Hydrogeological model of the Baltic Artesian Basin

Abstract

The Baltic Artesian Basin (BAB) is a complex multi-layered hydrogeological system in the south-eastern Baltic covering about 480,000 km². The aim of this study is to develop a closed hydrogeological mathematical model for the BAB. Heterogeneous geological data from different sources were used to build the geometry of the model, i.e. geological maps and stratigraphic information from around 20,000 boreholes. The finite element method was used for the calculation of the steady-state three-dimensional (3D) flow of unconfined groundwater. The 24-layer model was divided into about 1,000,000 finite elements. A simple recharge model was applied to describe the rate of infiltration, and the discharge was set at the water-supply wells. Variable hydraulic conductivities were used for the upper (Quaternary) deposits, while constant hydraulic conductivity values were assumed for the deeper layers. The model was calibrated on the statistically weighted borehole water-level measurements, applying L-BFGS-B (automatic parameter optimization method) for the hydraulic conductivities of each layer. The principal flows inside the BAB and the integral flow parameters were analyzed. The modeling results suggest that deeper aquifers are characterized by strong southeast–northwest groundwater flow, which is altered by the local topography in the upper, active water-exchange aquifers.

Résumé

Le bassin artésien de la Baltique (BAB) est un système hydrogéologique complexe multi-couches dans le Sud-Est de la Baltique couvrant environ 480 000 km². L’objectif de cette étude est de développer un modèle mathématique hydrogéologique fermé pour le BAB. Les données géologiques hétérogènes telles que cartes géologiques et informations stratigraphiques de quelques 20,000 forages, issues de différentes sources ont été utilisées pour construire la géométrie du modèle. Le modèle à éléments finis a été utilisé pour le calcul des écoulements en régime permanent et en 3D pour la partie aquifère libre. Le modèle à 24 couches a été divisé en environ 1,000,000 éléments finis. Un modèle simple de recharge a été appliqué afin de décrire la vitesse d’infiltration et le débit a été pris en considération au niveau des puits utilisés pour l’alimentation en eau. Des conductivités hydrauliques variables ont été utilisées pour les dépôts supérieurs (Quaternaire) alors que des valeurs de conductivité hydraulique constantes ont été attribuées pour les couches plus profondes. Le modèle a été calibré sur les données pondérées de niveau d’eau dans les forages en ayant recours à une méthode d’optimisation automatique des paramètres (L-BFGS-B) pour les conductivités hydrauliques de chaque couche. Les flux principaux au sein du BAB et les paramètres intégrés du flux ont été analysés. Les résultats du modèle suggèrent que les aquifères les plus profonds sont caractérisés par un fort écoulement Sud-Est Nord-Ouest, qui est perturbé localement par la topographie dans les parties aquifères supérieures caractérisés par des échanges d’eau actifs.

Resumen

La cuenca artesiana del Báltico (BAB) es un sistema hidrogeológico multicapa complejo en el sudeste del Báltico, que cubre alrededor de 480,000 km². El objetivo de este estudio es desarrollar un modelo matemático hidrogeológico cerrado para el BAB. Se usaron datos geológicos heterogéneos de diversas fuentes para construir la geometría del modelo, por ejemplo, mapas geológicos e información estratigráfica de alrededor de 20,000 perforaciones. Se usó el método de elementos finitos para el cálculo del flujo tridimensional (3D) en estado estacionario del agua subterránea no confinada. El modelo de 24 capas se dividió en cerca de 1,000,000 de elementos finitos. Se aplicó un modelo de recarga simple para describir la tasa de infiltración, y la descarga se estableció en los pozos de abastecimiento de agua. Se usaron conductividades hidráulicas variables para los depósitos superiores (Cuaternario) mientras que los valores de conductividad hidráulica se asumieron como constantes para las capas más profundas. El modelo fue calibrado en base a mediciones de niveles de agua de pozos estadísticamente ponderadas, aplicando el L-BFGS-B (método automático de automatización de parámetros) para las conductividades hidráulicas de cada capa. Se analizaron los flujos principales y los parámetros integrales de flujo dentro del BAB. Los resultados del modelado sugieren que los acuíferos más profundos están caracterizados por un fuerte flujo de agua subterránea sudeste–noroeste, que está alterado por la topografía local en la parte superior, de acuíferos con intercambios activos.

Resumo

A Bacia Artesiana do Báltico (BAB) é um sistema hidrogeológico multi-camada complexo, localizado no sudeste do Báltico, cobrindo uma área de cerca de 480,000 km². O objetivo desde estudo é o desenvolvimento de um modelo matemático hidrogeológico fechado da BAB. Foram usados dados geológicos heterogéneos provenientes de diferentes origens para construir a geometria do modelo, i.e., mapas geológicos e informação estratigráfica de cerca de 20,000 furos. Foi usado o método dos elementos finitos para o cálculo do estado estacionário tri-dimensional (3D) do fluxo de água subterrânea não confinada. O modelo com 24 camadas foi dividido em cerca de 1,000,000 de elementos finitos. Foi aplicado um modelo simples de recarga para descrever a taxa de infiltração, e a descarga foi definida nos poços para abastecimento de água. Foram usados diferentes valores de condutividade hidráulica nos depósitos superiores (Quaternário), enquanto que nas camadas mais profundas foram assumidos valores constantes de condutividade hidráulica. O modelo foi calibrado com base na estatística ponderada dos níveis de água medidos nos furos, aplicando o L-BFGS-B (método de optimização automática de parâmetros) para as condutividades hidráulicas de cada camada. Foram analisados os fluxos principais dentro da BAB, bem como todos os parâmetros de fluxo. Os resultados da modelação sugerem que os aquíferos mais profundos são caraterizados por um fluxo importante de direção sudeste–noroeste, que é modificado pela topografia local nos aquíferos superiores ativos onde se dá a troca de água.

