{"id":123,"date":"2022-12-30T18:21:09","date_gmt":"2022-12-30T18:21:09","guid":{"rendered":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/?post_type=part&p=123"},"modified":"2023-01-09T00:23:31","modified_gmt":"2023-01-09T00:23:31","slug":"quantifying-groundwater-flow-in-the-presence-of-density-variations","status":"publish","type":"part","link":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/part\/quantifying-groundwater-flow-in-the-presence-of-density-variations\/","title":{"raw":"3 Quantifying Groundwater Flow in The Presence of Density Variations","rendered":"3 Quantifying Groundwater Flow in The Presence of Density Variations"},"content":{"raw":"
\r\n

The description of groundwater flow in variable-density systems is considerably more complex than in constant-density systems. While field measurements of hydraulic head conveniently allow for an assessment of groundwater flow direction and magnitude for constant-density systems, hydraulic head measurements cannot be used for flow calculations without making allowance for buoyancy effects in variable-density systems. Without citing anyone or any literature, it was casually remarked in a Dutch report (DZRD, 1936) that \u201cIt is known that the measured head of water with a high chloride content requires a correction<\/em>\u201d [paraphrased translation from Dutch by the authors]. Nevertheless, the use of hydraulic head measurements in a variable-density groundwater system still leads to confusion or even misinterpretation to this day.<\/p>\r\n

Some basic intuition about the effect of density on the flow field can be gained by considering a confined aquifer with an impermeable base in which fresh water and salt water are separated by a vertical interface at time zero (Figure\u00a05a).<\/a><\/p>\r\n

\"Figure<\/p>\r\n

Figure\u00a0<\/strong>5<\/strong>\u00a0-\u00a0The effect of density on the flow field in a confined aquifer. a) Fresh and salt water are at rest and separated by a vertical interface at time zero. b) Rotating flow (clockwise) established under initially hydrostatic conditions. The base of the triangle in a) is much wider for the salt water than for the fresh water, to indicate that the freshwater pressure p<\/em>f<\/em><\/sub> at the aquifer bottom is greater than the saltwater pressure p<\/em>s<\/em><\/sub> because \u03c1<\/em>s<\/em><\/sub>\u00a0>\u00a0\u03c1<\/em>f<\/em><\/sub>. (After Santing, 1980).<\/p>\r\n

All the system boundaries are closed, so there can be no inflow or outflow of water. Under the prevailing hydrostatic conditions at time t<\/em> =\u00a00, the groundwater pressure p <\/span><\/em>increases with depth according to Equation\u00a01.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle p=p_{0}-\\rho gz[\/latex]<\/td>\r\n(1)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

where:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
p<\/em><\/td>\r\n=<\/td>\r\npressure (M\/(LT2<\/sup>)), e.g., kg\u00a0m\u22121<\/sup>\u00a0s\u22122<\/sup><\/td>\r\n<\/tr>\r\n
p<\/em>0<\/sub><\/td>\r\n=<\/td>\r\npressure at the top of the aquifer where z<\/em> = 0 (M\/(LT2<\/sup>)), e.g., kg\u00a0m\u22121<\/sup>\u00a0s\u22122<\/sup><\/td>\r\n<\/tr>\r\n
\u03c1<\/em><\/td>\r\n=<\/td>\r\ngroundwater density (M\/L3<\/sup>), e.g., kg\u00a0m\u22123<\/sup><\/td>\r\n<\/tr>\r\n
g<\/em><\/td>\r\n=<\/td>\r\ngravitational acceleration (L\/T2<\/sup>), e.g., m\u00a0s\u22122<\/sup><\/td>\r\n<\/tr>\r\n
z<\/em><\/td>\r\n=<\/td>\r\nelevation relative to a datum, which in this case is the top of the aquifer where z<\/em> = 0 and z<\/em> is positive upward, thus there is a minus sign in the equation because p<\/em> increases with depth (L), e.g., m<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Assume that the groundwater density on the freshwater side of the interface is \u03c1<\/em>f<\/em><\/sub>\u00a0=\u00a01000\u00a0kg\u00a0m\u22123<\/sup> and on the saltwater side it is \u03c1<\/em>s<\/em><\/sub>\u00a0=\u00a01025\u00a0kg\u00a0m\u22123<\/sup>. The pressure thus increases faster with depth on the saltwater side than on the freshwater side (Figure\u00a05a). The horizontal pressure difference gives rise to a flow, which in this example will be from the right to the left. Because the pressure difference between the freshwater and saltwater domains increases with depth, the initial flow will have the greatest magnitude at the base of the aquifer, and the interface will thus tend to rotate. As a result, the flow of fresh water above the interface will have to be from left to right to compensate for the displacement of the salt water along the aquifer base from right to left. The flow of the fresh water is thus in the opposite direction as the salt water (Figure\u00a05b), resulting in a clockwise rotation of the initially vertical interface.<\/p>\r\n

