Assessing groundwater quality trends in pumping wells using spatially varying transfer functions

Abstract

When implementing remediation programs to mitigate diffuse-source contamination of aquifers, tools are required to anticipate if the measures are sufficient to meet groundwater quality objectives and, if so, in what time frame. Transfer function methods are an attractive approach, as they are easier to implement than numerical groundwater models. However, transfer function approaches as commonly applied in environmental tracer studies are limited to a homogenous input of solute across the catchment area and a unique transfer compartment. The objective of this study was to develop and test an original approach suitable for the transfer of spatially varying inputs across multiple compartments (e.g. unsaturated and saturated zone). The method makes use of a double convolution equation accounting for transfer across two compartments separately. The modified transfer function approach was applied to the Wohlenschwil aquifer (Switzerland), using a formulation of the exponential model of solute transfer for application to subareas of aquifer catchments. A minimum of information was required: (1) delimitation of the capture zone of the outlet of interest; (2) spatial distribution of historical and future pollution input within the capture zone; (3) contribution of each subarea of the recharge zone to the flow at the outlet; (4) transfer functions of the pollutant in the aquifer. A good fit to historical nitrate concentrations at the pumping well was obtained. This suggests that the modified transfer function approach is suitable to explore the effect of environmental projects on groundwater concentration trends, especially at an early screening stage.

Résumé

Lors de la mise en œuvre de programmes de lutte contre les pollutions diffuses des aquifères, des outils sont nécessaires pour déterminer si les mesures mises en œuvre seront suffisantes pour atteindre les objectifs de qualité des eaux souterraines et, si oui, à quelle échéance temporelle. Les méthodes par fonction de transfert constituent une approche attractive car elles sont à priori plus faciles à mettre en œuvre que les modèles numériques de nappe. Cependant, les approches par fonction de transfert, appliquées couramment dans les études utilisant des traceurs environnementaux, sont limitées à un apport homogène de soluté sur tout le bassin versant et à un unique compartiment de transfert. L’objectif de cette étude est de développer et tester une approche originale adaptée pour le transfert de polluants entrant variant spatialement, et s’écoulant au travers de compartiments multiples (par exemple zones non saturée et saturée). La méthode utilise une équation de double convolution représentant séparément le transfert à travers deux compartiments. L’approche par fonction de transfert modifiée a été appliquée à l’aquifère de Wohlenschwil (Suisse), en utilisant une formulation du modèle exponentiel pour le transfert de soluté avec application à des sous-zones des bassins versants de l’aquifère. Un minimum d’information était requis : (1) délimitation de la zone d’appel du puits d’intérêt ; (2) distribution spatiale d’infiltrations passées et futures de pollution au sein de la zone d’appel ; (3) contribution de chaque sous-zone de la zone de recharge au débit au puits ; (4) fonctions de transfert du polluant dans l’aquifère. Un bon calage avec les historiques de concentrations en nitrate au puits de pompage a été obtenu. Cela suggère que l’approche par fonction de transfert modifiée est adaptée pour explorer les effets des projets environnementaux sur les tendances d’évolution des concentrations au sein des eaux souterraines, particulièrement dans une étape préliminaire au projet.

Resumen

Cuando se implementan programas de remediación para mitigar la contaminación de fuentes difusas en los acuíferos, se requieren herramientas para anticipar si las medidas son suficientes para satisfacer con los objetivos de la calidad de las aguas subterráneas y, en caso afirmativo, en qué plazos. Los métodos de función de transferencia son un enfoque atractivo, ya que son más fáciles de implementar que los modelos numéricos de agua subterránea. Sin embargo, el enfoque de la función de transferencia tal como son aplicadas comúnmente en estudios de seguimiento ambiental se limita a una entrada homogénea de solutos a través de la zona de captación y un único compartimento de transferencia. El objetivo de este estudio fue desarrollar y probar un enfoque original apropiado para la transferencia de las entradas espacialmente variables a través de múltiples compartimentos (por ejemplo zona no saturada y zona saturada). El método hace uso de una ecuación doble de convolución que tiene en cuenta la transferencia a través de dos compartimentos por separado. El enfoque de la función de transferencia modificada se aplicó al acuífero Wohlenschwil (Suiza), usando una formulación de un modelo exponencial de la transferencia de soluto para la aplicación a las subáreas de las cuencas de captación de acuíferos. Se requirió un mínimo de información: (1) la delimitación de la zona de captación de la salida de interés; (2) la distribución espacial de la entrada histórica y futuras de contaminantes dentro de la zona de captación; (3) la contribución de cada subzona de la zona de recarga para el flujo en la salida; (4) funciones de transferencia de contaminantes en el acuífero. Se obtuvo un buen ajuste a las concentraciones históricas de nitratos en el pozo de bombeo. Esto sugiere que el enfoque de la función de transferencia modificada es adecuado para explorar el efecto de los proyectos ambientales sobre las tendencias de concentración de las aguas subterráneas, especialmente en una etapa de detección temprana.