Appendix

Comparison with analytical solution

The model is validated against solutions of analytic problems. The first comparison is carried out with the analytical solution of the groundwater flow between two water reservoirs with fixed levels and uniform infiltration between them (Harr 1962). The analytical solution for the water level is predicted as

$$ {{h}^{2}} = h_{0}^{2} - \frac{{\left( {h_{0}^{2} - h_{{\text{L}}}^{2}} \right)x}}{L} + \frac{W}{{{{K}_{{\text{S}}}}}}\left( {L - x} \right)x, $$

(11)

where h is the water level, x is the position between the reservoirs, W is the recharge rate in m/s, K is the hydraulic conductivity in m/s, h ₀ and h _L are the water levels at x = 0 and x = L, respectively. Both water levels are equal and only half of the domain is modeled because the solution in this case is symmetric to the vertical line x = L/2. The length of the domain for the analytic solution (L) is 1.8 m, h ₀ = h _L = 0.1 m, W = 0.1 m/day and K = 1 m/day. The numeric domain starts at x = −0.1 m and ends at x = 1 m, the water head is set as a boundary condition h(−0.1–0) = 0.1 m, and the recharge rate W(0–0.91) = 0.1 m/day. In Fig. 14, the analytical (dashed line) and numerical (dotted line for K = 1 m/day and solid line for K = 0.71 m/day) solutions are shown in vertical cross-section with the numerical FE mesh as gray lines in the background.

Fig. 14

Groundwater level between reservoirs. Dashed line: analytical solution (Harr 1962); dotted line: MOSYS solution (K = 1 m/day); solid line: MOSYS solution (K = 0.71 m/day), gray lines: FE mesh

The analytical solution shows a slightly higher piezometric head distribution than the modeling results using the same conductivity, K = 1 m/day. This is expected due to the simplified treatment of the unconfined zones—in the model the water can also flow through the elements above the water table. If the conductivity in the numerical solution is decreased so that the maximum values of the piezometric head are equal (at K = 0.71 m/day), the numerical solution is closer to the analytical one (solid line in Fig. 14). As the conductivity values in the BAB are not exactly known and will be calibrated anyway, the agreement with the analytical solution is acceptable for the practical application.

The other case for comparison relates to the analytical solution in the confined multi-well aquifer. The analytical solution for the head change h’ caused by pumping wells is given in Yeo and Lee (2003):

$$ h\prime \left( {x,y} \right) = \frac{{ - 4}}{{{{\pi }^{2}}}}\sum\limits_{{m = 1}}^{\infty } {\sum\limits_{{n = 1}}^{\infty } {\frac{{\sum\limits_{{i = 1}}^{p} {{{q}_{{\text{i}}}}\sin \frac{{n\pi {{x}_{{\text{i}}}}}}{L}} \sin \frac{{m\pi {{y}_{{\text{i}}}}}}{H}}}{{\left( {{{T}_{{\text{x}}}}{{n}^{2}}\frac{H}{L} + {{T}_{{\text{y}}}}{{m}^{2}}\frac{L}{H}} \right)}}} } \sin \frac{{n\pi x}}{L}\sin \frac{{m\pi y}}{H}, $$

(12)

where L is the domain size along the x axis, H is the domain size along the y axis, T _x and T _y are the hydraulic transmissivities in the x and y directions, respectively, x _i and y _i are the co-ordinates of the pumping wells, p is the number of wells, q _i is the pumping rate of the i-th well.

Three wells with a pumping rate of 1 m³/day are positioned at (x,y) = (300, 500), (700, 300), (700, 700). The head is fixed to h = 0.5 m at x = 0 m and to h = 1 m at x = 1,000 m. No-flow boundaries are set at y = 0 m and y = 1,000 m. In the analytical solution, this boundary condition is realized extending the domain in the y-direction 5 times and using 12 imaginary wells, similar to the situation described in Yeo and Lee (2003). The thickness of the aquifer is 1 m and conductivity K _x = K _y = K _z = 1 m/day, resulting in T _x = T _y = 1 m²/day for the analytical solution. The mesh size is uniform with a target triangle area of 250 m². The triangle mesh (thin lines) and isolines of the piezometric head distribution (bold lines) are shown in Fig. 15. The comparison of the analytical (dashed line) and numerical (dotted line) solutions along line y = 500 m in Fig. 16 shows a good agreement between both. The differences in elements near the well can be reduced by local grid refinement (gray solid line), where the target triangle area is 50 m² within a radius of 50 m from the first well.

Fig. 15

Triangular mesh and isolines of piezometric head with step of 0.1 m. Head h = 0.5 m at x = 0 m and h = 1 m at x = 1,000 m

Fig. 16

Analytic solution (dashed line) and numerical results (dotted line; and solid gray line for the refined solution) at y = 500 m