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EAS 44600 Groundwater Hydrology Lecture 5: Hydraulic Head and Fluid Potential Dr. Pengfei Zhang Darcy’s Experiment The first studies of the movement of water through rock and sediment were performed by the French engineer Henry Darcy. Darcy used a column filled with sand to perform the experiments. The column had an inlet (influent) and an outlet (effluent) through which water was introduced and released from the column. The column also had two ports connected to two vertical tubes, called piezometers. The water level in the tubes rose to heights that reflected the energy of the water in the pipe at the location of the port (Figure 5-1). ha ha-hb hb A B L Figure 5-1. Horizontal pipe filled with sand to demonstrate Darcy’s experiment. Darcy found that the discharge of water, Q, through the pipe was proportional to the difference in the height of the water in the two “piezometers”, and was inversely proportional to the flow length, L. The flow was also proportional to the cross-sectional area, A, of the pipe, and was proportional to the ability of the sand to transmit water. This ability to transmit (or conduct) water is referred to as the hydraulic conductivity, K, of the sand. These observations were summed up as Darcy’s Law: h −h dh Q = − KA a b = − KA L dl (5-1) where dh/dl is known as the hydraulic gradient. To discuss hydraulic gradient, one must introduce the concept of hydraulic head. The hydraulic head is a means of expressing the potential energy of a unit weight of water. The total potential 5-1 energy in a unit weight of groundwater consists of both gravitational and pressure components. Another way of stating this is that the potential energy of the water is derived from both the elevation of the water above some datum, as well as the pressure exerted on the water by neighboring water. Datum refers to an arbitrary elevation from which the elevations of the water are measured (e.g., sea level). Work is done in order to lift a mass of fluid from a datum to an elevation of z. Recall that work (W) or energy is a force (F) multiplied by a distance: W = Fz = ( m ⋅ g ) ⋅ z (5-2) where the force is due to gravity, which is the product of the mass of the fluid (m) and the gravitational constant (g). Once lifted, the mass of fluid acquired energy equal to the work done in lifting the fluid. This is the gravitational potential energy (Eg): Eg = W = m ⋅ g ⋅ z (5-3) It is convenient to express this potential energy per unit volume of fluid, recognizing that the mass of the fluid is equal to its density (ρ) multiplied by volume (V): Eg volume = m ⋅ g ⋅ z (ρ ⋅V ) ⋅ g ⋅ z = = ρ⋅g⋅z V V (5-4) The above discussion on elevation head boils down to the statement that water flows downhill. However, groundwater flow is not so simple. Take for example Darcy’s experiment. In this experiment, a pump was used to force water through the column of sand (Figure 5-2). In this case, there is a pressure component to hydraulic head, as indicated by a column of water above the sand column in a piezometer (Figure 5-2). PUMP datum Figure 5-2. Schematic diagram showing a column with a pressure head. 5-2 A piezometer allows us to determine the pressure of the water inside the column where the piezometer is tapped (screened at the center of the column). The higher the pressure in the column, the higher the level of water in the piezometer will be. For a given pressure, the actual height of water achieved in a piezometer depends on the diameter of the piezometer, since pressure is force per area: pressure = force F m ⋅ g = = area A A (5-5) where m is the mass of water in the piezometer, and A is the cross-sectional area of the piezometer. Since m=ρV, the pressure can be expressed as: pressure = (ρ ⋅V ) ⋅ g ρ ⋅ ( A ⋅ hp ) ⋅ g = = ρ ⋅ g ⋅ hp A A (5-6) where hp is the height of water in the piezometer. Since the pressure of water in the piezometer reflects the pressure in the actual column, the pressure potential energy of the water in the piezometer also reflects the pressure potential energy in the column. Since energy is a force applied over a distance, the pressure potential energy (Ep) may be expressed as: E p = force ⋅ distance = ( pressure ⋅ A) ⋅ h p = ρ ⋅ g ⋅h p ⋅ A ⋅ h p (5-7) Notice that the product of A and hp is the volume of water in the piezometer, and that the pressure potential energy per unit volume in the piezometer is equal to the pressure potential energy of the water at that location in the column. Therefore, we have the following relationship: Ep volume = ρ ⋅ g ⋅ hp (5-8) Now we can sum up the elevation and pressure potential energies per unit volume: Eg Ep total potential energy = + = ρgz + ρgh p volume volume volume (5-9) The hydraulic head (h) is the means of expressing the potential energy of water per unit weight. Since the density and gravitational constant (ρ and g) are common to both terms in equation 5-9, these can be moved to the left hand side to give the hydraulic head: h= total potential energy 1 total potential energy ⋅ = = z + hp ρg volume weight (5-10) where z is the elevation above some datum, and hp is the height of water in the piezometer (the pressure that the fluid is under). 5-3 In deriving the above expression for hydraulic head we have ignored the fact that moving fluid tends to remain in motion due to its kinetic energy (Ek). This energy is equal to one-half the product of the fluid mass (m) and the square of the fluid velocity (v): Ek = 1 2 mv 2 (5-11) In groundwater, the fluid velocities are sufficiently slow that the kinetic energy contribution to hydraulic head is negligible. Groundwater does not necessarily flow downhill; it flows from areas of higher hydraulic head to areas of lower hydraulic head. The hydraulic head takes into account both elevation and pressure effects on the energy of water. 5-4