# 3.1 Equivalent Freshwater Head

To emphasize the location-dependence of the hydraulic head, pressure and density values, Lusczynski (1961) suggested adding a subscript *i* to the parameters in Equation 4 as shown in Equation 6.

(6) |

where:

h_{p,i} |
= | pressure head for a piezometer at location i containing a fluid column with density ρ_{i}, so it is the height of the fluid column above the z_{i} coordinate (L), e.g., m, which is typically taken as the center of the piezometer screen |

Lusczynski referred to this hydraulic head, *h*_{i}, as the *point water head*. The point water head is the hydraulic head that would be measured in a piezometer screened at point *i* and filled with groundwater having a density *ρ*_{i}. Groundwater often leaks into piezometers and observation wells along their length at unscreened locations, so the density of the standing water may not be the same as the groundwater at the screen (Post et al., 2018a; Post and von Asmuth, 2013), but for the discussion that follows, it is assumed that the water inside the piezometer has the same density as the groundwater at the screen. It is also useful to note that in systems with density variations caused by temperature differences, the density of the water inside of the piezometer will vary with depth because the stagnant water inside the standpipe achieves thermal equilibration with the surrounding rock, which gets warmer with increasing depth.

As shown in Figure 6, consider two piezometers completed at the same depth in an aquifer containing salt water, that sense the same pore water pressure (*p*_{i}). Figure 6a is a piezometer with a fluid column having the same density as the groundwater at the screen (*ρ*_{i}). Figure 6b is a piezometer that has a fictitious fluid column composed of fresh water (*ρ*_{f}). As illustrated by Equation 6, *h*_{p,i} = *p*_{i}_{ }*/**ρ*_{i}_{ }*g*, so the density of the water inside the piezometer determines the height to which the water rises. For a given groundwater pressure *p*, a taller water column is needed for fresh water than for salt water. Warm water requires a taller column than cold water. In freshwater systems, where the depth of investigation does not exceed a hundred meters or so below the surface, spatial density differences caused by temperature differences are usually small enough to be ignored. But in variable-density groundwater systems, such as coastal aquifers and near salt lakes, the relationship between *h*_{p} and *p* will not be the same for all piezometers.

**Figure ****6** – a) Point water head for a system with water density greater than freshwater density and b) equivalent freshwater head for the piezometer located at (a). The pressure and density at elevation *z*_{i} are *p*_{i} and *ρ*_{i}.

To compare measured hydraulic heads from different piezometers, they must all be referenced to the same density. The freshwater head *h*_{f} is the head that would be measured if the piezometer were filled with fresh water instead of groundwater with a higher density (Figure 6b). The choice of a reference density is arbitrary, but the density of fresh water is most commonly used. Analogous to the definition of hydraulic head, the freshwater head can be expressed as the sum of the elevation and the freshwater pressure head (Equation 7).

(7) |

In Equation 7, *ρ*_{f} is used to convert *p*_{i} to a length of freshwater column *h*_{p,f}. Rearranging Equation 6 results in Equation 8.

(8) |

Inserting Equation 8 into Equation 7 yields the relation between the point water head and the freshwater head (Equation 9).

(9) |

Since *h*_{f} − *z*_{i} = *h*_{p,f} and *h*_{i} − *z*_{i} = *h*_{p,i}, Equation 10 is valid.

(10) |

Equation 10 shows that fresh water would rise to a higher level in a piezometer if it replaced denser water (i.e., ). This result is consistent, as it should be, with the U-tube analogy in Figure 1.

With all hydraulic heads expressed as freshwater heads, care must still be taken to calculate the flow, as the freshwater head gradient does not indicate flow in the same way as the hydraulic head gradient in single-density systems. Consider for example the saltwater part of the aquifer in Figure 5 under the stagnant conditions at time *t* = 0. The equivalent freshwater head on the saltwater side is expressed by Equation 11.

(11) |

Because there is no flow, *h*_{i} is constant and Equation 11 can be differentiated with respect to *z* to obtain Equation 12.

(12) |

This outcome shows that there is a vertical freshwater head gradient in a stagnant body of saline groundwater. Based on the density values for seawater and fresh water, there is an increase of the freshwater head by 2.5 cm per meter of depth increase. It would therefore be a mistake to consider this value of the freshwater head gradient as an indication for vertical flow. One cannot expect to replace *h* with *h*_{f} in Equation 5 and expect to get the correct direction and magnitude of flow.

An implication that follows from this example is that in a variable-density system horizontal flow cannot be calculated from the horizontal freshwater head gradient if the piezometers are screened at different elevations. After all, the freshwater head increases with depth in a variable-density system even under stagnant conditions. The freshwater head gradient provides information about horizontal flow only if the screens are positioned on the same horizontal plane.

# Horizontal Flow

The correct use of the freshwater head gradient in flow calculations can be derived from Darcy’s law as given by Equation 2. To simplify the analysis, it is assumed that the permeability is homogeneous and isotropic so that *k* has no spatial or directional dependency and can be written as a scalar. Because the vector , which represents the Earth’s gravitational acceleration, has no components in the *x* and *y*-direction (i.e., it is zero in the horizontal plane), the horizontal components of the specific discharge vector , as defined by Equation 2, are shown by Equations 13 and 14.

