drain).

38. Predictive Models for Soil Hydraulic Conductivity
Now we can plug in the number of capillaries nj into Poiseuille’s law and introduced a multiplication by -L/Dz=1 (where Dz=0-L) to obtain a form similar to Darcy's law :
t Tortuosity Factor
Saturated Hydraulic Conductivity

39. Predictive Models for Soil Hydraulic Conductivity
As water content decreases from qs to qs- i x Dqv the tubes larger than Ri drain and do not contribute to water flow. The effect on the unsaturated hydraulic conductivity, which is a function of the water content, is given by
This equation, originally Developed by Childs and Collis-George (1950), serves as the basis for many predictive models of unsaturated hydraulic conductivity, most of which are based on SWC information and assume a capillary bundle geometry.

40. Predictive Models for Soil Hydraulic Conductivity
Even with a detailed SWC, there is still an unknown, i.e. the tortuosity (t). One way to circumvent this problem is to measure the saturated hydraulic conductivity (which is relatively easy to measure) and use it to solve for t, assuming it is constant. One such formulation was proposed by Jackson (1972):
where qj=(qs-j x Dqv), c is a constant usually taken as c=1, and the water content is divided to M equal increments (of qv)

41. Jackson Model - Example
The matric potential-soil water content function (SWC) for Sarpy loam soil is given by the VG model expressed for the matric potential (h) as:

42. Jackson Model - Example
First we split the SWC into 10 equal increments and calculate water content q and corresponding matric potential h for the midpoint of each increment.

43. Jackson Model - Example
Now we can apply Jackson’s equation and calculate K(q) for the midpoints of each increment. If we consider for example the calculation of K(0.289):
* j is a pointer whereas i is a counter
** /2=0.37/(10x2)=0.0185; note that hj corresponds to j, which is less than the water content indicated in the table by the amount of /2.

44. Jackson Model - Example
When we compare our results with the analytically derived van Genuchten-Mualem (VGM) K(q) model (see next slides) we see good agreement.

45. Unsteady and Unsaturated Flow – The Richards Equation15
Hillel, pp
Unsteady and Unsaturated Flow – The Richards Equation

46. Unsaturated Nonsteady Flow
Unsaturated nonsteady flow is the most common flow regime under field situations, where water content q, the matric potential h, and the flux Jw vary in time and space.
The application of Buckingham-Darcy's law to describe the flow is no longer sufficient as the flux varies from one location to another, a condition called flux divergence.
The difference between the flux entering a volume element of soil and that exiting the same volume element is the change in water content, or storage (CONSERVATION OF MASS = CONTINUITY PRINCIPLE).

47. The Richards Equation del operator (gradient of J)
The principal of continuity may be formalized mathematically for one-dimensional vertical flow as:
Note that the partial derivatives indicate possible variations of flux and water content in space and in time.
The three-dimensional form of the continuity equation is given as:
del operator
(gradient of J)

48. Unsaturated Nonsteady Flow – Richards Equation
To obtain a flow equation for unsaturated nonsteady flow we have to combine Buckingham-Darcy's law for the flux (Jw) with the continuity equation.
For one-dimensional vertical flow we obtain:
For three-dimensional flow we obtain:
This equation is known as the RICHARDS EQUATION, and is the most general unsaturated flow equation used in practice.
Note that this equation is based on the assumption of isothermal conditions (uniform temperature) and flow through a rigid matrix (no swelling or shrinking of the soil).

49. Unsaturated & Unsteady Horizontal Flow
Unsaturated and nonsteady horizontal flow in one dimension may be represented by the simplest form of the Richards equation, with no gravitational component, as:
In this case we have two variables, q and h, which vary in time and space, but only one equation.
In some cases it is possible to reduce the equation to only one variable by using the interrelationship q-h (the SWC curve).
For water infiltration into a homogeneous soil having uniform initial water content it is advantageous to eliminate the matric potential (h).

50. Unsaturated Nonsteady Horizontal Flow
We first apply the chain rule to the matric potential gradient to obtain:
Reciprocal of the
SPECIFIC WATER CAPACITY Cw(q)
Cw(q) is the slope of the SWC curve at any particular water content.
Next we express K(h[q]) = K(q), and define a new hydraulic function, the Soil Water Diffusivity D(q) as:
The Soil Water Diffusivity is the ratio of unsaturated hydraulic conductivity to the specific water capacity.

