Analytical models for the groundwater tidal prism and associated benthic water flux

Jeffrey N. King; Ashish J. Mehta; Robert G. Dean

doi:10.1007/s10040-009-0519-y

Hydrogeology Journal

February 2010, Volume 18, Issue 1, pp 203–215 | Cite as

Analytical models for the groundwater tidal prism and associated benthic water flux

Authors
Authors and affiliations

Jeffrey N. KingEmail author
Ashish J. Mehta
Robert G. Dean

Paper

First Online: 02 October 2009

Received: 15 December 2008

Accepted: 26 August 2009

234 Downloads
17 Citations

Abstract

The groundwater tidal prism is defined as the volume of water that inundates a porous medium, forced by one tidal oscillation in surface water. The pressure gradient that generates the prism acts on the subterranean estuary. Analytical models for the groundwater tidal prism and associated benthic flux are presented. The prism and flux are shown to be directly proportional to porosity, tidal amplitude, and the length of the groundwater wave; flux is inversely proportional to tidal period. The duration of discharge flux exceeds the duration of recharge flux over one tidal period; and discharge flux continues for some time following low tide. Models compare favorably with laboratory observations and are applied to a South Atlantic Bight study area, where tide generates an 11-m³ groundwater tidal prism per m of shoreline, and drives 81 m³ s ⁻¹ to the study area, which describes 23% of an observational estimate. In a marine water body, the discharge component of any oscillatory benthic water flux is submarine groundwater discharge. Benthic flux transports constituents between groundwater and surface water, and is a process by which pollutant loading and saltwater intrusion may occur in coastal areas.

Keywords

Benthic flux Coastal aquifers Groundwater/surface-water relations Submarine groundwater discharge (SGD) Subterranean estuary

Notation

²²⁶Ra: Radium-226
A: Tidal amplitude [L]
H: Depth to base of the hydrogeologic unit [L]
K: Hydraulic conductivity [LT ⁻¹]
L: Wave length in a porous medium [L]
P: Groundwater tidal prism [L ³]
Open image in new window: Dimensionless groundwater tidal prism
T: Wave period [T]
$ {\hat V_{\text{gw}}} $: Volume of water in a hydrogeologic unit [L ³]
$ \Delta {\hat V_{\text{gw}}} $: Volume of water fluxing into or out of a hydrogeologic unit between t ₁ and t ₂ [L ³]
$ \partial {\hat V_{\text{gw}}}/\partial t $: q_bf.P integrated across the inclined bed [L³T⁻¹]
d_ts: Inland over-height elevation [L]
h: Elevation of the phreatic surface within the porous medium [L]
q_bd: Benthic discharge flux, property specific units: for example $ \left[ {{L^3}{T^{ - 1}}{L^{ - 2}} = L{T^{ - 1}}} \right] $ for a benthic volume discharge flux, [M ³ T ⁻¹ L ⁻²] for a benthic mass discharge flux
q_bf: Benthic flux, property specific units
q_br: Benthic recharge flux, property specific units
q_bd.P: P-forced benthic discharge flux [LT⁻¹]
q_bf.P: P-forced benthic flux [LT⁻¹]
q_br.P: P-forced benthic recharge flux [LT⁻¹]
s_b: Slope of inclined bed [LL ⁻¹]
t: Time [T]
t/T: Dimensionless time
x: Cartesian horizontal (offshore) dimension [L]
$ \hat x $: x-coordinate on the inclined bed [L]
Open image in new window: x-coordinate on the inclined bed at high tide [L]
Open image in new window: x-coordinate on the inclined bed at low tide [L]
y: Cartesian horizontal (along-shore) dimension [L]
z: Cartesian vertical dimension [L]
$ \hat z $: Vertical coordinate on the inclined bed [L]
Φ: Dimensionless q _bf.P integrated across the inclined bed
Ω: (Surface-water) tidal prism [L ³]
β: Angle of inclined bed, with respect to horizontal
ε: Nielsen’s (1990) perturbation parameter
η: Elevation of the free-water surface in the tidally-forced, surface-water body [L]
λ: Wave number in a porous medium, $ \lambda = 2\pi /L $ [L ⁻¹]
λ_B: Boussinesq wave number in a porous medium [L ⁻¹]
λ_r: Real component of wave number in a porous medium [L ⁻¹]
λ_ι: Imaginary component of wave number in a porous medium [L ⁻¹]
n: Porosity [L ³ L ⁻³]
σ: Radial tidal frequency, $ \sigma = 2\pi /T $ [T ⁻¹]

