Tide-induced head fluctuations in a coastal aquifer: effects of the elastic storage and leakage of the submarine outlet-capping

Abstract

This paper considers the tidal head fluctuations in a single coastal confined aquifer which extends under the sea for a certain distance. Its submarine outlet is covered by a silt-layer with properties dissimilar to the aquifer. Recently, Li et al. (2007) gave an analytical solution for such a system which neglected the effect of the elastic storage (specific storage) of the outlet-capping. This article presents an analytical solution which generalizes their work by incorporating the elastic storage of the outlet-capping. It is found that if the outlet-capping is thick enough in the horizontal direction, its elastic storage has a significant enhancing effect on the tidal head fluctuation. Ignoring this elastic storage will lead to significant errors in predicting the relationship of the head fluctuation and the aquifer hydrogeological properties. Quantitative analysis shows the effect of the elastic storage of the outlet-capping on the groundwater head fluctuation. Quantitative conditions are given under which the effect of this elastic storage on the aquifer’s tide-induced head fluctuation is negligible. Li, H.L., Li, G.Y., Chen, J.M., Boufadel, M.C. (2007) Tide-induced head fluctuations in a confined aquifer with sediment covering its outlet at the sea floor. [Fluctuations du niveau piézométrique induites par la marée dans un aquifère captif à décharge sous-marine.] Water Resour. Res 43, doi:10.1029/2005WR004724

Résumé

Cet article considère les fluctuations piézométriques dues à la marée dans un aquifère côtier captif simple s’étendant à une certaine distance sous la mer. Son exutoire sous-marin est recouvert par un dépôt silteux de propriétés différentes de celles de l’aquifère. Récemment, Li et autres (2007) ont donné une représentation analytique d’un tel système tenant compte de l’effet d’emmagasinement élastique du réservoir sous le toit à l’exutoire. Cet article présente une solution analytique qui généralise le modèle en introduisant l’emmagasinement élastique à l’exutoire. Il démontre que si la couveture à l’exutoire est assez épaisse en direction, l’emmagasinement élastique a un effet amplificateur important sur la fluctuation piézométrique due à la marée. Ignorer cet emmagasinement élastique conduirait à des erreurs importantes sur le rapport entre la hauteur piézométrique réelle et la hauteur telle qu’elle ressort des caractéristiques de l’aquifère. Le modèle montre donc l’effet de l’emmagasinement élastique sur la fluctuation du niveau de l’aquifère. Il indique les seuils en dessous desquels l’effet de cet emmagasinement élastique sur la fluctuation de l’aquifère induite par la marée est négligeable. Li, H.L., Li, G.Y., Chen, J.M., Boufadel, M.C. (2007) Tide-induced head fluctuations in a confined aquifer with sediment covering its outlet at the sea floor. [Fluctuations du niveau piézométrique induites par la marée dans un aquifère captif à décharge sous-marine.] Water Resour. Res 43, doi:10.1029/2005WR004724

Resumen

Este trabajo considera las fluctuaciones piezométricas debidas a la marea en un único acuífero costero confinado que se extiende bajo el mar a través de una cierta distancia. Su descarga submarina en el lecho del mar está cubierta por una capa de limosa con propiedades diferentes del acuífero. Recientemente, Li et al. (2007) dieron una solución analítica para tal sistema que despreciaba el efecto del almacenamiento elástico (almacenamiento específico) de la cubierta en la descarga. Este trabajo presenta una solución analítica que generaliza el trabajo de estos autores, incorporando el almacenamiento elástico de la cubierta de la descarga. Se encontró que si la cubierta tiene suficiente espesor en la dirección horizontal, su almacenamiento elástico tiene un efecto enriquecedor significativo sobre la fluctuación piezométrica debido a la marea. Ignorar este almacenamiento elástico conduce a errores significativos en la predicción de la relación entre la fluctuación piezométrica y las propiedades hidrogeológicas del acuífero. Los análisis cuantitativos muestran el efecto del almacenamiento elástico de la cubierta sobre la fluctuación del agua subterránea. Se dan condiciones cuantitativas bajo las cuales los efectos de este almacenamiento elástico sobre las fluctuaciones piezométricas inducidas por la marea son insignificantes. Li, H.L., Li, G.Y., Chen, J.M., Boufadel, M.C. (2007) Tide-induced head fluctuations in a confined aquifer with sediment covering its outlet at the sea floor. [Fluctuations du niveau piézométrique induites par la marée dans un aquifère captif à décharge sous-marine.] Water Resour. Res 43, doi:10.1029/2005WR004724