51. Unsaturated Nonsteady Horizontal Flow
Soil Water Diffusivity D(q):
Since the K(q) and Cw(q) functions are highly nonlinear, the diffusivity is also nonlinear. However, the range of variation in D(q) is much smaller than the range of K(q) variation because the nonlinearity in numerator and denominator cancel out to some degree. The resulting water content or diffusivity form of the Richards equation for one dimensional horizontal flow is then:
The driving force is the gradient in water content
“water concentration”
This is not in accordance with the basic physical principle of flow occurring due to potential gradients !!
Special care is required when applying the diffusivity form of the Richards equation!!

52. Unsaturated Nonsteady Horizontal Flow

53. Unsaturated Nonsteady Horizontal Flow
Graphical presentation of the K(q), Cw(q), and D(q) functions
K(q)
CW(q)
D(q)

54. Boltzmann transformation for horizontal flow

55. ODE for Unsaturated Nonsteady Horizontal Flow
Analytical solution with constant diffusivity D(q)=D:

56. Diffusivity Form of the Richards Equation
Special care is required when applying the diffusivity form of the Richards equation!!
It should be applied only to homogeneous soils, where there is a unique relation between h and q, neglecting hysteresis, and to describe horizontal flow where there is no gravitational component.
What happens in layered soils?
We cannot apply diffusivity form of Richards equation!!
(the derivative q/x is not defined across the discontinuity)
Matric potential is continuous across layers
Water Content is discontinuous across layers

57. Diffusivity Form of the Richards Equation
The diffusivity form of Richards equation is applicable:
In homogenous soils for conditions where the vertical gravitational component is very small in comparison with matric influences (translated into a water content "gradient").
Examples:
Early phases of infiltration into relatively dry soil,
Evaporation of water from the soil surface.
For special cases the diffusivity may be assumed as constant over a small range of water content.
This facilitates analytical solutions of the Richards equation.

58. Parametric Forms of the Diffusivity Function
Diffusivity is defined as:
We can express K(q) with the van Genuchten – Mualem relationship and substitute with the VG model for Q:

59. Parametric Forms of the Diffusivity Function
Applying the same approach for the Brooks and Corey model yields:

60. Numerical Solutions of Richards Equation
The high degree of nonlinearity in the coefficients of the Richards equation for unsaturated flow makes it impractical to obtain analytical solutions.
It might be considered as one of the most challenging equations in environmental physics (considerably different than flow of heat or electricity).
Except for a few special cases, obtaining solutions to practical flow problems represented by the Richards equation is feasible by means of successive numerical approximations, generally using computers.

61. Numerical Solutions of Richards Equation
Numerical Approximations (Finite Difference Method) are based on:
Discretization of the flow domain into a grid: "layers" for one dimensional flow or a mesh for two-dimensional flow.
Discretization of the continuous partial differential equation into a series of approximated difference equations, one for each of the spatial nodes.
Application of initial and boundary conditions.
Formulation of the problem into a computer code which is able to solve the equations for the entire flow domain at various time steps.

62. Numerical Solutions of Richards Equation
Example: A Numerical Approximation to the Richards Equation
Find a numerical approximation to changes in water content with time under vertical unsaturated and nonsteady flow into a soil with a known initial water content, and known hydraulic functions q(y) and K(y).
Solution: We first discretize the soil profile (the spatial domain) into units of Dz, and the time into units of Dt. We denote the water content at the i-th depth increment and at the j-th time step as qij; similarly we express the corresponding matric head as hij (see graph previous slide) The hydraulic conductivity, which is a function of the matric head (or of the water content), relates transfer between two soil layers and thus must be averaged to represent both layers as: K([hij + hji+1]/2) = Kji+1/2. The resulting numerical approximation for Buckingham-Darcy’s law for vertical flow is then given by:

63. Numerical Solutions of Richards Equation
Example – continued:
Combining terms and rearranging allows us to solve for the only unknown in this equation, which is qij+1, the water content for the i-th depth increment at the next (future) time step:
Because this is a discrete approximation of a continuous process, increments Dz and Dt should be kept small. In addition, initial soil water contents with respect to depth and the conditions at the boundaries of the flow domain need to be specified.