In reality, the description of the flow in an aquifer with a non-horizontal interface is far more complex (Bakker et al., 2004; Verruijt, 1980), but this simple example demonstrates that the presence of density differences gives rise to rotational flow. If this is the case, the hydraulic head is no longer suitable to describe the groundwater flow. The formal proof for this, which relies on determining whether the so-called curl (a property of a vector field that indicates rotational movement) of the force field is zero, can be found in Hubbert (1957).<\/p>\r\n

The fact that the flow field is not irrotational means that measurements of hydraulic head cannot be directly compared to infer flow directions or magnitudes. Instead, flow calculations must be based on the general form of Darcy\u2019s law, which is shown in Equation\u00a02 (Bear, 1972).<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle \\vec{q}=-\\frac{k}{\\mu }(\\nabla p-\\rho \\vec{g})[\/latex]<\/td>\r\n(2)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

where:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]\\vec{q}[\/latex]<\/td>\r\n=<\/td>\r\nspecific discharge (L\/T), e.g., m\u00a0s-<\/sup>1<\/sup><\/td>\r\n<\/tr>\r\n
k<\/em><\/td>\r\n=<\/td>\r\nintrinsic permeability (L2<\/sup>), e.g., m2<\/sup><\/td>\r\n<\/tr>\r\n
\u03bc<\/em><\/td>\r\n=<\/td>\r\ndynamic viscosity of the groundwater (M\/(LT)), e.g., kg\u00a0m\u22121<\/sup>\u00a0s\u22121<\/sup><\/td>\r\n<\/tr>\r\n
[latex]\\nabla[\/latex]<\/td>\r\n=<\/td>\r\ngradient operator and represents the rate of change of a variable (p<\/em> in this case) per unit of distance in the x<\/em>, y<\/em> and z<\/em> direction<\/td>\r\n<\/tr>\r\n
[latex]\\vec{g}[\/latex]<\/td>\r\n=<\/td>\r\ngravitational acceleration vector (L\/T2<\/sup>), e.g., m\u00a0s\u22122<\/sup><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

When \u03c1 <\/em>is spatially constant, Equation\u00a02 may be written as Equation\u00a03.<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle \\vec{q}=-\\frac{k}{\\mu }\\nabla (p-\\rho gz)[\/latex] [latex]\\displaystyle =-\\frac{\\rho gk}{\\mu }\\nabla \\left ( \\frac{p}{\\rho g} +z\\right )=-K\\nabla \\left ( \\frac{p}{\\rho g} +z\\right )[\/latex]<\/td>\r\n(3)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

In the expression after the third equal sign, the parameters before the gradient operator were replaced by K<\/em>\u00a0(m\u00a0s\u22121<\/sup>) the hydraulic conductivity. The term within the last set of parentheses is the hydraulic head h<\/em>\u00a0(m) as shown in Equation\u00a04.<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle h=z+h_{p}=z+\\frac{p}{\\rho g}[\/latex]<\/td>\r\n(4)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

where:<\/p>\r\n\r\n\r\n\r\n\r\n
h<\/em>p<\/em><\/sub><\/td>\r\n=<\/td>\r\npressure head (L), e.g., m<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The first term in Equation\u00a04, z<\/em>, is the elevation head, which determines the groundwater\u2019s potential energy (per unit of weight) due to its position in the Earth\u2019s gravitational field. In practice it is simply the elevation of the piezometer screen relative to a standard datum, usually mean sea level. The pressure head is a measure of the energy of groundwater because of its pressure above a reference pressure. The value of p<\/em> is gage pressure, for which the atmospheric pressure is taken as the reference pressure. Inserting Equation\u00a04 into Equation\u00a03 gives Equation\u00a05.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle \\vec{q}=-K\\nabla h[\/latex]<\/td>\r\n(5)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Equation\u00a05 is the familiar form of Darcy\u2019s law based on the hydraulic head gradient. However, given that the step from Equation\u00a02 to 3 relies on the assumption that the density is spatially constant, the hydraulic head gradient is not suitable to quantify groundwater flow in variable-density groundwater flow systems. The hydraulic head form of Darcy's law (Equation\u00a05) is strictly applicable to constant density systems.<\/p>\r\n\r\n<\/div>","rendered":"