摘要

本文考虑在海底延展一定距离的单层滨海承压含水层中的潮汐水头波动。含水层的海底露头覆盖有一层与含水层性质不同的淤泥层。最近, Li等 (2007) 给出了一个忽略露头盖层弹性储水 (贮水率) 的解析解。本文通过考虑露头盖层弹性储存推广了该解析解。结果表明, 若露头盖层在水平方向足够厚, 则弹性储存会显著增强潮汐水头波动。若忽略弹性储存, 在预测水头波动和含水层水文地质特征的关系时, 会造成显著误差。定量分析了露头盖层的弹性储存对地下水位波动的效应。给出了可忽略弹性储存对含水层潮汐水头波动影响的量化情景。 Li, H.L., Li, G.Y., Chen, J.M., Boufadel, M.C. (2007) Tide-induced head fluctuations in a confined aquifer with sediment covering its outlet at the sea floor. [Fluctuations du niveau piézométrique induites par la marée dans un aquifère captif à décharge sous-marine.] Water Resour. Res 43, doi:10.1029/2005WR004724

Resumo

O presente artigo considera as variações piezométricas induzidas pelas marés num aquífero costeiro confinado que se extende debaixo do mar ao longo de determinada distância. A porção submarina deste aquífero está coberta por uma camada de silte com propriedades diferentes das do aquífero. Recentemente, Li et al. (2007) definiram uma solução analítica para este tipo de sistemas que não tem em conta o efeito do armazenamento elástico (armazenamento específico) da camada submarina confinante. O presente artigo apresenta uma solução analítica que generaliza este trabalho, incorporando o armazenamento elástico da camada confinante submarina. Constatou-se que, se a camada confinante é suficientemente extensa horizontalmente, o seu armazenamento elástico acentua significativamente as flutuações piezométricas devidas às marés. Ao ignorar este armazenamento elástico introduzem-se erros significativos na estimação da relação entre as flutuações piezométricas e as propriedades hidrogeológicas do aquífero. A análise quantitativa mostra o efeito do armazenamento elástico da camada submarina confinante nas flutuações dos níveis de água subterrânea. As condições quantitativas são dadas de tal forma que o efeito deste armazenamento elástico nas flutuações piezométricas é negligenciável. Li, H.L., Li, G.Y., Chen, J.M., Boufadel, M.C. (2007) Tide-induced head fluctuations in a confined aquifer with sediment covering its outlet at the sea floor. [Fluctuations du niveau piézométrique induites par la marée dans un aquifère captif à décharge sous-marine.] Water Resour. Res 43, doi:10.1029/2005WR004724

چکیده

دراین مقاله نوسانات ارتفاع آب، ناشی ازجزر و مد در یک آبخوان محصور منفرد که تا طول مشخصی در زیر دریا امتداد یافته است، بررسی می شود. خروجی این آبخوان به کف دریا با یک لایه سیلت (لای) که خواص متفاوتی با آبخوان دارد، پوشیده شده است. به تازگی، Li و همکارانش (2007Li et. al., ) یک راه حل تحلیلی برای این مسأله با صرفنظر کردن از اثر ذخیره الاستیک (ذخیره مخصوص) ارائه کرده اند. در مقاله حاضر راه حلی تحلیلی ارائه می گردد که با اضافه کردن اثر ذخیره الاستیک پوشش خروجی، نتایج تحقیق Li تعمیم داده می شود. بر اساس نتایج بدست آمده، اگر پوشش خروجی به قدر کافی در راستای افقی ضخیم باشد، ذخیره الاستیک آن تأثیر زیادی بر نوسانات ارتفاع آب ناشی از جزر و مد خواهد داشت. نادیده گرفتن این ذخیره الاستیک می تواند منجر به ایجاد خطای زیادی در پیش بینی رابطه بین نوسان ارتفاع آب و خصوصیات هیدرولوژیکی آبخوان شود. همچنین با استفاده از تحلیلهای کمی، اثر ذخیره الاستیک پوشش خروجی بر روی نوسان تراز آب زیرزمینی نشان داده شده است. به علاوه شرایطی کمی برای مواردی که می توان اثر ذخیره الاستیک روی نوسانات ارتفاع آب در آبخوان تحت تأثیر جزر و مد را نادیده گرفت نیز ارائه شده است. Li, H.L., Li, G.Y., Chen, J.M., Boufadel, M.C. (2007) Tide-induced head fluctuations in a confined aquifer with sediment covering its outlet at the sea floor. [Fluctuations du niveau piézométrique induites par la marée dans un aquifère captif à décharge sous-marine.] Water Resour. Res 43, doi:10.1029/2005WR004724