Subscripts

Open image in new window: High tide
Open image in new window: Low tide
P: Related to the groundwater tidal prism
bd: Benthic discharge
bf: Benthic flux
br: Benthic recharge
0: Roots of Eq. (9)

Modèles analytiques pour le prisme de marée d’eaux souterraine et le flux associé d’eau benthique

Résumé

Le prisme de marée d’eaux souterraine est défini comme le volume brut d’eau qui inonde le milieu poreux, contraint par une oscillation de marée de la masse d’eau de surface. Le gradient de pression qui engendre le prisme agit sur l’estuaire souterrain. Des modèles analytiques du prisme de marée d’eaux souterraine et du flux benthique associé sont présentés. Il est montré que prisme et flux sont directement proportionnels à la porosité, à l’amplitude de la marée et à la longueur de la vague d’eau souterraine; le flux est inversement proportionnel à la période de la marée. La durée du flux de vidange excède celle du flux de recharge au cours d’un cycle de marée ; et un flux de vidange se produit encore quelque temps après la marée basse. La confrontation des modèles aux observations de laboratoire est favorable. Ces modèles sont appliqués à une zone d’étude de la Baie Atlantique Sud où la marée génère un prisme de marée d’eau souterraine de 11 m³ par mètre de côte, et pousse 81 m³ s⁻¹ vers la zone d’étude, ce qui rend compte de 23% d’une estimation issue de l’observation. Dans une masse d’eau marine, la composante de vidange d’un flux oscillatoire d’eau benthique est une vidange sous-marine d’eau souterraine. Le flux benthique transporte des éléments entre eaux souterraines et eaux de surface: il s’agit d’un processus par lequel pollution et intrusion d’eau de mer en zone côtière peuvent se produire.

Modelos analíticos para el prisma de mareas de aguas subterráneas y el flujo de agua bentónico asociado

Resumen

El prisma de marea de agua subterránea se define como un volumen total de agua que inunda el medio poroso, forzado por una oscilación de marea en un cuerpo de agua de superficie. El gradiente de presión generado por el prisma actúa en el estuario subterráneo. Se presentan modelos analíticos para el prisma de marea de aguas subterráneas y el flujo bentónico asociado. Se demuestra que el prisma y los flujos son directamente proporcionales a la porosidad, amplitud de la marea y la longitud de la onda del agua subterránea; el flujo es inversamente proporcional al período de marea. La duración del flujo de descarga excede la duración del flujo de recarga sobre un período de marea; y un flujo de descarga es inducido para un período siguiente de bajamar. Los modelos se comparan favorablemente con las observaciones de laboratorio y son aplicados al South Atlantic Bight donde la marea genera un prisma de marea de agua subterránea de 11-m³, por metro de costa, y fuerza 81 m³ s⁻¹ al área de estudio, lo cual describe el 23% de una estimación observacional. En un cuerpo de agua marino, el componente de una descarga de un flujo de agua bentónico oscilatorio es la descarga submarina subterránea. El flujo bentónico transporta componentes entre agua subterránea y agua superficial, y es un proceso por el cual la carga de contaminantes y la intrusión de agua salina pueden ocurrir en las áreas costeras.

地下水纳潮量及相应的海底水通量的分析模型

摘要

地下水纳潮量为地表水体一次潮汐振动作用下淹没多孔介质的水的总体积。产生纳潮量的压力梯度作用于地下河口。本文提出了地下水纳潮量和相关海底水通量的解析模型。研究表明, 纳潮量和水通量与孔隙度、潮汐幅度以及地下水波长成正比关系 ; 通量与潮汐周期成反比。一次潮汐周期中, 排泄通量的持续时间超过补给通量的持续时间 ; 且落潮后排泄通量会持续一段时间。模型与实验观测表现出很好的一致性, 且已应用于南大西洋海湾研究区, 该区每米海岸线潮汐能产生11m³的地下水纳潮量, 驱使水以81 m³ s⁻¹至研究区, 相当于观测估计量约23%的。在海水中, 一次振荡的海底水通量中的排泄量部分就是海底地下水的排泄量。海底通量使组分在地下水和地表水之间迁移, 这也是造成滨海地区污染物累积和海水入侵的过程之一。