\n

The description of groundwater flow in variable-density systems is considerably more complex than in constant-density systems. While field measurements of hydraulic head conveniently allow for an assessment of groundwater flow direction and magnitude for constant-density systems, hydraulic head measurements cannot be used for flow calculations without making allowance for buoyancy effects in variable-density systems. Without citing anyone or any literature, it was casually remarked in a Dutch report (DZRD, 1936) that \u201cIt is known that the measured head of water with a high chloride content requires a correction<\/em>\u201d [paraphrased translation from Dutch by the authors]. Nevertheless, the use of hydraulic head measurements in a variable-density groundwater system still leads to confusion or even misinterpretation to this day.<\/p>\n

Some basic intuition about the effect of density on the flow field can be gained by considering a confined aquifer with an impermeable base in which fresh water and salt water are separated by a vertical interface at time zero (Figure\u00a05a).<\/a><\/p>\n

\"Figure<\/p>\n

Figure\u00a0<\/strong>5<\/strong>\u00a0–\u00a0The effect of density on the flow field in a confined aquifer. a) Fresh and salt water are at rest and separated by a vertical interface at time zero. b) Rotating flow (clockwise) established under initially hydrostatic conditions. The base of the triangle in a) is much wider for the salt water than for the fresh water, to indicate that the freshwater pressure p<\/em>f<\/em><\/sub> at the aquifer bottom is greater than the saltwater pressure p<\/em>s<\/em><\/sub> because \u03c1<\/em>s<\/em><\/sub>\u00a0>\u00a0\u03c1<\/em>f<\/em><\/sub>. (After Santing, 1980).<\/p>\n

All the system boundaries are closed, so there can be no inflow or outflow of water. Under the prevailing hydrostatic conditions at time t<\/em> =\u00a00, the groundwater pressure p <\/span><\/em>increases with depth according to Equation\u00a01.<\/a><\/p>\n\n\n\n
<\/td>\n\"\displaystyle p=p_{0}-\rho gz\"<\/td>\n(1)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n\n\n\n\n
p<\/em><\/td>\n=<\/td>\npressure (M\/(LT2<\/sup>)), e.g., kg\u00a0m\u22121<\/sup>\u00a0s\u22122<\/sup><\/td>\n<\/tr>\n
p<\/em>0<\/sub><\/td>\n=<\/td>\npressure at the top of the aquifer where z<\/em> = 0 (M\/(LT2<\/sup>)), e.g., kg\u00a0m\u22121<\/sup>\u00a0s\u22122<\/sup><\/td>\n<\/tr>\n
\u03c1<\/em><\/td>\n=<\/td>\ngroundwater density (M\/L3<\/sup>), e.g., kg\u00a0m\u22123<\/sup><\/td>\n<\/tr>\n
g<\/em><\/td>\n=<\/td>\ngravitational acceleration (L\/T2<\/sup>), e.g., m\u00a0s\u22122<\/sup><\/td>\n<\/tr>\n
z<\/em><\/td>\n=<\/td>\nelevation relative to a datum, which in this case is the top of the aquifer where z<\/em> = 0 and z<\/em> is positive upward, thus there is a minus sign in the equation because p<\/em> increases with depth (L), e.g., m<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Assume that the groundwater density on the freshwater side of the interface is \u03c1<\/em>f<\/em><\/sub>\u00a0=\u00a01000\u00a0kg\u00a0m\u22123<\/sup> and on the saltwater side it is \u03c1<\/em>s<\/em><\/sub>\u00a0=\u00a01025\u00a0kg\u00a0m\u22123<\/sup>. The pressure thus increases faster with depth on the saltwater side than on the freshwater side (Figure\u00a05a). The horizontal pressure difference gives rise to a flow, which in this example will be from the right to the left. Because the pressure difference between the freshwater and saltwater domains increases with depth, the initial flow will have the greatest magnitude at the base of the aquifer, and the interface will thus tend to rotate. As a result, the flow of fresh water above the interface will have to be from left to right to compensate for the displacement of the salt water along the aquifer base from right to left. The flow of the fresh water is thus in the opposite direction as the salt water (Figure\u00a05b), resulting in a clockwise rotation of the initially vertical interface.<\/p>\n

