Delineating spring recharge areas in a fractured sandstone aquifer (Luxembourg) based on pesticide mass balance

Abstract

A simple method to delineate the recharge areas of a series of springs draining a fractured aquifer is presented. Instead of solving the flow and transport equations, the delineation is reformulated as a mass balance problem assigning arable land in proportion to the pesticide mass discharged annually in a spring at minimum total transport cost. The approach was applied to the Luxembourg Sandstone, a fractured-rock aquifer supplying half of the drinking water for Luxembourg, using the herbicide atrazine. Predictions of the recharge areas were most robust in situations of strong competition by neighbouring springs while the catchment boundaries for isolated springs were extremely sensitive to the parameter controlling flow direction. Validation using a different pesticide showed the best agreement with the simplest model used, whereas using historical crop-rotation data and spatially distributed soil-leaching data did not improve predictions. The whole approach presents the advantage of integrating objectively information on land use and pesticide concentration in spring water into the delineation of groundwater recharge zones in a fractured-rock aquifer.

Résumé

Une méthode simple pour définir des aires de recharge d’un ensemble de sources drainant un aquifère fracturé est présentée. Au lieu de résoudre les équations de l'écoulement et de tranport, la délimitation est reformulée comme un problème de bilan massique, assignant à la surface cultivable la masse de pesticide observée annuellement à la source en considérant le plus court chemin. L'approche a été appliquée au grès de Luxembourg, un aquifère fracturé fournissant la moitié de l'eau potable du Luxembourg, en utilisant l'herbicide atrazine. La prédiction des zones d'alimentation est apparue plus robuste lorsqu’elles sont en situation de forte concurrence dans le cas de sources voisines, tandis que les limites des bassins versants des sources isolées sont apparues extrêmement sensibles au paramètre contrôlant la direction de l’écoulement. Une validation à partir d’un autre pesticide a montré une bonne concordance avec le modèle utilisé le plus simple, alors que l’utilisation d’un historique de données de rotation des cultures et des données distribuées de lixiviation des sols n’a pas amélioré les prédictions. L’approche complète présente l’avantage d’intégrer objectivement les informations sur l’occupation du sol et les concentration en pesticide dans l’eau de la source dans la définition des zones de recharge d’un aquifère fracturé.

Resumen

Se presenta un método simple para delimitar las áreas de recarga de una serie de manantiales que drenan un acuífero fracturado. En lugar de resolver las ecuaciones de flujo y transporte, la delimitación se reformula como un problema de balance de masa asignando la tierra cultivable en proporción a la masa de pesticida descargada anualmente en un manantial del transporte total a un costo mínimo. El enfoque fue aplicado en la Arenisca Luxemburgo, un acuífero en roca fracturada que suministra la mitad de agua potable a Luxemburgo, usando el herbicida atrazina. Las predicciones de zonas de capturas fueron más robustas en situaciones de fuerte competencia entre manantiales vecinos mientras que los límites de la cuenca para manantiales aislados fueron extremadamente sensibles a los parámetros que controlan la dirección de flujo. La validación usando pesticidas diferentes mostraron una buena concordancia con el modelo más simple utilizado, mientras que el uso de datos históricos de rotación de cultivos y datos de lixiviación de suelos distribuidos espacialmente no mejoraron las predicciones. El enfoque presenta la ventaja de la integrar objetivamente la información del uso de la tierra y la concentración de pesticidas en el agua del manantial para la delimitación de las zonas de recarga en un acuífero de rocas fracturadas.

Resumo

Apresenta-se um método simples para a delineação de áreas de recarga de uma série de nascentes que drenam um aquífero fraturado. Em vez de resolver as equações de fluxo e transporte, a delineação é reformulada como um problema de balanço de massa, definindo a área de terreno arável proporcionalmente à massa de pesticida que é descarregada anualmente numa nascente, à custa de um transporte total mínimo. Esta metodologia foi aplicada no Arenito do Luxemburgo, que é um aquífero de rocha fraturada que fornece metade da água de abastecimento do Luxemburgo, usando o herbicida atrazina. As previsões em termos de zona de captura revelaram-se mais robustas em zonas onde há uma forte competição entre nascentes vizinhas, enquanto as fronteiras das bacias de nascentes isoladas se revelaram extremamente sensíveis ao parâmetro que controla a direção do fluxo subterrâneo. A validação, usando um pesticida diferente, revelou um bom ajuste com o modelo mais simples que foi usado, enquanto o uso de dados históricos de rotação de culturas e dados espacialmente distribuídos de percolação no solo não contribuíram para melhorar as previsões. A abordagem geral apresenta a vantagem de integrar objetivamente informação sobre o uso do solo e a concentração de pesticidas na água de nascentes na delineação de zonas de recarga de água subterrânea num aquífero fraturado.

