# Box 3 Foundation for Understanding Hydraulic Head and Force Potentials

**David B. McWhorter, Eileen P. Poeter, William W. Woessner**

As illustrated in the main body of this book, Darcy’s Law indicates the rate of groundwater flow is proportional to the gradient of a scalar quantity called hydraulic head as shown again here in Equation Box 3-1. A scalar quantity is one that can be represented by a single number describing magnitude (as opposed to a vector that requires two numbers, magnitude and direction (an angle)). Scalar quantities include entities such as pressure, concentration and temperature; while velocity is an example of a vector quantity.

(Box 3-1) |

where:

q |
= | specific discharge (L/T) |

Q |
= | volumetric flow rate (L^{3}/T) |

A |
= | cross-section area of flow perpendicular to the direction of flow (L^{2}) |

K |
= | hydraulic conductivity (L/T) |

= | derivative of hydraulic head in direction L (dimensionless) |

Science is rife with instances in which the rate of flow of some quantity of interest is expressed in this form. Heat conductance in response to a temperature gradient, flow of electricity in response to differences in voltage, and diffusion of a solute due to a concentration gradient are familiar examples. The simple proportionality of such flux laws often mask the deeper connection between the scalar quantity and the factors responsible for the flux under consideration. Hubbert (1940), in his classic treatise on groundwater flow, makes this point in reference to hydraulic head by stating that *“to adopt it without further investigation would be like reading the length of a mercury column of a thermometer without knowing that temperature is the physical quantity being indicated”*. Hubbert went on to show that hydraulic head is a contribution to the total mechanical energy of the water, a quantity he referred to as *“fluid potential”*. Water flow from high to low hydraulic head is tantamount to flow from high to low mechanical energy. Hubbert points out that mechanical energy is converted to thermal energy by the friction generated as the viscous fluid flows through the interstitial pore space. But is hydraulic head a physically meaningful concept under all circumstances? If not, is there an alternative way to calculate groundwater flow when hydraulic head is not a meaningful quantity? These questions can be answered by an analysis that begins with a consideration of the mechanical forces acting on groundwater.

Consider flow along a curved path as would occur when groundwater passes under an obstruction as depicted in Figure 77 in the main body of this book. Forces acting on flowing groundwater fall into one of two categories: 1) those that resist the fluid motion and 2) those that drive the fluid motion. The dominant resistive force is the drag on the fluid that results from viscous shear as the fluid makes its way through the tiny openings among the solid grains. A second resistive force, almost always negligible, is an inertial force that occurs because the water is following a curved path (convective acceleration in fluid mechanics). Inertial forces act on the groundwater when the flow is non-uniform (as in Figure 77) or unsteady, or both. Viscous and inertial forces only exist when the groundwater is in motion. Groundwater motion is caused by the driving forces. The driving forces are related to hydraulic head.

**Force on Groundwater Due to Pressure Difference**

Variable water pressure and gravity are the fundamental forces that drive flow through the porous medium in all common circumstances of groundwater occurrence. These forces are the same as the forces that drive flow in a pressurized pipe and in a stream channel. Although fluid motion results from the combined effect of these forces, it is useful to consider the individual effects of each force before combining them. We consider the force due to variable pressure first.

Water contained within the volume element of porous medium depicted in Figure Box 3-1 experiences a force due to the difference in pressure acting on either end at points 1 and 2. The pressure at point 2 exerts a force on the water in the element equal to the product of pressure, *p*_{2}, and the area of exposed water, *nA*, where *n* is porosity and *A* is the area of the base of the cylinder of the volume element at point 2. Similarly, the pressure at point 1 exerts the force *nAp*_{1}. These forces are indicated by the bold red arrows in Figure Box 3-1.

The difference between these two forces is the driving force on the water in the coordinate direction *L* due to variable water pressure (Equation Box 3-2). By convention, forces exerted in positive coordinate directions are considered positive, so the negative sign in Equation Box 3-2 assures compliance with that convention.

Driving force due to pressure difference = = |
(Box 3-2) |

where:

p_{i} |
= | water pressure at location i (F/L^{2}) |

n |
= | porosity (dimensionless) |

A |
= | cross-sectional area of volume element (L^{2}) |

Δp |
= | difference in water pressure between points 1 and 2 (F/L^{2}) |

ΔL |
= | distance between points 1 and 2 (L) |

Divide the force in Equation Box 3-2 by the volume of water (*A* ∆*L* *n*) to obtain the force per unit volume of fluid, *f*_{p}, due to the pressure difference Equation Box 3-3 expresses the result in derivative form.

(Box 3-3) |

The derivative in Equation Box 3-3 evaluates the rate of change of pressure in the direction of *L* (from point 1 to point 2) and is called a directional derivative.

**Forces on Groundwater Due to Gravity**

It was natural and convenient to adopt a unit volume of water as our reference when considering the force resulting from spatial variation of pressure. This force is a surface force and can be computed without any need to know the mass of the reference element. In contrast, gravitational force on groundwater is a body force that is proportional to mass, and it is natural to adopt a unit mass for our reference entity in this case. However, forces referenced to different entities cannot be added or compared, so a common reference entity must be selected. We elect to express the force on groundwater due to gravity as a force per unit volume of water so it can be added directly to the force expressed in Equation Box 3-3. As demonstrated in a later paragraph, it is a simple matter to convert force per unit volume to force per unit weight or mass of fluid.

Throughout this book, *z* is the vertical coordinate measured as a positive value upward from some convenient datum as in Figure Box 3-2. Gravitational force acts vertically downward, so is considered negative in accordance with convention. Thus, the gravitational force per unit volume of fluid is –*ρg*, where *ρ* is the fluid mass density (mass per unit volume) and *g* is the gravitational constant. Note that this is simply the familiar weight per unit volume (also called unit or specific weight) with dimensions of force per unit volume. The component of this vertical force acting in the coordinate direction *L* is shown in Figure Box 3-2 and is calculated by Equation Box 3-4.