In reality, the description of the flow in an aquifer with a non-horizontal interface is far more complex (Bakker et al., 2004; Verruijt, 1980), but this simple example demonstrates that the presence of density differences gives rise to rotational flow. If this is the case, the hydraulic head is no longer suitable to describe the groundwater flow. The formal proof for this, which relies on determining whether the so-called curl (a property of a vector field that indicates rotational movement) of the force field is zero, can be found in Hubbert (1957).<\/p>\n

The fact that the flow field is not irrotational means that measurements of hydraulic head cannot be directly compared to infer flow directions or magnitudes. Instead, flow calculations must be based on the general form of Darcy\u2019s law, which is shown in Equation\u00a02 (Bear, 1972).<\/a><\/p>\n\n\n\n
<\/td>\n\"\displaystyle \vec{q}=-\frac{k}{\mu }(\nabla p-\rho \vec{g})\"<\/td>\n(2)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n\n\n\n\n
\"\vec{q}\"<\/td>\n=<\/td>\nspecific discharge (L\/T), e.g., m\u00a0s–<\/sup>1<\/sup><\/td>\n<\/tr>\n
k<\/em><\/td>\n=<\/td>\nintrinsic permeability (L2<\/sup>), e.g., m2<\/sup><\/td>\n<\/tr>\n
\u03bc<\/em><\/td>\n=<\/td>\ndynamic viscosity of the groundwater (M\/(LT)), e.g., kg\u00a0m\u22121<\/sup>\u00a0s\u22121<\/sup><\/td>\n<\/tr>\n
\"\nabla\"<\/td>\n=<\/td>\ngradient operator and represents the rate of change of a variable (p<\/em> in this case) per unit of distance in the x<\/em>, y<\/em> and z<\/em> direction<\/td>\n<\/tr>\n
\"\vec{g}\"<\/td>\n=<\/td>\ngravitational acceleration vector (L\/T2<\/sup>), e.g., m\u00a0s\u22122<\/sup><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

When \u03c1 <\/em>is spatially constant, Equation\u00a02 may be written as Equation\u00a03.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle \vec{q}=-\frac{k}{\mu }\nabla (p-\rho gz)\" \"\displaystyle =-\frac{\rho gk}{\mu }\nabla \left ( \frac{p}{\rho g} +z\right )=-K\nabla \left ( \frac{p}{\rho g} +z\right )\"<\/td>\n(3)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

In the expression after the third equal sign, the parameters before the gradient operator were replaced by K<\/em>\u00a0(m\u00a0s\u22121<\/sup>) the hydraulic conductivity. The term within the last set of parentheses is the hydraulic head h<\/em>\u00a0(m) as shown in Equation\u00a04.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle h=z+h_{p}=z+\frac{p}{\rho g}\"<\/td>\n(4)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n
h<\/em>p<\/em><\/sub><\/td>\n=<\/td>\npressure head (L), e.g., m<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The first term in Equation\u00a04, z<\/em>, is the elevation head, which determines the groundwater\u2019s potential energy (per unit of weight) due to its position in the Earth\u2019s gravitational field. In practice it is simply the elevation of the piezometer screen relative to a standard datum, usually mean sea level. The pressure head is a measure of the energy of groundwater because of its pressure above a reference pressure. The value of p<\/em> is gage pressure, for which the atmospheric pressure is taken as the reference pressure. Inserting Equation\u00a04 into Equation\u00a03 gives Equation\u00a05.<\/a><\/p>\n\n\n\n
<\/td>\n\"\displaystyle \vec{q}=-K\nabla h\"<\/td>\n(5)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Equation\u00a05 is the familiar form of Darcy\u2019s law based on the hydraulic head gradient. However, given that the step from Equation\u00a02 to 3 relies on the assumption that the density is spatially constant, the hydraulic head gradient is not suitable to quantify groundwater flow in variable-density groundwater flow systems. The hydraulic head form of Darcy’s law (Equation\u00a05) is strictly applicable to constant density systems.<\/p>\n<\/div>\n","protected":false},"parent":0,"menu_order":3,"template":"","meta":{"pb_part_invisible":false,"pb_part_invisible_string":""},"contributor":[],"license":[],"class_list":["post-123","part","type-part","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/pressbooks\/v2\/parts\/123","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/pressbooks\/v2\/parts"}],"about":[{"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/wp\/v2\/types\/part"}],"version-history":[{"count":17,"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/pressbooks\/v2\/parts\/123\/revisions"}],"predecessor-version":[{"id":364,"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/pressbooks\/v2\/parts\/123\/revisions\/364"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/wp\/v2\/media?parent=123"}],"wp:term":[{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/wp\/v2\/contributor?post=123"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/wp-json\/wp\/v2\/license?post=123"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}