Appendices

Appendix A: cost-distance matrix

As infinitely many possible cost-distance functions can be imagined, a cost-distance function based on physical consideration was derived from a random walk model. Additionally to regional faults, the Luxembourg Sandstone is fractured along two main directions approximately orthogonal to each other (NE–SW and NW–SE; Colbach 2005). As a first approximation, these two directions are assumed to be the only hydraulically significant ones and the hydraulic conductivity to be an anisotropic property of the aquifer that can be described by a first-order tensor spanning an ellipsoid in the 2D plane (Scheidegger 1957). Two related hydraulic properties are relevant for the problem:

• The hydraulic conductivity describing the ability of the rock medium as a whole to transmit water

• The connectivity of the fracture network

It is postulated that the probability for two given points in the 2D space to be connected decreases with distance and with deviation from the major semi-axis of the anisotropy ellipsoid. Let x(r, θ) be any point in the plane with position expressed in polar coordinates (r = distance from the spring and θ = angle from the major semi-axis), and P(x) the probability that x is connected to the spring, so that

$$ P(x)\sim \frac{1}{r} $$

(9)

$$ P(x)\sim \frac{1}{\theta } $$

(10)

Now consider a 2D random walk on a square grid oriented along the major and minor axis of anisotropy and a Manhattan metrics, so that only steps parallel to one of these axes are allowed. The step size is constant, and the random walk is directed towards the spring, so that at each point, there are only two possible choices, either “down” parallel to the major semi-axis (the ordinate) or “right” parallel to the minor semi-axis (the abscissa) and towards the major semi-axis. The probability for a molecule to flow along one of these two directions is

$$ P\left( {s=\delta d} \right)=p $$

(11)

and

$$ P\left( {s=\delta r} \right)=1-p=q $$

(12)

where s is the step size and δd and δr are the respective directions (“down” and “right”).

p and q can be equal, in which case the aquifer is isotropic. Any path taken by a water molecule during its random walk to the spring can be decomposed into its two components along each axis. In Cartesian coordinates, the probability for any x(i, j) to be hydraulically connected to the spring is

$$ P(x)={q^i}{p^j} $$

(13)

Changing to a cylindrical coordinate system by setting i = r cos θ and j = r sin θ yields

$$ P(x)=p{q^r}\times {p^{{\sin \theta }}}\times {q^{{\cos \theta }}} $$

(14)

The cost-distance function implicitly describes the probability that a point is hydraulically connected to the spring, thus the higher the probability, the lower the cost. The cost-distance function is defined thus

$$ d\left( {g,s} \right)=P{(x)^{-1 }}={{\left( {pq} \right)}^{{-\mathrm{r}}}}\times {p^{{-\sin \theta }}}\times {q^{{-\cos \theta }}} $$

(15)

Groundwater flows parallel to the hydraulic gradient in isotropic media only. In two dimensions, anisotropy causes a deviation of flow direction towards the semi-major axis of the anisotropy ellipse.

Let i ₁ and j ₁ be the components of the hydraulic gradient along the major and minor semi-axes, i ₂ the component of groundwater flow along the major semi-axis, θ ₁ the angle between the hydraulic gradient and the major semi-axis, and θ ₂ the angle between the groundwater flow direction and the major semi-axis and s the anisotropy factor.

$$ {i_1}={j_1}=\frac{1}{\mathrm{s}}{i_2} $$

(16)

$$ \tan \left( {{\theta_1}} \right)=\left( {\frac{{{j_1}}}{{{i_1}}}} \right)=1 $$

(17)

$$ \tan \left( {{\theta_2}} \right)=\left( {\frac{{{j_1}}}{{{i_2}}}} \right)=\frac{1}{\mathrm{s}}\left( {\frac{{{j_1}}}{{{i_1}}}} \right) $$

(18)

$$ \begin{array}{*{20}c} {{\tan^{-1 }}\left[ { \tan \left( {{\theta_2}} \right)} \right]={\tan^{-1 }}\left[ {\frac{1}{\mathrm{s}}\left( {\frac{{{j_1}}}{{{i_1}}}} \right)} \right]={\tan^{-1 }}\left[ {\frac{1}{\mathrm{s}}} \right]} \hfill \\ {\Rightarrow {\theta_2}={\tan^{-1 }}\left[ {\frac{1}{\mathrm{s}}} \right]} \hfill \\\end{array}$$

(19)

Thus, deviation from the steepest slope can be computed for different anisotropy factors s. This factor is related to the random walk probabilities introduced above by the relation

$$ s=\frac{p}{q} $$

(20)

The correspondence between s and p is given in Table 3.

Table 3 Deviation of groundwater flow from the hydraulic gradient for different anisotropy factors s. p is the probability for water to flow towards the spring defined in Eq. 11

Anisotropy was estimated from measurements of fracture orientation on outcrops around the plateau by setting s equal to the ratio of the frequency of the two dominant directions. This approach yields s = 1.7, so the maximum deviation of the groundwater flow from the hydraulic gradient was assumed to be about 15° and p was set equal to 0.63 in Eq. 7.

Appendix B: groundwater residence times

Table 4 Estimated groundwater residence times