摘要

当实施修复项目以减缓含水层扩散源污染时,如果采取的措施足够满足地下水水质目标,就需要工具进行预测,要是这样,就需要在什么样的时间框架下进行预测。传递函数方法是一个有吸引力的方法,因为其方法比数值地下水模型更容易实施。然而,通常应用在环境示踪研究的传递函数方法受限于汇水区溶质的均匀性输入及独特的传递区划。本研究的目的就是开发和测试适合穿过多区划(例如非饱和带和饱和带)空间上变化的输入项传递的由独创性的方法。方法使用一个双卷积方程,这个双卷积方程可导出分别穿过两个区划的传递。采用应用在含水层汇水区亚区的溶质传递指数模型公式,把改进的传递函数方法用在了(瑞士)Wohlenschwil含水层上。需要的至少信息:1)相关出口捕获带的界定;2)捕获带内过去和将来污染输入的空间分布;3)补给带每个亚区对出口处水流的贡献量;4)含水层中污染物的传递函数。发现对抽水井历史硝酸盐含量拟合的非常好。这表明改进的传递函数方法适合探索环境项目对地下水污染物含量趋势的影响,特别是在屏蔽早期。

Resumo

Na implementação de programas de remediação para mitigar contra fontes difusas de contaminação em aquíferos, são necessárias ferramentas para antecipar se as medidas são suficientes para alcançar os padrões de qualidade da água e, nesse caso, em qual período de tempo. Métodos de funções de transferência são uma abordagem atraente, uma vez que eles são mais simples de se implementar do que modelos numéricos de águas subterrâneas. Entretanto, abordagens de funções de transferência comumente aplicadas como estudos ambientais de traçadores são limitados a entradas homogêneas de solutos através da área de captação e um único compartimento de transferência. O objetivo desse estudo foi desenvolver e testar uma abordagem original apropriada para a transferência de entradas espacialmente variante através de múltiplos compartimentos (p. ex. zonas não saturadas e saturadas). O método faz uso de uma equação de dupla convolução considerada para transferência através de dois compartimentos separados. A abordagem de função de transferência modificada foi aplicada ao aquífero Wohlenschwil (Suíça), utilizando a formulação de um modelo exponencial para transferência de soluto para aplicação a subáreas de captação do aquífero. A mínima informação requerida foi: (1) delimitação da zona de captura da saída de interesse; (2) distribuição espacial das entradas de poluição históricas e futuras da saída de interesse; (3) contribuição e cada subárea da zona de recarga para o fluxo na saída; (4) funções de transferência do poluente no aquífero. Um bom ajuste às concentrações históricas de nitrato no poço de bombeamento foi obtido. Isso sugere que a abordagem de função de transferência modificada é apropriada para explorar o efeito de projetos ambientais em tendências de concentrações de águas subterrâneas, especialmente nos estágios de rastreamento iniciais.

Appendix: Mathematical development of the exponential model for the transfer function at localized input zone and outlet

A saturated aquifer system of length L is assumed with a uniform thickness e, a uniform porosity θ and a uniform surface infiltration rate I (illustration in Fig. 1). This conceptual case has already been introduced by Kazemi et al. (2006), and one part of the mathematical development is taken from this reference—Eqs. (13), (14), (16), (17) and (20). According to the hypothesis of Dupuit, assuming horizontal velocity constant over depth, the Darcy flux over the distance corresponds to q(x) = Ix/e, and the real velocity over the distance x is v(x) = Ix/eφ. The travel time t(x) required for a solute from a point x to reach the outlet at distance L is:

$$ t(x)={\displaystyle \underset{x}{\overset{L}{\int }}}\frac{\mathrm{d}x}{v(x)}=\frac{e\phi }{I}{\displaystyle \underset{x}{\overset{L}{\int }}}\frac{\mathrm{d}x}{x}=\frac{e\phi }{I} \ln \left(\frac{L}{x}\right)={T}_0 \ln \left(\frac{L}{x}\right) $$

(13)

with

$$ {T}_0=\frac{e\phi }{I} $$

(14)

The cumulated flow Q _0z at the outlet (located at distance L) from the surface (z = 0) to a certain depth z is equal to Q _0z = zIL/e. The cumulated recharge Q _xL at the surface from an input point x to the outlet (x = L) is Q _xL = I(L − x). If z(x) is the depth reached by flows from an input point located at distance x, Q _0z and Q _xL are equal and the following relationship can be written as: z(x)IL/e = I(L − x). Equation (15) can be deduced by isolating z(x) from the last relationship:

$$ z(x)=e\left(1-\frac{x}{L}\right) $$

(15)

