Ghyben-Herzberg equation

The Ghyben-Herzberg equation , also known as the Ghijben-Herzberg equation , is an equation in hydrogeology that describes how sweet groundwater and salty seawater relate to one another on islands or near the sea. The Ghyben-Herzberg equation is used, for example, to estimate the dimensions of freshwater lenses on islands or to prevent saltwater intrusions when building wells near the sea.

The equation is based on Archimedes' principle and was published in 1888 by Willem Badon Ghyben (1845-1907) and in 1901 by Alexander Herzberg (1841-1912).

Basics

Fresh water and salt water differ in their salinity and therefore also have different densities . The density of fresh water, depending on the temperature, is approx.

${\displaystyle \rho _{\text{süß}}={\text{0,99 kg/l ... 1,00 kg/l}}}$,

while the density of sea ​​water depends on the salinity and temperature

${\displaystyle \rho _{\text{Salz}}={\text{1,02 kg/l ... 1,03 kg/l}}}$ lies.

If there is a body of fresh water in the subsurface , it will float on the underlying seawater groundwater of higher density. The surface of the body of fresh water rises and is thus raised above the surface of the sea water.

A similar effect occurs - much more strongly - with icebergs : These typically consist of freshwater ice with a density of around 0.92 kg / l, so they swim in the surrounding seawater that has a greater density of around 1.03 kg / l. The top of an iceberg therefore rises - about a ninth in volume due to an 11 percent difference in density - above the seawater surface all around. Ice floes in a freshwater lake with a density of around 1.00 kg / l are almost the same, but only by 8% of its volume.

The Ghyben-Herzberg equation now relates the height of the fresh water body above the salt water body to how deep the fresh water body lies in the salt water body. Applied to the iceberg, this corresponds to determining the size of the iceberg underwater by measuring its size above water. The prerequisite for this is knowledge of the density of the iceberg and sea water.

formulation

The groundwater situation in the Ghyben-Herzberg equation. That corresponds to the one in the equation, that corresponds to the . ${\displaystyle z}$ ${\displaystyle h_{\text{uMS}}}$ ${\displaystyle h}$ ${\displaystyle h_{\text{oMS}}}$

The Ghyben-Herzberg equation is:

${\displaystyle h_{\text{uMS}}={\frac {\rho _{\text{Süß}}}{\rho _{\text{Salz}}-\rho _{\text{Süß}}}}h_{\text{oMS}}}$.

Here is

• ${\displaystyle h_{\text{uMS}}}$the position of the fresh water below the sea level (UMS) in meters, that is the vertical distance from the salt water surface level to the interface of the fresh water and salt water
• ${\displaystyle h_{\text{oMS}}}$ the height of the fresh water above sea level (oMS) in meters, i.e. the vertical distance from the top of the fresh water to the salt water level
• ${\displaystyle \rho _{\text{Süß}}}$ the density of fresh water
• ${\displaystyle \rho _{\text{Salz}}}$ the density of the salt water.

In part, there is only the relationship that applies to normal saline salt water and fresh water

${\displaystyle h_{\text{uMS}}\approx 38\cdot h_{\text{oMS}}}$

referred to as the Ghyben-Herzberg equation. This is further derived below.

Calculation example

The mean density of fresh water is one kilogram per liter, so is

${\displaystyle \rho _{\text{Süß}}=1\,{\text{g/cm}}^{3}}$.

The mean density of sea water is approx.

${\displaystyle \rho _{\text{Salz}}={\text{1,025 g/cm}}^{3}}$.

Inserting it into the formula gives:

${\displaystyle {\frac {\rho _{\text{Süß}}}{\rho _{\text{Salz}}-\rho _{\text{Süß}}}}={\frac {1}{1{,}025-1}}={\frac {1}{0{,}025}}=40}$

Thus, under the common conditions of fresh water and salt water:

${\displaystyle h_{\text{uMS}}\approx 40\cdot h_{\text{oMS}}}$,

depending on the assumed salinity, this value can fluctuate slightly. Every meter of fresh water above sea level corresponds to approximately 40 meters of fresh water below sea level.

Is z. B. If the groundwater level is encountered at a height of three meters above sea level when drilling near the sea, the fresh water is down to below sea level. The total thickness of the fresh water is thus . ${\displaystyle 40\cdot 3\,{\text{m}}=120\,{\text{m}}}$ ${\displaystyle 3\,{\text{m}}+120\,{\text{m}}=123\,{\text{m}}}$

With high salinity

In the dead sea , the salinity is much higher than in the open sea, the density there is up to , ${\displaystyle \rho _{\text{Salz}}=1{,}240\,{\text{g/cm}}^{3}}$

thus results

${\displaystyle {\frac {\rho _{\text{Süß}}}{\rho _{\text{Salz}}-\rho _{\text{Süß}}}}={\frac {1}{1{,}240-1}}={\frac {1}{0{,}240}}\approx 4{,}15}$

Thus applies there

${\displaystyle h_{\text{uMS}}\approx 4{,}15\,h_{\text{oMS}}}$,

every meter above sea level there corresponds to a little more than four meters of fresh water.

If fresh water were to be found three meters above sea level when drilling, this would correspond to a fresh water thickness of only below sea level and thus a total fresh water thickness of . ${\displaystyle 4{,}15\cdot 3\,{\text{m}}=12{,}45\,{\text{m}}}$ ${\displaystyle 3\,{\text{m}}+12{,}45\,{\text{m}}=15{,}45\,{\text{m}}}$

literature

• Willem Badon Ghyben: Nota in verband met de vorrgenomen put boring nabij Amsterdam. 1889.
• Alexander Herzberg: The water supply for some North Sea baths. 1901.

Individual evidence

1. ↑ ^{a b c} Georg Mattheß, Károl Ubell: General hydrogeology, groundwater balance . In: Georg Mattheß (Hrsg.): Textbook of Hydrogeology . 2nd Edition. tape 1 . Bornträger Brothers, Berlin / Stuttgart 2003, ISBN 3-443-01049-0 .
2. ^ Bernward Hölting, Wilhelm Georg Coldewey: Hydrogeology . Introduction to General and Applied Hydrogeology. 8th edition. Springer-Verlag, Berlin / Heidelberg 2013, ISBN 978-3-8274-2353-5 , pp. 314-315 , doi : 10.1007 / 978-3-8274-2354-2 .