Surface and Groundwater Hydrology

Darcy's law and seepage velocity

Darcy's law states that the discharge through a porous medium is proportional to the hydraulic gradient, Q = KAi, where K is the hydraulic conductivity in m/s, A is the cross-sectional area, and i = \Delta h / L is the gradient. The Darcy velocity (specific discharge) is q = Ki. It is an apparent velocity averaged over the whole cross-section, including the solid grains. Water moves only through the pore space, so the true average velocity is larger. This is the seepage velocity, v_s = q/n = Ki/n, where n is the porosity. When the porosity, gradient and seepage velocity are known, the conductivity follows from K = v_s n / i. This relation is the key step in such problems. Under advection, a conservative solute crossing a length L therefore takes t = L/v_s to travel that distance. The hydraulic conductivity is also linked to the intrinsic permeability k by K = kg/\nu, where \nu is the kinematic viscosity. For the same porous medium and the same gradient, the discharge is inversely proportional to \nu, so a more viscous fluid moves more slowly.

Phi-index infiltration loss estimation

The \phi-index is a constant loss rate, expressed in cm/h or mm/h, that is assumed to remove water from the storm at all times when the rainfall intensity exceeds it. Rainfall excess is computed interval by interval. In an interval where the intensity i is greater than \phi, the excess depth is (i - \phi)\Delta t. In an interval where i \le \phi, all the rain is lost and the excess is zero. The unknown \phi is found by trial. One chooses the value for which the total losses equal the total rainfall minus the observed direct runoff. For a catchment split into parts with different losses, the excess is computed for each part and then weighted by the fraction of area. When an initial loss is also mentioned, one must check whether the question intends it to be deducted before the \phi calculation. The answer key of this problem computes the excess from the intensities and the \phi value alone, so the stated initial loss is not subtracted there.

Transmissivity and hydraulic conductivity

Hydraulic conductivity K is the flow per unit area under unit gradient, with units of m/s or m/day. Transmissivity T is the flow per unit width of the full saturated thickness of an aquifer under unit gradient. Its units are \text{m}^2/\text{day} or \text{m}^2/\text{s}. The two are linked by T = Kb, where b is the saturated thickness, as a plain product of conductivity and depth. Layers in a stratified aquifer combine differently depending on flow direction. For flow parallel to the bedding, the layers act as parallel conductors, and the equivalent horizontal conductivity is the thickness-weighted arithmetic mean K_x = \sum K_i h_i / \sum h_i. Flow perpendicular to the bedding gives the thickness-weighted harmonic mean instead. The two must not be interchanged.

Flood routing methods

Flood routing predicts how a flood wave changes as it moves through a reservoir or a channel. Hydrologic routing uses the continuity equation together with an empirical storage–discharge relation. It does not use the momentum equation. Examples are the Goodrich and Puls methods for reservoirs and the Muskingum method for channels. Hydraulic routing solves the continuity equation together with the dynamic (momentum) equation. These are the St. Venant equations, which describe unsteady gradually varied flow in channels. The energy equation is not the basis of either class of method. A common mistake is to say that hydrologic routing uses momentum, or that hydraulic routing uses energy.

Unit hydrograph

A unit hydrograph is the direct runoff hydrograph produced by one unit (usually 1 cm) of rainfall excess, spread uniformly over the catchment and falling in a specified duration D. Its volume must equal the depth of excess times the catchment area, with the area converted to \text{m}^2, so V = R \times A. For a triangular unit hydrograph with peak Q_p and base time T_b, the volume is V = \tfrac{1}{2} Q_p T_b. Hence Q_p = 2V/T_b, with time in seconds. An S-hydrograph is obtained by summing a series of unit hydrographs of duration D displaced by D hours. Its equilibrium discharge equals the rate of rainfall excess supply, Q_e = AR/D, where R is the excess depth. For R = 1 cm, Q_e = 2.78\,A/D in m^3/s when A is in km^2 and D is in hours.

Dissolved oxygen sag in rivers

The Streeter–Phelps model describes the dissolved oxygen (DO) deficit D = C_s - C downstream of a waste discharge. Its basic assumptions are first-order deoxygenation, in which the deoxygenation rate is directly proportional to the oxidizable organic material L remaining, r_d = k_d L, and reoxygenation that is directly proportional to the DO deficit, r_r = k_2 D. The deoxygenation rate therefore increases with L, not inversely. The reaeration term depends on the deficit D, and it is not simply proportional to the saturation concentration C_s. The self-purification capacity of a river is governed by the ratio of the reaeration rate constant to the deoxygenation rate constant, k_2/k_d. A higher ratio means the river recovers its oxygen more effectively. A high reaeration constant also indicates greater turbulence or velocity, while a high deoxygenation constant indicates that the pollutants are more biodegradable.