Hydrology 2

Hydrology Notes

Volume of Voids

• Volume of voids over total volume is represented by n, where n > 0.
• This value is always positive.
• It relates to the movement of water within the voids of the soil.
• It considers the composition of water in the soil.
• Changes in height or depth are averaged to understand the overall change.

Green Ampt Method

• Illustrates the distribution of soil moisture during downward movement.
• Identifies moisture zones: saturated zone, transmission zone, wetting zone, and wetting front.
• These zones change over time.

Moisture Zones

• Saturation Zone: 100% saturation.
• Transmission Zone: Zone through which water moves.
• Wetting Zone: Soil area that is getting wet.
• Wetting Front: The boundary between the wet and dry soil.
• Transition Zone: Gradual change in moisture content.

Infiltration Rate and Cumulative Infiltration

• Infiltration rate (f) is the rate at which water enters the soil surface (in/hr or cm/hr).
• Potential infiltration rate occurs when water is ponded on the surface; the actual rate is less if no ponding occurs.
• Most equations describe potential infiltration rate.
• Cumulative infiltration (F(t)) is the accumulated depth of water infiltrated:
  F(t) = \int f(x) dx
• Infiltration rate is the time derivative of cumulative infiltration:
  f = \frac{dF}{dt}

Rainfall and Infiltration

• Rainfall hyetograph shows rainfall pattern as a function of time.
• Cumulative infiltration at time t is F(t); at time t + \Delta t is F(t + \Delta t).
• Increase in cumulative infiltration is F(t + \Delta t) - F(t).
• Rainfall excess is rainfall that is not retained or infiltrated.

Green and Ampt Simplified Picture of Infiltration

• Wetting front is a sharp boundary.
• Moisture content is \theta_i below the front and saturated soil with porosity n above.
• Wetting front penetrates to depth L in time t. Water is ponded to a small depth h_0.

Cumulative Depth of Water Infiltrated

• For a unit horizontal cross-sectional area, the increase in water stored is L(n - \theta_i).
• This increase equals F, the cumulative depth of water infiltrated:
  F(t) = L(n - \thetai) = L\Delta\theta where \Delta\theta = (n - \thetai).

Darcy's Law

• Darcy's Law is expressed as:
  q = -K \frac{dh}{dz}, where q is the Darcy flux, K is the hydraulic conductivity and dh/dz is the hydraulic gradient.

• The Darcy flux q is constant throughout the depth and is equal to -f, because q is positive upward while f is positive downward.

• Approximation of Darcy's law:
  f = K \frac{[h2 - h1]}{L}

• h1 at the surface is equal to the ponded depth h0. h_2 in dry soil below the wetting front equals - \psi - L (

• Darcy's law written as:
  f = K \frac{[h_0 - (-\psi - L)]}{L}

• If ponded depth h_0 is negligible:
  f = K \frac{\psi + L}{L}

• Wetting front depth is L = \frac{F}{\Delta\theta}; assuming h_0 = 0, substitution into equation gives:
  f = K \frac{\psi \Delta\theta + F}{F}

• Since f = \frac{dF}{dt}, equation can be expressed as:
  \frac{dF}{dt} = K \frac{\psi \Delta\theta + F}{F}

Green-Ampt Equation

• To solve for F, cross-multiply to obtain:
  F dF = K dt [F + \psi \Delta\theta]
  \frac{F + \psi \Delta\theta - \psi \Delta\theta}{F + \psi \Delta\theta} dF = K dt

• Integrate:
  \int \frac{F + \psi \Delta\theta - \psi \Delta\theta}{F + \psi \Delta\theta} dF = \int K dt

• Leads to:
  F(t) - \psi \Delta\theta { ln[F(t) + \psi \Delta\theta] - ln(\psi \Delta\theta) } = Kt

• Or:
  F(t) - \psi \Delta\theta ln \Big(1 + \frac{F(t)}{\psi \Delta\theta} \Big) = Kt

• This is the Green-Ampt equation for cumulative infiltration.

• Infiltration rate f can be obtained from:
  f = K \Big[ \frac{\psi \Delta\theta}{F(t)} + 1 \Big]

• When h0 is not negligible, substitute \psi + h0 for \psi in the equations.

• The equation is nonlinear and can be solved by successive substitution:
  F(t) = Kt + \psi \Delta\theta ln \Big(1 + \frac{F(t)}{\psi \Delta\theta} \Big)

Green-Ampt Infiltration Parameters

• Table provides Green-Ampt parameters for various soil classes.
• Includes porosity, effective porosity, wetting front soil suction head, and hydraulic conductivity.
• The numbers in parentheses represent one standard deviation around the parameter value.
• Effective porosity is defined as:
  \thetae = n - \thetar

Effective Saturation

• Effective saturation has the range 0 \leq Se \leq 1.0 provided \thetar \leq \theta \leq n
  \Delta \theta = - \frac{\psi}{1 + Se} \Delta\theta = (1 - Sr)\theta_e
• Solve for Se first.
• Constants yr and ye are obtained by draining a soil in stages, measuring the values of s_e and \psi at each stage, and fitting equation to the resulting data.

Sample Problem

• Evaluate infiltration rate and cumulative infiltration depth for a silty clay soil at 0.1-hr increments up to 6 hr.
• Assume an initial effective saturation of 20 percent and continuous ponding.

Solution Approach

1. Identify infiltration rate and infiltration depth per time t.
2. Start with time = 0.1 hr.
3. Use the value of Kt for F(t) as an initial value.
4. Iterate until the value of F(t) on the left equation is equivalent to the value of F(t) on the right equation.