(13) | ||

(14) |

Rearranging and differentiating Equation 7 with respect to* x *and* y *and inserting the result into Equations 13 and 14 yields the horizontal flow components as a function of the freshwater head gradient as shown in Equations 15 and16.

(15) | ||

(16) |

For many applications of practical interest , so it can be omitted, although this approximation is not likely to be accurate in geothermal systems with elevated temperature because water viscosity decreases by nearly a factor of 4 from 20 °C to 100 °C. The parameter is the freshwater hydraulic conductivity, for which deviation of a few percent from the hydraulic conductivity at the ambient groundwater density is an acceptable simplification given the large uncertainty about the magnitude of *K* in a field setting.

# Vertical Flow

Recalling, once again, that the positive *z*-direction is taken upward (against gravity), the component of the specific discharge vector in the vertical direction is given by Equation 17, where *q*_{z} is positive in the + *z* direction.

(17) |

For flow in the vertical direction, Equation 7 can be rearranged and differentiated with respect to *z*. Inserting the result into Equation 17 yields Equation 18.

(18) |

The term between the parentheses is the sum of the freshwater head gradient and a buoyancy term . It is useful to note that in Equation 18 differs from that of Equation 12 which applied to a stagnant system where *q*_{z} = 0. The practical application of Equation 18 can be illustrated by the example (depicted in Figure 7 and Table 1) of a freshwater aquifer that overlies a saltwater aquifer and is separated from it by an aquitard. Flow across the aquitard is upward. The vertical flow rate is rather small (0.1 m yr^{–}^{1}) and lateral groundwater flow along the base of the freshwater aquifer is sufficiently high to keep the groundwater fresh. The results shown in the figure were calculated using a numerical model built into SEAWAT (Langevin and Guo, 2006), so this is a synthetic case, but it serves to illustrate some important points. First, it is useful to note how the point water head varies vertically just below the top of the aquitard: it decreases with depth in the first meter or so. Without knowledge of the groundwater density, taking the hydraulic head gradient at face value as an indicator of the flow direction would lead to the incorrect conclusion that flow was downward in the aquitard. Next, it is useful to note that the vertical point water head gradient in the saltwater aquifer is notably smaller than in the aquitard because of the aquifer’s higher hydraulic conductivity (*K*_{f}* *= 1 m d^{−1} versus *K*_{f} = 0.01 m d^{−}^{1}). The freshwater head increases with depth at all depths, but at a lower rate in the lower aquifer than in the aquitard. This example once more shows that a simple comparison of freshwater or point-water heads in two wells may not provide the correct direction of vertical flow.

z_{i} |
h_{i} |
ρ_{i} |
h_{f} |

m | m | kg m^{−3} |
m |

0 | 2 | 1000 | 2 |

−5 | 2.06 | 1024.7 | 2.24 |

Table 1 lists data required to do the hydraulic head corrections for two piezometers, one that is screened at the base of the freshwater aquifer and the other screened directly below the aquitard. The aquitard is five meters thick. The freshwater heads were calculated using Equation 9. The specific discharge *q*_{z} can be calculated using the difference form of Equation 18 to obtain Equation 19.

(19) |

The difficulty is that the density *ρ* in the buoyancy term is the mean density over the interval Δ*z*, which has been emphasized by replacing *ρ* from Equation 18 with *ρ*_{mean} in Equation 19. For this synthetic example, *ρ*_{mean} is known as it can be calculated from the given density distribution with depth (). The green line in Figure 7 indicates that over the interval 0 to −5 m, *ρ*_{mean }= 1019.5 kg m^{–}^{3}, so *q*_{z} can be calculated as follows.

Negative elevations are used in the calculation of the freshwater head gradient and *q*_{z} is a positive number because the flow is directed upward.

More often than not, the mean density distribution between two piezometer screens is not known, and the mean density is typically approximated by the mean of the densities of the two measurement points. In this case, this would yield = 1012.4 kg m^{−3}, which would give *q*_{z} = 3.6 × 10^{−4} m d^{−1}. The estimated flux is in error by ~20 percent, but it could be argued that such an error dwarfs in comparison to the uncertainty of *K*_{f}. Post and others (2007) recommended taking the uncertainty of *ρ*_{mean} into account by considering the conceivable end members of the density distribution between the points of measurement, and Post and others (2018a) used borehole resistivity logs to determine *ρ*_{mean}*.*

Lusczynski (1961) introduced the concept of environmental head to facilitate the determination of vertical flow, which is indicated by the environmental head gradient. Despite this advantage, determining how to calculate the environmental head is not always trivial and its application has sometimes led to confusion. Readers may find more information about the environmental head in the original Lusczynski (1961) paper as well as in Post and others (2007). Guo and Langevin (2002) derived versions of Equations 15 through 18 for the general case when gravity is not perpendicular to the *x*–*y* plane. Use of these alternative equations is required when the coordinate system needs to be aligned with principal directions of permeability that are askew to the horizontal and vertical axes. Density effects in sloping aquifers further complicate the analysis and are discussed in Bachu (1995) and Bachu and Michael (2002). Moreover, the theoretical framework presented in this section can only be practically applied when the density field is in a steady state. If the fluid densities change in time, as in the examples of variable-density flow presented later in this book, numerical models can assist in the interpretation of flow and transport processes as discussed in Section 6.