Determination of the parameter pattern and values for a one-dimensional multi-zone unconfined aquifer

Abstract

In an aquifer, heterogeneity plays an important role in governing groundwater flow. Hence, aquifer characterization should involve both the pattern and values of the hydrogeological parameters. A new analytical solution describing the one-dimensional groundwater flow in a multi-zone unconfined aquifer is presented, and a methodology developed from the analytical solution and a heuristic approach for determining the pattern and values of the aquifer parameters are proposed. The analytical solution demonstrates that the hydraulic head varies spatially and is influenced by aquifer heterogeneity. Simulated annealing, a heuristic approach, is incorporated with the solution to simultaneously identify the pattern and values of the hydraulic conductivity for a horizontal multi-zone unconfined aquifer. This approach may be used to give an approximate result for a two-dimensional problem by dividing the model area into a number of transects along the transverse direction, identifying the parameter values along the longitudinal direction for each transect, and then smoothing the identified results.

Résumé

Dans un aquifère, l’hétérogénéité joue un rôle important en gouvernant l’écoulement des eaux souterraines. En conséquence, la caractérisation de l’aquifère devrait inclure et l’espace et les valeurs des paramètres hydrogéologiques. Une nouvelle solution analytique décrivant l’écoulement des eaux souterraines uni-dimensionnel dans un aquifère non-confiné à plusieurs couches est présenté, et la méthodologie développée à partir de la solution analytique et une approche heuristique pour déterminer l’espace et les valeurs des paramètres de l’aquifère sont proposées. La solution analytique démontre que la charge hydraulique varie spatialement et est influencée par l’hétérogénéité de l’aquifère. Le « recuit simulé », une approche heuristique d’optimisation, est incorporé avec la solution pour identifier simultanément l’espace et les valeurs de conductivités hydrauliques pour un aquifère libre horizontal et multi-zones. Cette approche peut être utilisée pour donner un résultat approximatif pour un problème bidimensionnel en divisant l’aire du modèle en une suite de plusieurs sections, identifiant les valeurs de paramètre le long de la direction longitudinale pour chaque section, et ensuite en arrondissant les résultats.

Resumen

En un acuífero, la heterogeneidad juega un papel importante en el flujo del agua subterránea. Así, la caracterización de un acuífero debe tener en cuenta tanto la disposición como los valores de los parámetros hidráulicos. Se presenta un nueva solución analítica que describe el flujo del agua subterránea en una dimensión en un acuífero libre multi-zona, y se propone una metodología desarrollada a partir de la solución analítica y la aproximación heurística para determinar la disposición y los valores de los parámetros del acuífero. La solución analítica demuestra que el nivel piezométrico varía espacialmente y está influenciado por la heterogeneidad del acuífero. Un templado simulado, una aproximación heurística, se ha incorporado a la solución para identificar simultáneamente la disposición y los valores de la conductividad hidráulica para un acuífero libre horizontal multi-zona. Esta aproximación puede ser utilizada para proporcionar un resultado aproximado en un problema bidimensional dividiendo el área del modelo en un número de transectos a lo largo de la dirección transversa, identificando los valores del parámetro a lo largo de la dirección longitudinal para cada transecto, y entonces suavizando los resultados identificados.

Appendix

The equation for one-dimensional steady-state groundwater flow in a heterogeneous unconfined aquifer with recharge is

$$ \frac{\partial } {{\partial x}}{\left( {k_{i} h_{i} \frac{{\partial h_{i} }} {{\partial x}}} \right)} + w = 0,i = 1,2,...,n $$

(6)

where h _i is the hydraulic head at x in the ith zone, x is the distance from the origin, K _i is the hydraulic conductivity in the ith zone, n is the number of zones and w is the recharge rate. Integrating Eq. (1) yields the expression

$$ h^{2}_{i} = - \frac{w} {{K_{i} }}x^{2} + C_{1} x + C_{2} $$

(7)

where C ₁ and C ₂ are constants of integration. The left-hand side and right-hand side boundary conditions of the problem domain are, respectively,

$$ h\left| {_{{x = 0}} = h_{0} } \right. $$

(8)

and

$$ h\left| {_{{x = L}} } \right. = h_{L} $$

(9)

where h ₀ is the hydraulic head at the origin, h _L is the hydraulic head at L, and L is the distance from the origin at the point h _L is measured. In addition, the continuity requirements for the hydraulic head and the flow rate per unit width at x _i are, respectively,

$$ h_{i} {\left( {x_{i} } \right)} = h_{{i + 1}} {\left( {x_{i} } \right)} $$

(10)

and

$$ q^{\prime }_{i} {\left( {x_{i} } \right)} = q^{\prime }_{{i + 1}} {\left( {x_{i} } \right)} $$