By combining Eqs. (13) and (15), it follows that:

$$ t(z)={T}_0 \ln \left(\frac{1}{1-\frac{z}{e}}\right) $$

(16)

$$ x(z)=\left(1-\frac{z}{e}\right)L $$

(17)

$$ x(t)=L \exp \left(-\frac{t}{T_0}\right) $$

(18)

$$ z(t)=e\left[1- \exp \left(-\frac{t}{T_0}\right)\right] $$

(19)

As reported by Kazemi et al. (2006), the transit time probability density function is defined as:

$$ \varphi (t)=\frac{1}{Q_0}\frac{\mathrm{d}Q(t)}{\mathrm{d}t} $$

(20)

where Q ₀ is the total discharge flow and Q(t) is the portion of discharge flow that has a residence time in the system of less than or equal to t. If solute enters in the same conditions as flow, and if the solute travel has the same characteristics as flow transit, the transit time probability density function ρ(t) is the transfer function g(t) of solute in the aquifer.

The transfer function of solute reaching the outlet and coming from the area bounded by the distances x = x ₁ and x = x ₂, can be defined as:

$$ \begin{array}{ll}g(t)=\frac{1}{Q_{0_{{\mathrm{X}}_1{\mathrm{X}}_2}}}\cdot \frac{\mathrm{d}{Q}_{{\mathrm{X}}_1{\mathrm{X}}_2}(t)}{\mathrm{d}t}\hfill &, {t}_2\le t\le {t}_1\hfill \end{array} $$

(21)

$$ \begin{array}{ll}g(t)=0\hfill & \hfill \end{array} $$

(22)

; otherwise, with

$$ \begin{array}{ll}\hfill & {t}_1={T}_0 \ln \left(\frac{L}{x_1}\right)\hfill \end{array} $$

(23)

$$ {t}_2={T}_0 \ln \left(\frac{L}{x_2}\right) $$

(24)

where ${Q}_{0_{{\mathrm{X}}_1{\mathrm{X}}_2}}$ is the total discharge flow at the outlet for water from the subarea of the recharge zone delimited by x ₁ and x ₂, and ${Q}_{{\mathrm{X}}_1{\mathrm{X}}_2}(t)$ is the part of the discharge flow that has a residence time in the system in the interval [t ₂; t].

Based on Eq. (18), ${Q}_{0_{{\mathrm{X}}_1{\mathrm{X}}_2}}$ and ${Q}_{{\mathrm{X}}_1{\mathrm{X}}_2}(t)$ are:

$$ \begin{array}{ll}{Q}_{0_{{\mathrm{X}}_1{\mathrm{X}}_2}}=I\left({x}_2-{x}_1\right)=IL\left[ \exp \left(-\frac{t_2}{T_0}\right)- \exp \left(-\frac{t_1}{T_0}\right)\right]\hfill &, {t}_2\le t\le {t}_1\hfill \end{array} $$

(25)

$$ \begin{array}{ll}{Q}_{{\mathrm{X}}_1{\mathrm{X}}_2}(t)=I\left({x}_2-x(t)\right)=IL\left[ \exp \left(-\frac{t_2}{T_0}\right)- \exp \left(-\frac{t}{T_0}\right)\right]\hfill &, {t}_2\le t\le {t}_1\hfill \end{array} $$

(26)

The transfer function ${g}_{{\mathrm{t}}_1{\mathrm{t}}_2}(t)$ of solute reaching the outlet in the travel time interval [t ₂; t ₁] can be derived from the discretization of Eq. (21), inserting Eqs. (25) and (26):

$$ \begin{array}{ll}{g}_{{\mathrm{t}}_1{\mathrm{t}}_2}(t)=\frac{ \exp \left(-\frac{t}{T_0}\right)}{T_0\left[ \exp \left(-\frac{t_2}{T_0}\right)- \exp \left(-\frac{t_1}{T_0}\right)\right]}\hfill &, {t}_2\le t\le {t}_1\hfill \end{array} $$

(27)

$$ \begin{array}{ll}{g}_{{\mathrm{t}}_1{\mathrm{t}}_2}(t)=0\hfill &, \mathrm{otherwise}\hfill \end{array} $$

(28)

Inserting Eq. (16) in Eq. (27), the transfer function ${g}_{{\mathrm{z}}_1{\mathrm{z}}_2}(t)$ of solute reaching the outlet at the depth interval [z ₂; z ₁] is:

$$ \begin{array}{ll}{g}_{{\mathrm{z}}_1{\mathrm{z}}_2}(t)=\frac{ \exp \left(-\frac{t}{T_0}\right)}{T_0\left(\frac{z_1}{e}-\frac{z_2}{e}\right)}\hfill &, {\mathrm{t}}_2\le \mathrm{t}\le {\mathrm{t}}_1\hfill \end{array} $$