(Box 3-4) |

where:

f_{g} |
= | component of force per unit volume of fluid due to gravity acting along L coordinate (F/L^{3}) |

ρ |
= | fluid density (M/L^{3}) |

g |
= | gravitational constant (acceleration of gravity) (L/T^{2}) |

z |
= | elevation above a horizontal datum (L) |

**Net Driving Force**

The forces in Equations Box 3-3 and Box 3-4 are added to arrive at the net driving force, *F*_{v}, exerted in the arbitrary direction, *L*, on a unit volume of groundwater as shown in Equation Box 3-5.

(Box 3-5) |

where:

F_{v} |
= | driving force per unit volume of fluid (F/L^{3}) |

The subscript, *v*, denotes volume as the reference entity. The individual forces appearing in Equation Box 3-5 may point in the same or opposite directions along *L*. The net driving force is zero when the individual forces are equal in magnitude but point in opposite directions. In that case, there is no flow and Equation Box 3-5 becomes Equation Box 3-6.

(Box 3-6) |

This is the familiar equation of hydrostatics and is the same equation one would use to compute the pressure in any column of standing water (e.g., a reservoir or standpipe).

It is a simple matter to express the net driving force in reference to either unit mass or unit weight. To obtain the net driving force per unit mass of fluid, divide Equation Box3-5 by the mass density to obtain Equation Box 3-7:

(Box 3-7) |

where:

F_{m} |
= | driving force per unit mass of fluid (F/M) |

Equation Box 3-8 calculates the force per unit weight of fluid and is obtained by dividing Equation Box 3-5 by the unit weight, *ρg*.

(Box 3-8) |

where:

F_{w} |
= | driving force per unit weight of fluid (dimensionless) |

Few restrictions were invoked to enable the development of these expressions for the forces that drive the motion of groundwater. In particular, these expressions can vary and still remain applicable. The force equations remain valid for groundwater of variable density due to fluid compressibility, non-uniform solute concentration or non-uniform temperature. Furthermore, these results apply to a wide variety of other liquids (e.g., petroleum products, solvents) and even to gases.

**Force Potentials and Hydraulic Head**

A *force potential* associated with a particular force field is a scalar function, the negative gradient of which is the force in question. The word potential in the present context is related to the familiar concept of potential energy. The common example is the potential energy of a fluid particle (e.g., a unit mass) that is dependent upon the position of the particle in the gravitational force field. But a fluid particle may also possess potential energy owing to its position in force fields other than the gravitational field. There is potential energy associated with the convective force field that exists in non-uniform flow and the force field due to variable pressure, for example.

The potential energy possessed by a fluid particle at a particular point is the work done on the particle by the force field when the particle is repositioned from a reference point to the point in question. This is a meaningful definition only if the work done on the particle by the force field is the same for every path that can be taken in route from the reference point to the point in question. Otherwise, the value of potential energy would not be unique.

The work done on a fluid particle when it is moved a differential distance, *dL*, in the direction of a force *F*, is the differential work, *FdL*. Integration of the differential work between the reference point and any point of interest is the potential energy at the point of interest, provided the integration is independent of the path taken between the points. This condition is satisfied if the differential work is an exact differential (see any elementary calculus book or on the web at Wolfram MathWorld for more information on exact differentials). *If the fluid density is constant, the differential work is an exact differential for each of the three net driving forces in Equations Box 3-5, Box 3-7, and Box 3-8.*

For example, consider the net driving force per unit weight of fluid as expressed in Equation Box 3-8. Because both *ρ* and *g* are constant, their product can be placed inside the differential operator, *d*, to obtain Equation Box 3-9.

(Box 3-9) |

where:

F_{w}dL |
= | Differential work on a unit weight (FL) |

Equation Box 3-9 can be integrated up to an arbitrary constant of integration that is set to zero for convenience to provide an expression for the potential energy per unit weight of fluid, also known as hydraulic head shown in Equation Box 3-10.

(Box 3-10) |

where:

h |
= | hydraulic head (L) |

Thus, we see that hydraulic head at a point can be viewed as the potential energy per unit weight of fluid. It is, also, a force potential in the sense that the negative gradient of hydraulic head is the driving force on the fluid per unit weight. Force potentials, particularly hydraulic head, are of more than academic interest. Hydraulic head is determined by a single measurement of water level relative to an arbitrary datum, a valuable attribute in groundwater field work (Figure 19 in the main body of book). Further, the negative derivative of hydraulic head in any direction evaluates the net driving force in that direction such that when *h* is constant throughout the system, flow is zero (i.e., there is no flow). None of this requires explicit consideration of the individual forces. In addition, the validity of using hydraulic head to determine groundwater flow does not require negligible inertial forces. For these reasons, hydraulic head enjoys an exalted status among the potentials relevant to groundwater flow. However, use of hydraulic head to evaluate flow systems requires that the fluid has constant density, even though the basic force equations that led up to the calculation of hydraulic head remain valid for fluids of variable density.

**What If the Density Is Variable?**

Hydraulic head is not a meaningful concept when the fluid density varies due to variable solute concentration or temperature. However, Equations Box 3-5, Box 3-7, and Box 3-8 remain valid and provide a means of calculating specific discharge in such cases. In particular Equation Box 3-1, valid for water of constant density, can be replaced by Equation Box 3-11 for cases of variable density.

(Box 3-11) |

Equation Box 3-11 is obtained by using Equation Box 3-8 for the driving force and provides a valid starting point for the analysis of the complex problems of variable density.