Assessing the effect of construction-induced consolidation on groundwater travel time distribution under unconfined conditions

Abstract

Estimation of groundwater travel time distribution (TTD) is critical to environmental assessment and successful urban water management. This paper presents a novel analytical approach to account for the groundwater TTD in an unconfined aquifer, considering the impervious urban structure and the modified aquifer hydraulic properties due to consolidation processes. The analytical solution follows the Dupuit-Forchheimer assumption and is verified through comparisons with numerical results. Sensitivity study reveals a potential increase in the mean travel time when the aquifer intersects with an impervious underground structure, while the degree of change depends on the interplay of two mechanisms (i.e., the increase in upgradient groundwater storage and the decrease in pore space of the consolidated zone). Global sensitivity analysis suggests that urban groundwater TTD systems are strongly dependent on both construction-related parameters and aquifer compressibility, indicating that adaptive strategies of groundwater management should be taken for different aquifer types. The proposed analytical model has great implications for the management of water resources in urban areas, the utilization of urban groundwater for human consumption, and the quality of urban aquifers.

Résumé

L’estimation de la distribution du temps de transit des eaux souterraines (DTT) est essentielle pour l’évaluation environnementale et la gestion réussie des eaux urbaines. Cet article présente une nouvelle approche analytique pour prendre en compte la DTT des eaux souterraines dans un aquifère libre, en considérant la structure urbaine imperméable et les propriétés hydrauliques modifiées de l’aquifère dues aux processus de consolidation. La solution analytique suit l’hypothèse de Dupuit-Forchheimer et est vérifiée par comparaison avec les résultats numériques. L’étude de sensibilité révèle une augmentation potentielle du temps de parcours moyen lorsque l’aquifère intercepte une structure souterraine imperméable, tandis que le degré de changement dépend de l’interaction de deux mécanismes (c’est-à-dire l’augmentation du stockage des eaux souterraines en amont et la diminution de l’espace poreux de la zone consolidée). L’analyse de sensibilité globale suggère que les systèmes de DTT des eaux souterraines urbaines dépendent fortement à la fois des paramètres liés à la construction et de la compressibilité de l’aquifère, ce qui indique que des stratégies adaptatives de gestion des eaux souterraines doivent être prises en considérant les différents types d’aquifères. Le modèle analytique proposé a de grandes implications pour la gestion des ressources en eau dans les zones urbaines, l’utilisation des eaux souterraines urbaines pour la consommation humaine, et pour la qualité des aquifères en milieu urbain.

Resumen

La estimación de la distribución del tiempo de tránsito de las aguas subterráneas (TTD) es fundamental para la evaluación medioambiental y para el éxito de la gestión del agua urbana. Este trabajo presenta un nuevo enfoque analítico para considerar el TTD de las aguas subterráneas en un acuífero no confinado, teniendo en cuenta la estructura urbana impermeable y las propiedades hidráulicas modificadas del acuífero debido a los procesos de consolidación. La solución analítica sigue la hipótesis de Dupuit-Forchheimer y se verifica mediante comparaciones con resultados numéricos. El estudio de sensibilidad revela un aumento potencial del tiempo medio de tránsito cuando el acuífero se cruza con una estructura subterránea impermeable, mientras que el grado de cambio depende de la interacción de dos mecanismos (es decir, el aumento del almacenamiento de agua subterránea ascendente y la disminución del espacio poroso de la zona consolidada). El análisis de sensibilidad global sugiere que los sistemas de TTD de aguas subterráneas urbanas dependen en gran medida tanto de los parámetros relacionados con la construcción como de la compresibilidad del acuífero, lo que indica que deben adoptarse estrategias adaptativas de gestión de las aguas subterráneas para los distintos tipos de acuíferos. El modelo analítico propuesto tiene importantes implicancias para la gestión de los recursos hídricos en las zonas urbanas, la utilización de las aguas subterráneas para el consumo humano y la calidad de los acuíferos urbanos.

摘要

地下水运移时间分布(TTD)对于环境评估和成功的城市水管理至关重要。本文提出了一种新的分析方法，结合不透水城市结构和固结过程导致的含水层水力特性变化，计算非承压含水层中的地下水运移时间分布(TTD)。分析方法遵循Dupuit-Forchheimer假设，并通过与数值结果的比较进行验证。敏感性研究表明，当含水层与不透水地下结构相交时，地下水运移的平均时间可能增加，而变化程度取决于两种机制的相互作用(即，上游地下水储量增加和固结区孔隙空间减少)。全球敏感性分析表明，城市地下水运移时间分布系统强烈依赖于施工相关参数和含水层压缩性，这表明应针对不同含水层类型采取地下水管理的对应策略。本次提出的分析模型对城市水资源管理、城市地下水开发利用以及城市含水层质量具有重要意义。