(29)

$$ \begin{array}{ll}{g}_{{\mathrm{z}}_1{\mathrm{z}}_2}(t)=0\hfill & \hfill \end{array} $$

(30)

; otherwise, with

$$ \begin{array}{ll}\hfill & {t}_1={T}_0 \ln \left(\frac{1}{1-\frac{z_1}{e}}\right)\hfill \\ {}\hfill & {t}_2={T}_0 \ln \left(\frac{1}{1-\frac{z_2}{e}}\right)\hfill \end{array} $$

Inserting Eq. (13) in Eq. (27), the transfer function ${g}_{{\mathrm{x}}_1{\mathrm{x}}_2}(t)$ of solute from the subarea of the recharge zone delimited by the distance interval [x ₂; x ₁] is:

$$ \begin{array}{ll}{g}_{{\mathrm{x}}_1{\mathrm{x}}_2}(t)=\frac{L \exp \left(-\frac{t}{T_0}\right)}{T_0\left({x}_2-{x}_1\right)}\hfill &, {t}_2\le t\le {t}_1\hfill \end{array} $$

(31)

$$ \begin{array}{ll}{g}_{{\mathrm{x}}_1{\mathrm{x}}_2}(t)=0\hfill & \hfill \end{array} $$

(32)

; otherwise, with

$$ \begin{array}{ll}\hfill & {t}_1={T}_0 \ln \left(\frac{L}{x_1}\right)\hfill \end{array} $$

(33)

$$ {t}_2={T}_0 \ln \left(\frac{L}{x_2}\right) $$

(34)

Applying the mean value theorem, the average time of travel ${\overline{t(z)}}_{{\mathrm{z}}_1{\mathrm{z}}_2}$ over the depth interval [z ₁; z ₂] is the mean value of the function t(z) (Eq. 16) in the interval [z ₁; z ₂], which corresponds to:

$$ {\overline{t(z)}}_{{\mathrm{z}}_1{\mathrm{z}}_2}=\frac{1}{z_1-{z}_2}{\displaystyle \underset{z_2}{\overset{z_1}{\int }}}t(z)\mathrm{d}z=\frac{1}{z_1-{z}_2}{\displaystyle \underset{z_2}{\overset{z_1}{\int }}}{T}_0 \ln \left(\frac{1}{1-\frac{z}{e}}\right)\mathrm{d}z $$

(35)

$$ {\overline{t(z)}}_{{\mathrm{z}}_1{\mathrm{z}}_2}=\frac{e}{z_1-{z}_2}{T}_0\left[\left(1-\frac{z_1}{e}\right) \ln \left(1-\frac{z_1}{e}\right)-\left(1-\frac{z_2}{e}\right) \ln \left(1-\frac{z_2}{e}\right)+\frac{z_1-{z}_2}{e}\right] $$

(36)

Similarly, the average travel time ${\overline{t(x)}}_{{\mathrm{x}}_1{\mathrm{x}}_2}$ for solute from an input zone defined in the interval [x ₁; x ₂] is:

$$ {\overline{t(x)}}_{{\mathrm{x}}_1{\mathrm{x}}_2}=\frac{1}{x_2-{x}_1}{\displaystyle \underset{x_1}{\overset{x_2}{\int }}}t(x)\mathrm{d}x=\frac{1}{x_2-{x}_1}{\displaystyle \underset{x_1}{\overset{x_2}{\int }}}{T}_0 \ln \left(\frac{L}{x}\right)\mathrm{d}x $$

(37)

$$ {\overline{t(x)}}_{{\mathrm{x}}_1{\mathrm{x}}_2}=\frac{T_0}{x_2-{x}_1}\left[{x}_2\left( \ln \left(\frac{L}{x_2}\right)+1\right)-{x}_1\left( \ln \left(\frac{L}{x_1}\right)+1\right)\right] $$

(38)

$$ {\overline{t(x)}}_{{\mathrm{x}}_1{\mathrm{x}}_2}={T}_0\left[1 + \ln (L)+\frac{x_1 \ln \left({x}_1\right)-{x}_2 \ln \left({x}_2\right)}{x_2-{x}_1}\right] $$

(39)

From a practical point of view, Eq. (39) is relevant to calculate the mean travel time of solute pumped at a pumping well, where the screen section of the well corresponds to a certain depth interval of the aquifer. The mean travel time provides a first approximation of the order of magnitude of reaction time of the aquifer system (Fenton et al. 2011; Sousa et al. 2013).