Appendices

Appendix A: Derivation of the solution

Suppose

$$h{\left( {x,t} \right)} = {\text{Re}}{\left[ {H{\left( {x,t} \right)}} \right]} = A\operatorname{Re} {\left[ {X{\left( x \right)}e^{{i\omega t}} } \right]},$$

(22)

$$h_1 \left( {x,t} \right) = \operatorname{Re} \left[ {H_1 \left( {x,t} \right)} \right] = A\operatorname{Re} \left[ {X_1 \left( x \right)e^{{\text{i}}\omega {\text{t}}} } \right],$$

(23)

where X(x), X ₁ (x) are complex functions. Re denotes the real part of the followed complex expression, ${\text{i}} = \sqrt { - 1}$. Substituting Eqs. (22) and (23) back into Eqs. (1), (2), (5), (7)–(12) and then extending the equations into complex ones with respect to X(x), yields

$${\text{i}}\omega S_{s} X{\left( x \right)} = KX\prime \prime {\left( x \right)},\quad \quad x > 0,$$

(24)

$${\text{i}}\omega S_{\text{s}} X\left( x \right) = KX\prime \prime \left( x \right) + {\text{i}}\omega L_{\text{e}} S_{{\text{s, }}} - L, < x < 0,\,$$

(25)

$${\text{i}}\omega S_{{\text{s1}}} X_1 \left( x \right) = KX_1 \prime \prime \left( x \right) + {\text{i}}\omega L_{{\text{e1}}} S_{{\text{s1}}} ,\quad - \left( {L + m} \right) \leqslant x < - L,$$

(26)

$$\mathop {\lim }\limits_{x \to \infty } X\prime \left( x \right) = 0,$$

(27)

$$X_1 \left( x \right)\left| {_{x = - \left( {m + L} \right)} = 1,} \right.$$

(28)

$$\mathop {\lim }\limits_{x \to 0^ - } X\left( x \right) = \mathop {\lim }\limits_{x - 0^ + } X\left( x \right),{\text{ }}$$

(29)

$$\mathop {\lim }\limits_{x \to 0^ - } X\prime \left( x \right) = \mathop {\lim }\limits_{x \to 0^ + } X\prime \left( x \right),$$

(30)

$${\mathop {\lim }\limits_{x \to - L^{ - } } }X_{1} {\left( x \right)} = {\mathop {\lim }\limits_{x \to - L^{ + } } }X{\left( x \right)},$$

(31)

$${\mathop {\lim }\limits_{x \to - L^{ - } } }K_{1} X_{1} \prime {\left( x \right)} = {\mathop {\lim KX}\limits_{x \to - L^{ + } } }\prime {\left( x \right)}.$$

(32)

The general solutions to Eqs. (33)–S(35) are

$$X\left( x \right) = C_1 e^{a\left( {1 + {\text{i}}} \right)x} + C_2 e^{ - a\left( {1 + {\text{i}}} \right)x} + L_{\text{e}} ,\quad \quad - L < x < 0,$$

(33)

$$X\left( x \right) = C_3 e^{a\left( {1 + {\text{i}}} \right)x} + C_4 e^{ - a\left( {1 + {\text{i}}} \right)x} ,\quad \quad x > 0,$$

(34)

$$X_1 \left( x \right) = C_5 e^{\frac{\theta }{m}\left( {1 + {\text{i}}} \right)x} + C_6 e^{ - \frac{\theta }{m}\left( {1 + {\text{i}}} \right)x} + L_{\text{e}} ,\quad \quad - \left( {L + m} \right) \leqslant x < - L,$$