Modelos analíticos para o prisma de maré de água subterrânea e escoamentos bênticos associados

Resumo

O prisma de maré de água subterrânea é definido como sendo o volume de água que invade o meio poroso, em consequência de uma oscilação da maré num corpo de água superficial. O gradiente de pressão que origina o prisma intervém no estuário subterrâneo. São apresentados modelos analíticos para cálculo de prismas de maré de água subterrânea e escoamentos bênticos associados. Mostra-se que, tanto o prisma como o fluxo, são directamente proporcionais à porosidade, à amplitude da maré e ao comprimento da onda da água subterrânea; o fluxo é inversamente proporcional ao período da maré. A duração do fluxo de descarga excede a duração do fluxo de recarga num período de maré; e um fluxo de descarga acontece durante algum tempo após o período de maré baixa. Os resultados dos modelos foram comparados favoravelmente com observações laboratoriais e foram aplicados num caso de estudo em Bight (Atlântico Sul), onde a maré provoca um prisma de maré de água subterrânea de 11-m³ por metro de costa, ocasionando um caudal de 81 m³s⁻¹ na área de estudo, descrevendo 23% do fenómeno observado. Num corpo de água marinho, a componente de descarga de um fluxo hídrico bêntico oscilatório corresponde à descarga submarina de água subterrânea. Os fluxos bênticos transportam componentes entre a água subterrânea e a água superficial, e correspondem a um processo pelo qual a poluição e a intrusão salina podem ocorrer nas áreas costeiras.

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Notes

Acknowledgements

Comments from L.K. Brakefield-Goswami, P.A. Howd, D.F. Payne, C.G. Smith, and three anonymous reviewers improved the manuscript. C.I. Voss, E. Abarca Cameo, M.G. Deacon, S.C. Cooper, S. Duncan, and S. Schemann edited the manuscript. This work was partially funded by the US Geological Survey, Water Resources Discipline.

Appendix

Elements of Eq. (6)

Volume $ {\hat V_{{\text{gw}}.1}} $ is the volume per meter of shoreline between the phreatic surface h(x,t) and the still-water surface, from point $ \hat x $ on the inclined bed to x → ∞, as shown in Fig. 2a.

$$ \begin{array}{*{20}{c}} {{{\hat V}_{{\text{gw}}.1}} = n\int_{\hat x}^\infty {h\left( {x,t} \right)dx} } \\ { = n\int_{\hat x}^\infty {A\cos \left( {\sigma t - \lambda x} \right){e^{ - \lambda x}}dx} } \\ {\quad + n\int_{\hat x}^\infty {\frac{{\varepsilon A}}{2}dx} } \\ {\quad + n\int_{\hat x}^\infty {\frac{{\varepsilon A\sqrt 2 }}{2}\cos \left( {2\sigma t + \frac{\pi }{4} - \sqrt 2 \lambda x} \right){e^{ - \sqrt 2 \lambda x}}dx} } \\ \end{array} $$

(22)

Each term in Eq. (22) is evaluated separately:

$$ \begin{array}{*{20}{c}} {n\int_{\hat x}^\infty {A\cos \left( {\sigma t - \lambda x} \right){e^{ - \lambda x}}dx = \frac{{nA}}{{2\lambda }}{e^{ - \lambda \hat x}}\left[ {\sin \sigma t\left( {\sin \lambda \hat x + \cos \lambda \hat x} \right)} \right.} } \\ {\left. {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad - \cos \sigma t\left( {\sin \lambda \hat x - \cos \lambda \hat x} \right)} \right]} \\ \end{array} $$

(23)

$$ n\int_{\hat x}^\infty {\frac{{\varepsilon A}}{2}dx = \frac{{n\varepsilon A}}{2}\left( {\infty - \hat x} \right)} $$