Resumo

A estimativa da distribuição do tempo de trânsito (DTT) das águas subterrâneas é fundamental para a avaliação ambiental e a gestão bem-sucedida das águas urbanas. Este artigo apresenta uma nova abordagem analítica para contabilizar a DTT das águas subterrâneas em um aquífero livre, considerando a estrutura urbana impermeável e as propriedades hidráulicas do aquífero modificadas devido aos processos de consolidação. A solução analítica segue a hipótese de Dupuit-Forchheimer e é verificada por meio de comparações com resultados numéricos. O estudo de sensibilidade revela um aumento potencial no tempo de trânsito médio quando o aquífero cruza com uma estrutura subterrânea impermeável, enquanto o grau de mudança depende da interação de dois mecanismos (ou seja, o aumento do armazenamento de águas subterrâneas ascendentes e a diminuição do espaço poroso da zona consolidada). A análise de sensibilidade global sugere que os sistemas de DTT de águas subterrâneas urbanas são fortemente dependentes tanto dos parâmetros relacionados à construção quanto da compressibilidade do aquífero, indicando que estratégias adaptativas de gestão de águas subterrâneas devem ser adotadas para diferentes tipos de aquíferos. O modelo analítico proposto tem grandes implicações para a gestão dos recursos hídricos em áreas urbanas, a utilização das águas subterrâneas urbanas para consumo humano e a qualidade dos aquíferos urbanos.

Appendices

Appendix 1

Nomenclature

x :

[L] Horizontal position

a :

[T] Groundwater age

ρ(a):

[-] Probability density function (pdf) of travel time distribution (TTD)

R :

[LT^–1] Diffuse recharge rate

K :

[LT^–1] Saturated hydraulic conductivity

L :

[L] Length of the aquifer

θ :

[-] Effective porosity

h _L :

[L] Fixed downstream head

τ _p :

[T] Mean travel time (MTT) in the preurban case

H _p :

[L] Mean saturated aquifer thickness in the preurban case

w :

[L] Half of the underground structure’s length

l _c :

[L] Distance from the underground structure’s center to the discharge zone

d _c :

[L] Depth of the underground structure

s _c :

[L] Consolidation settlement

C _c :

[-] Compression index of the porous media

Δσ:

[M T^–2] Surcharge load of the underground structure

h _d, h_u:

[-] Void ratio of the unconsolidated and the consolidated porous media, respectively

θ _c :

[L] Effective porosity of the consolidated porous media

K _c :

[LT^–1] Hydraulic conductivity of the consolidated porous media

h _d, h_u:

[L] Hydraulic head at the right and left end of the consolidated zone, respectively

v :

[LT^–1] Fluid velocity

Q :

[L² T^–1] Groundwater discharge in the horizontal direction

q :

[LT^–1] Darcy flux

H _d, H_u:

[L] Mean saturated thickness for the downgradient and upgradient zone, respectively

a₀, a₁, a_c:

[T] Travel time across the downgradient zone/lag time of the upgradient zone/travel time across the consolidated zone

τ ₀, τ₁:

[T] MTT for water recharged from the downgradient and upgradient zone, respectively

τ :

[T] MTT for the whole domain in the urban case

τ _d :

[T] $\frac{\theta {H}_{\textrm{d}}}{R}$

τ _u :

[T] $\frac{\theta {H}_{\textrm{u}}}{R}$

${w}_{\textrm{c}}^{\ast }$, ${l}_{\textrm{c}}^{\ast }$, ${d}_{\textrm{c}}^{\ast }$, Δσ^∗, a^∗, τ^∗:

[-] Normalized parameters and variables

Appendix 2

Expressions for H _p, H _d, and H _u

Under unconfined conditions, the hydraulic head (h) can be expressed using the equation of the Dupuit-Forchheimer ellipse (Dupuit 1863). The mean of hydraulic heads between two random horizontal locations (x = x₁ and x = x₂), $\overline{H}\left({x}_1,{x}_2\right)$, can be calculated by:

$$\overline{H}\left({x}_1,{x}_2\right)=\frac{\sqrt{\frac{R}{K}}}{x_2-{x}_1}\left(\frac{x_2}{2}\sqrt{\psi^2-{x}_2^2}+\frac{\psi^2}{2}\arcsin \frac{x_2}{\psi }-\frac{x_1}{2}\sqrt{\psi^2-{x}_1^2}-\frac{\psi^2}{2}\arcsin \frac{x_1}{\psi}\right)$$

(34)

where $\psi =\sqrt{L^2+\frac{K}{R}{h}_{\textrm{L}}^2}$.

The mean of the saturated thickness in the preurban case (H_p) can be calculated by integrating Eq. (1):

$${H}_{\textrm{p}}=\frac{h_{\textrm{L}}}{2}+\sqrt{\frac{R}{K}}\left(\frac{L}{2}+\frac{K{h}_{\textrm{L}}^2}{2 RL}\right)\arcsin \frac{L}{\sqrt{L^2+\frac{K{h}_{\textrm{L}}^2}{R}}}$$

(35)

In the urban case, hydraulic heads in the downgradient zone follow the Dupuit-Forchheimer ellipse similar to the preurban case:

$$h(x)=\sqrt{\frac{R}{K}\left[{\left(L-2{w}_{\textrm{c}}\right)}^2-{\left(x-2{w}_{\textrm{c}}\right)}^2\right]+{h}_{\textrm{L}}^2},\kern0.5em L-l+{w}_{\textrm{c}}\le x<L$$