Ponding Time

• Ponding time is the elapsed time between the time rainfall begins and the time water begins to pond on the soil surface.

• Develop an equation for ponding time under a constant rainfall intensity i, using the Green-Ampt infiltration equation.

• i=K \Big[ \frac{\psi \Delta\theta}{F(t_p)} + 1 \Big]

• Solving for tp: tp = \frac{K \psi \Delta\theta}{i(i-K)}

Other Infiltration Methods

• Φ-index: Constant rate of abstraction.

• Horton's equation: Empirical relation assuming infiltration begins at some rate and exponentially decreases until reaching a constant rate f_c.

  • Infiltration capacity:
    f=fc+(f0-f_c)e^{-kt}
  • Cumulative infiltration capacity:
    Ft=fc t + \frac{(f0 - fc)(1-e^{-kt})}{k}
    where k is a decay constant.

• Richard's equation:

• Philip used the Boltzmann transformation B(\theta) = t^{-1/2} to convert Richard's equation into an ordinary differential equation in B and solved to obtain an intimate series for cumulative infiltration F, approximated as
  F = St^{1/2} + Kt

  • Differentiating f(t)=\frac{df}{dt}, the infiltration rate is defined as
    f(t) = \frac{1}{2}St^{-1/2}
    where S is the sorptivity, a parameter that is a function of the soil suction potential, and K is the hydraulic conductivity. As t \rightarrow \infty , f(t) \rightarrow K. The two terms S and K represent the effects of soil suction head and gravity head.

Subsurface Flow: Groundwater

• Most rocks below a particular depth are saturated.
• At equal pressure (zero pressure), water flows towards lower elevation (downhill).
• At equal elevation, water flows towards lower pressure.
• Water flows at different rates through different materials: larger holes -> faster flow.

Advantages of Using Groundwater

• Much less subject to seasonal variations in availability than surface water.
• Slow movement leads to high biological purity.

Other Groundwater Concepts

• Capillarity: Movement of water due to a medium for upward movement; similar matter concentrates energy.
• Wells: Vertical structures for pumping water.
• Unsaturated Zone: Area above the water table.
• Saturated Zone: Area below the water table.
• Physical Weathering: Changes the size and shape of soil particles.
• Artesian Well: Pressurized well that freely flows water.

Groundwater Contamination

• Plumes are created when a contaminant contacts the aquifer.
• Contaminants can be released at the ground surface, into the unsaturated zone, or directly into the aquifer.

Processes Affecting Plume Movement

1. Advection: Moves compound into the aquifer.
2. Dispersion: Spreading of contaminants due to advection.
3. Retardation: Contaminants are held to the surface of aquifer solids.
4. Chemical Precipitation: Chemicals change form or are destroyed.
5. Biotransformation: Chemicals change form or are destroyed.

• Each compound is affected differently by these processes.
• Retardation factor: Ratio of groundwater movement to the movement of organic compounds.

Runoff and Streamflow

• Surface Runoff: Water that travels over the ground surface to a channel.

Peak Runoff Estimation by Rational Method

• Relates peak runoff to rainfall intensity.

• Assumes entire catchment contributing, uniform rainfall distribution, and all losses are in the coefficient.
  Q_p = CIA

  where C is the runoff coefficient, i is rainfall intensity (m/s), A is watershed area (m^2), and Q_p is peak runoff (cms).

Streamflow

• Streamflow is a measure of the water volume transported by a stream.

• Streamflow or discharge is the volume of water that moves through a specific point in a stream during a given period of time. Measured in cubic feet per second (cfs).

• Gaining Stream - stream gains water

• Losing Stream - stream loses water

Measurement of Streamflow

• The amount of water flowing in a river is called the discharge.
• Devices like stilling wells and weirs are used to measure discharge.
• Current meters measure river velocity.

Watershed

• Area draining to a stream.
• Streamflow is generated by water entering surface channels.
• Affected by physical, vegetative, climatic features, and geologic considerations.

Discharge Hydrographs

• Graph showing the flow rate as a function of time at a given location on a stream.
• An integral expression of the physiographic and climatic characteristics that govern the relations between rainfall and runoff of a particular drainage basin.
• Two types: annual hydrograph and storm hydrograph

Baseflow Separation Techniques

• Straight line method
• Fixed Base Method
• Variable Slope Method

Direct Runoff and Excess Rainfall

• Excess (effective) rainfall is rainfall that is not retained or infiltrated that becomes direct runoff.
• Excess rainfall hyetograph (excess rainfall vs time).

Abstraction Estimation: Phi Index Method

• Represents the difference between total and excess rainfall hyetographs, a constant rate of abstraction.
  \phi = \frac{a}{\Sigma(R_m - \Delta t)}

Example

• The text provides a step-by-step example to estimate the parameters needed for models.

Example - Abstraction Method

• The text provided a step-by-step example/ solution for abstraction method using: Basin area (A), Baseflow (straight line method), and volume of direct runoff

Runoff Coefficients

• The most common definition of a runoff coefficient is that it is the ratio of the peak rate of direct runoff to the average intensity of rainfall in a storm.

• A runoff coefficient can also be defined to be the ratio of runoff to rainfall over a given time period.

• Equation:
  C = \frac{\Sigma rd}{\Sigma Rm}

• With = 4.80 in and
  \Sigma R = 1.33 + 2.20 + 2.08 + 0.20 + 0.09 = 3.90 in then:

• C = \frac{4.80}{3.90} = 0.81$$