(24)

$$ \begin{array}{*{20}{c}} {n\int_{\hat x}^\infty {\frac{{\varepsilon A\sqrt 2 }}{2}\cos \left( {2\sigma t + \frac{\pi }{4} - \sqrt 2 \lambda x} \right){e^{ - \sqrt 2 \lambda x}}dx} } \hfill \\ {\,\quad \quad \quad \quad \quad \quad \quad \quad = \frac{{n\varepsilon A}}{{4\lambda }}{e^{ - \sqrt 2 \lambda \hat x}}\left[ {\sin \left( {2\sigma t + \frac{\pi }{4}} \right)\left( {\sin \sqrt 2 \lambda \hat x + \cos \sqrt 2 \lambda \hat x} \right)} \right.} \hfill \\ {\left. {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad - \cos \left( {2\sigma t + \frac{\pi }{4}} \right)\left( {\sin \sqrt 2 \lambda \hat x - \cos \sqrt 2 \lambda \hat x} \right)} \right]} \hfill \\ \end{array} $$

(25)

Volume $ {\hat V_{{\text{gw}}.2}} $ is the volume per meter of shoreline between the inclined bed and the base of the hydrogeologic unit, from point $ \hat x $ on the inclined bed to the intersection of the inclined bed and the impermeable base of the hydrogeologic unit, as shown in Fig. 2a.

$$ \begin{array}{*{20}{c}} {{{\hat V}_{{\text{gw}}.2}} = n\int_{ - H/{s_{\text{b}}}}^{\hat x} {\left( {{s_{\text{b}}}x + H} \right)dx} } \\ { = \frac{{n{H^2}}}{{2{s_{\text{b}}}}}{{\left( {1 + \frac{A}{H}\cos \sigma t} \right)}^2}} \\ \end{array} $$

(26)

Volume $ {\hat V_{{\text{gw}}.3}} $ is the volume per meter of shoreline between the still-water surface and the impermeable base of the hydrogeologic unit, from point $ \hat x $ on the inclined bed to x → ∞, as shown in Fig. 2a.

$$ {\hat V_{{\text{gw}}.3}} = nH\left( {\infty - \hat x} \right) $$

(27)

Elements of Eq. (14)

Volume $ {\hat V_{{\text{gw}}.4}} $ is the volume per meter of shoreline between the inclined bed and the phreatic surface h(x,t ₁), from point $ {\hat x_1} $ to point $ {\hat x_2} $, both located on the inclined bed, as shown in Fig. 2b.

$$ \begin{array}{*{20}{c}} {{{\hat V}_{{\text{gw}}.4}} = n\int_{{{\hat x}_1}}^{{{\hat x}_2}} {\left[ {\left( {{s_{\text{b}}}x} \right) - h\left( {x,{t_1}} \right)} \right]dx} } \\ { = n\int_{{{\hat x}_1}}^{{{\hat x}_2}} {\left[ {{s_{\text{b}}}x - A{e^{ - \lambda x}}\cos \left( {\sigma {t_1} - \lambda x} \right)} \right.} } \\ {\quad - \varepsilon A\left. {\left[ {\frac{1}{2} + \frac{{\sqrt 2 }}{2}{e^{ - \sqrt 2 \lambda x}}\cos \left( {2\sigma {t_1} + \frac{\pi }{4} - \sqrt 2 \lambda x} \right)} \right]} \right]dx} \\ { = n\left[ {\frac{{{s_{\text{b}}}}}{2}\left( {\hat x_2^2 - \hat x_1^2} \right) - \frac{{\varepsilon A}}{2}\left( {{{\hat x}_2} - {{\hat x}_1}} \right)} \right.} \\ { + \frac{A}{{2\lambda }}\left[ {\left( {\sin \sigma {t_1} - \cos \sigma {t_1}} \right)\left( {{e^{ - \lambda {{\hat x}_2}}}\sin \lambda {{\hat x}_2} - {e^{ - \lambda {{\hat x}_1}}}\sin \lambda {{\hat x}_1}} \right)} \right.} \\ {\left. {\quad + \left( {\sin \sigma {t_1} + \cos \sigma {t_1}} \right)\left( {{e^{ - \lambda {{\hat x}_2}}}\cos \lambda {{\hat x}_2} - {e^{ - \lambda {{\hat x}_1}}}\cos \lambda {{\hat x}_1}} \right)} \right]} \\ { + \frac{{\sqrt 2 }}{4}\frac{{\varepsilon A}}{\lambda }\left[ {\sin \left( {2\sigma {t_1}} \right)\left( {{e^{ - \sqrt 2 \lambda {{\hat x}_2}}}\sin \sqrt 2 \lambda {{\hat x}_2} - {e^{ - \sqrt 2 \lambda {{\hat x}_1}}}\sin \sqrt 2 \lambda {{\hat x}_1}} \right)} \right.} \\ {\quad \left. {\left. { + \cos \left( {2\sigma {t_1}} \right)\left( {{e^{ - \sqrt 2 \lambda {{\hat x}_2}}}\cos \sqrt 2 \lambda {{\hat x}_2} - {e^{ - \sqrt 2 \lambda {{\hat x}_1}}}\cos \sqrt 2 \lambda {{\hat x}_1}} \right)} \right]} \right]} \\ \end{array} $$