(36)

H_d can be calculated by combining Eqs. (34) and (36):

$${H}_\mathrm{d}=\frac{\sqrt{\frac{R}{K}}}{l_{\textrm{c}}-{w}_{\textrm{c}}}{\displaystyle \begin{array}{c}\left(\frac{L-2{w}_{\textrm{c}}}{2}\sqrt{\psi_\mathrm{d}^2-{\left(L-2{w}_{\textrm{c}}\right)}^2}+\frac{\psi_{\textrm{d}}^2}{2}\arcsin \frac{L-2{w}_{\textrm{c}}}{\psi_{\textrm{d}}}\right.\\ {}-\frac{L-{l}_{\textrm{c}}-{w}_{\textrm{c}}}{2}\sqrt{\psi_{\textrm{d}}^2-{\left(L-{l}_{\textrm{c}}-{w}_{\textrm{c}}\right)}^2}+\frac{\psi_{\textrm{d}}^2}{2}\arcsin \frac{L-{l}_{\textrm{c}}-{w}_{\textrm{c}}}{\psi_{\textrm{d}}}\end{array}}$$

(37)

where ${\psi}_{\textrm{d}}=\sqrt{{\left(L-2{w}_{\textrm{c}}\right)}^2+\frac{K}{R}{h}_{\textrm{L}}^2}$.

Similarly, hydraulic heads in the upgradient zone is given by:

$$h(x)=\sqrt{\frac{R}{K}\left[{\left(L-{l}_{\textrm{c}}-{w}_{\textrm{c}}\right)}^2-{x}^2\right]+{h}_{\textrm{u}}^2},\kern0.5em 0\le x<L-{l}_{\textrm{c}}-{w}_{\textrm{c}}$$

(38)

The mean saturated thickness for the upgradient recharge zone (H) is given by:

$${H}_{\textrm{u}}=\frac{h_{\textrm{u}}}{2}+\sqrt{\frac{R}{K}}\left(\frac{L-{l}_{\textrm{c}}-{w}_{\textrm{c}}}{2}+\frac{K{h}_{\textrm{u}}^2}{2R\left(L-{l}_{\textrm{c}}-{w}_{\textrm{c}}\right)}\right)\arcsin \frac{L-{l}_{\textrm{c}}-{w}_{\textrm{c}}}{\sqrt{{\left(L-{l}_{\textrm{c}}-{w}_{\textrm{c}}\right)}^2+\frac{K{h}_{\textrm{u}}^2}{R}}}$$

(39)

Appendix 3

Simulated hydraulic heads and velocity fields for model verification

Simulated hydraulic heads and velocity fields are shown in Figs. 10 and 11.

Fig. 10

a–h Comparison between simulated hydraulic heads and analytical solutions

Fig. 11

a–h Simulated velocity fields for eight verification scenarios

Appendix 4

Sobol variance-based sensitivity analysis

Sobol sensitivity analysis evaluates the contributions from the variability in model parameters to the variance in the total entity through Monte Carlo integrations (Saltelli et al. 1999, 2010; Sobol 2001). Consider a problem with k parameters:

$$y=f\left(\boldsymbol{\uptheta} \right)=f\left({\theta}_1,{\theta}_2,\cdots, {\theta}_k\right)$$

(40)

in which y is the vector of objective functions for evaluating model performance, and θ = {θ₁, θ₂, ⋯, θ_k} is the vector of parameters controlling the variance of the objective function y.

The Sobol approach decomposes the variance of the objective functions, V(y), into a sequence of independent terms following an increasing dimensionality:

$$f\left({\theta}_1,{\theta}_2,\cdots, {\theta}_k\right)={f}_0+{\sum}_{i=1}^k{f}_i\left({\theta}_i\right)+{\sum}_{1\le i\le j\le k}{f}_{ij}\left({\theta}_i,{\theta}_j\right)+\dots +{f}_{1,2,\dots, k}\left({\theta}_1,{\theta}_2,\cdots, {\theta}_k\right)$$

(41)

The total variance V(y) can be derived based on Eq. (41), which can be seen as:

$$V(y)={\sum}_i{V}_i+{\sum}_{i<j}{V}_{ij}+\dots +{V}_{1,2,\dots, k}$$

(42)

in which V_i denotes the fraction of the total variance contributed by the parameter θ_i, and V_ij denotes the fraction corresponding to the interaction between the ith and jth parameters (i.e., θ_i and θ_j). Correspondingly, the individual contribution from V_i to the total variance V(y) is defined as the first-order sensitivity index S_i (Rosolem et al. 2012; Sobol 2001):

$${S}_i=\frac{V_i}{V(y)}$$

(43)

The total-order indices S_Ti can be seen as:

$${S}_{Ti}=1-\frac{V_{\sim i}}{V(y)}$$

(44)

where V_~i is the averaged variance calculated by fixing θ_i and sampling other parameters from the parameter space. Accordingly, the total-order indices S_Ti accounts for both direct and indirect impacts of parameter θ_i on the total variance V(y).