(11)

where h _i (x _i ) is the hydraulic head, x _i is the ith zone boundary, and $q^{\prime }_{i} {\left( {x_{i} } \right)}$ is the flow rate per unit width at x _i in the ith zone. By solving the above equations, the hydraulic head and the flow rate per unit width in the ith zone of the horizontal multi-zone unconfined aquifer with recharge are therefore, respectively,

$$ \begin{aligned} & h_{i} {\left( x \right)} = {\sqrt {{ - wx^{2} } \mathord{\left/ {\vphantom {{ - wx^{2} } {{K_{i} + \lambda x} \mathord{\left/ {\vphantom {{K_{i} + \lambda x} {K_{i} + {\sum\limits_{j = 1}^i {\Delta k_{j} } }\lambda x_{{j - 1}} + w{\sum\limits_{j = 1}^i {\Delta k_{j} x^{2}_{{j - 1}} + h^{2}_{0} } }}}} \right. \kern-\nulldelimiterspace} {K_{i} + {\sum\limits_{j = 1}^i {\Delta k_{j} } }\lambda x_{{j - 1}} + w{\sum\limits_{j = 1}^i {\Delta k_{j} x^{2}_{{j - 1}} + h^{2}_{0} } }}}}} \right. \kern-\nulldelimiterspace} {{K_{i} + \lambda x} \mathord{\left/ {\vphantom {{K_{i} + \lambda x} {K_{i} + {\sum\limits_{j = 1}^i {\Delta k_{j} } }\lambda x_{{j - 1}} + w{\sum\limits_{j = 1}^i {\Delta k_{j} x^{2}_{{j - 1}} + h^{2}_{0} } }}}} \right. \kern-\nulldelimiterspace} {K_{i} + {\sum\limits_{j = 1}^i {\Delta k_{j} } }\lambda x_{{j - 1}} + w{\sum\limits_{j = 1}^i {\Delta k_{j} x^{2}_{{j - 1}} + h^{2}_{0} } }}}} } \\ & \\ \end{aligned} $$

(12)

and

$$ q^{\prime }_{i} {\left( x \right)} = wx - \lambda \mathord{\left/ {\vphantom {\lambda 2}} \right. \kern-\nulldelimiterspace} 2 $$

(13)

where

$$ \lambda = \frac{{{\left[ {{\sum\limits_{i = 1}^n {{w{\left( {x^{2}_{i} - x^{2}_{{i - 1}} } \right)}} \mathord{\left/ {\vphantom {{w{\left( {x^{2}_{i} - x^{2}_{{i - 1}} } \right)}} {k_{i} }}} \right. \kern-\nulldelimiterspace} {k_{i} }} }} \right]} + {\left( {h^{2}_{L} - h^{2}_{0} } \right)}}} {{{\sum\limits_{i = 1}^n {{{\left( {x_{i} - x_{{i - 1}} } \right)}} \mathord{\left/ {\vphantom {{{\left( {x_{i} - x_{{i - 1}} } \right)}} {K_{i} }}} \right. \kern-\nulldelimiterspace} {K_{i} }} }}} $$

(14)

and

$$ \Delta k_{j} = {\left( {1 \mathord{\left/ {\vphantom {1 {K_{j} }}} \right. \kern-\nulldelimiterspace} {K_{j} } - 1 \mathord{\left/ {\vphantom {1 {K_{{j + 1}} }}} \right. \kern-\nulldelimiterspace} {K_{{j + 1}} }} \right)} $$

(15)

Equation (12) shows that the hydraulic head varies spatially and is influenced by the heterogeneous hydraulic conductivity of the multi-zone unconfined aquifer. When the aquifer is homogeneous, Eqs. (12) and (13) can, respectively, be reduced to (Fetter 1994)

$$ h = {\sqrt {h^{2}_{0} - \frac{{{\left( {h^{2}_{0} - h^{2}_{L} } \right)}}} {L}x + \frac{w} {K}{\left( {L - x} \right)}x} } $$

(16)

and

$$ q\prime = - Kh\frac{{\partial h}} {{\partial x}} = \frac{{K{\left( {h^{2}_{0} - h^{2}_{L} } \right)}}} {{2L}} - w{\left( {\frac{L} {2} - x} \right)} $$

(17)

Note that Eq. (16) can be employed to find the elevation of the water table everywhere between two points located L distance apart if the saturated thickness of the aquifer is known at the two end points.