(28)

Volume $ {\hat V_{{\text{gw}}.5}} $ is the volume per meter of shoreline between the phreatic surfaces h(x,t ₁) and h(x,t ₂), from point $ {\hat x_2} $ on the inclined bed to x → ∞, as shown in Fig. 2b.

$$ \begin{array}{*{20}{c}} {{{\hat V}_{{\text{gw}}.5}} = n\int_{{{\hat x}_2}}^\infty {\left[ {h\left( {x,{t_2}} \right) - h\left( {x,{t_1}} \right)} \right]dx} } \\ {{{\hat V}_{{\text{gw}}.5}} = n\int_{{{\hat x}_2}}^\infty {\left[ {A\cos \left( {\sigma {t_2} - \lambda x} \right){e^{ - \lambda x}} + \varepsilon A\left[ {\frac{1}{2} + \frac{{\sqrt 2 }}{2}\cos \left( {2\sigma {t_2} + \frac{\pi }{4} - \sqrt 2 \lambda x} \right){e^{ - \sqrt 2 \lambda x}}} \right]} \right.} } \\ {\left. { - A\cos \left( {\sigma {t_1} - \lambda x} \right){e^{ - \lambda x}} - \varepsilon A\left[ {\frac{1}{2} + \frac{{\sqrt 2 }}{2}\cos \left( {2\sigma {t_1} + \frac{\pi }{4} - \sqrt 2 \lambda x} \right){e^{ - \sqrt 2 \lambda x}}} \right]} \right]dx} \\ { = n\left[ {\frac{A}{{2\lambda }}{e^{ - \lambda {{\hat x}_2}}}\left[ {\left( {\sin \sigma {t_2} - \sin \sigma {t_1}} \right)\left( {\sin \lambda {{\hat x}_2} + \cos \lambda {{\hat x}_2}} \right)} \right.} \right.} \\ {\left. {\quad \quad - \left( {\cos \sigma {t_2} - \cos \sigma {t_1}} \right)\left( {\sin \lambda {{\hat x}_2} - \cos \lambda {{\hat x}_2}} \right)} \right]} \\ {\quad + \frac{{\sqrt 2 }}{4}\frac{{\varepsilon A}}{\lambda }{e^{ - \sqrt 2 \lambda {{\hat x}_2}}}\left[ {\cos \left( {\sqrt 2 \lambda {{\hat x}_2}} \right)\left( {\cos 2\sigma {t_2} - \cos 2\sigma {t_1}} \right)} \right.} \\ {\left. {\left. {\quad \quad + \sin \left( {\sqrt 2 \lambda {{\hat x}_2}} \right)\left( {\sin 2\sigma {t_2} - \sin 2\sigma {t_1}} \right)} \right]} \right]} \\ \end{array} $$

(29)

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Copyright information

Authors and Affiliations

Jeffrey N. King
- 1
Email author
Ashish J. Mehta
- 2
Robert G. Dean
- 2

1.Florida Integrated Science CenterUS Geological SurveyFort LauderdaleUSA
2.Department of Civil and Coastal EngineeringUniversity of FloridaGainesvilleUSA

Analytical models for the groundwater tidal prism and associated benthic water flux

Abstract

Keywords

Notation

Subscripts

Modèles analytiques pour le prisme de marée d’eaux souterraine et le flux associé d’eau benthique

Résumé

Modelos analíticos para el prisma de mareas de aguas subterráneas y el flujo de agua bentónico asociado

Resumen

地下水纳潮量及相应的海底水通量的分析模型

摘要

Modelos analíticos para o prisma de maré de água subterrânea e escoamentos bênticos associados

Resumo

Notes

Acknowledgements

References

Copyright information

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