(35)

where C ₁, C ₂, C ₃, C ₄, C ₅ and C ₆ are six unknown complex constants. By means of Eqs. (27)–(32), after some routine calculation, one obtains

$$C_1 = - \frac{{L_{\text{e}} }}{{\text{2}}},$$

(36)

$$C_3 = 0,$$

(37)

$$C_4 = C_2 + \frac{1}{2}L_{{\text{e,}}} $$

(38)

$$C_5 = \frac{1}{{2\theta \sigma }}e^{\frac{L}{m}\Delta } \left[ {\left( {\theta \sigma - 1} \right)C_2 e^{aL\left( {1 + {\text{i}}} \right)} - \frac{1}{2}\left( {\theta \sigma + 1} \right)L_{\text{e}} e^{ - aL\left( {1 + {\text{i}}} \right)} + \theta \sigma \left( {L_{\text{e}} - L_{{\text{e1}}} } \right)} \right],$$

(39)

$$C_6 = \left[ {\left( {1 - L_{{\text{e1}}} } \right) - C_5 e^{ - \left( {1 + \frac{L}{m}} \right)\Delta } } \right]e^{ - \left( {1 + \frac{L}{m}} \right)\Delta } ,$$

(40)

and C ₂ is given by Eq. (19)

Substituting Eqs. (36)–(37), Eq. (19) into Eqs. (24)–(35), one can obtain X(x), X ₁(x). Substituting the resultant expression of X ₁(x) into H ₁(x,t), and calculating the real part of H ₁(x,t), one finally obtains the solution:

$$\begin{array}{*{20}l} {{h_{1} {\left( {x,t} \right)} = e^{{{\left( {\frac{\theta }{m}{\left( {x + L} \right)}} \right)}}} {\left| {C_{5} } \right|}\cos {\left( {\omega t + \frac{\theta }{m}{\left( {x + L} \right)} - \arg C_{5} } \right)}} \hfill} \\ {{ + e^{{{\left( { - \frac{\theta }{m}{\left( {x + L} \right)}} \right)}}} {\left| {C_{6} } \right|}\cos {\left( {\omega t - \frac{\theta }{m}{\left( {x + m + L} \right)} - \arg C_{6} } \right)} + L_{{{\text{e}}1}} \cos {\left( {\omega t} \right)},} \hfill} \\ \end{array} - {\left( {L + m} \right)} < x < - L.$$

(41)

Substituting the resultant expression of X(x) into H(x,t), and calculating the real part of H(x,t), one finally obtains the Eqs. (16) and (17).

Appendix B: Derivation of the maximum relative error

Using Eqs. (16) and (18), the difference E of the Eq. (16) and that of Li et al. (2007) at the interface x = –L can be expressed as

$$\begin{array}{*{20}c} {{E \mathord{\left/ {\vphantom {E A}} \right. \kern-\nulldelimiterspace} A} = \mathop {\lim }\limits_{\theta \to 0^ + } \left[ {\left| {C_2 } \right|e^{aL} \cos \left( {\omega t - ax + \arg C_2 } \right)} \right] - \left| {C_2 } \right|e^{aL} \cos \left( {\omega t - ax + \arg C_2 } \right)} \\ { = e^{aL} \operatorname{Re} \left\{ {\mathop {\lim }\limits_{\theta \to 0^ + } C_2 e^{{\text{i}}\left( {\omega t - ax} \right)} - C_2 e^{{\text{i}}\left( {\omega t{\text{ - }}ax} \right)} } \right\} \leqslant e^{aL} \left| {\mathop {\lim }\limits_{\theta \to 0^ + } C_2 e^{{\text{i}}\left( {\omega t - ax} \right)} - C_2 e^{{\text{i}}\left( {\omega t - ax} \right)} } \right|} \\ { = e^{aL} \left| {\mathop {\lim }\limits_{\theta \to 0^ + } C_2 - C_2 } \right|.} \\ \end{array} $$

(42)

One is always able to chose the value of t so that the imaginary part of $\mathop {\lim }\limits_{\theta \to 0^ + } C_2 e^{{\text{i}}\left( {\omega t - ax} \right)} - C_2 e^{{\text{i}}\left( {\omega t - ax} \right)}$ vanishes and the inequality (Eq. 42) becomes an equality. So the maximum error $E_{{\max }} {\left( {\theta ,\sigma ,aL,L_{{\text{e}}} ,L_{{{\text{e1}}}} } \right)}$ is given by Eq. (21).