The use of spectral analysis-based exact solutions to characterize topography-controlled groundwater flow

Abstract

Spectral analysis enhances the ability to analyze groundwater flow at a steady state by separating the top boundary condition into its periodic forms. Specifically, spectral analysis enables comparisons of the impact of individual spatial scales on the total flow field. New exact spectral solutions are presented for analyzing 3D groundwater flow with an arbitrarily shaped top boundary. These solutions account for depth-decaying, anisotropic and layered permeability while utilizing groundwater flux or the phreatic surface as a top boundary condition. Under certain conditions, groundwater flow is controlled by topography. In areas where the groundwater flow is controlled by the topography, the unknown water table is often approximated by the topography. This approximation induces a systematic error. Here, the optimal resolution of digital elevation models (DEMs) is assessed for use as a top boundary in groundwater flow models. According to the analysis, the water-table undulation is smoother than the topography; therefore, there is an upper limit to the resolution of DEMs that should be used to represent the groundwater surface. The ability to represent DEMs of various spectral solutions was compared and the results indicate that the fit is strongly dependent on the number of harmonics in the spectral solution.

Résumé

L’analyse spectrale accroît la capacité d’analyse d’écoulement de nappe en régime permanent en décomposant la condition limite au toit en ses composantes périodiques. En particulier, elle rend possible la comparaison de l’impact des diverses échelles spatiales sur le champ d’écoulement total. Des nouvelles solutions spectrales exactes sont présentées pour l’analyse 3D de l’écoulement souterrain avec une forme arbitraire en limite au toit. Ces solutions rendent compte de la diminution de la perméabilité avec la profondeur, de son anisotropie en milieu stratifié quand on utilise le flux de nappe ou sa surface comme condition aux limites. Dans certaines conditions, le flux souterrain est contrôlé par la topographie. Dans des zones où le flux souterrain est contrôlé par la topographie, la surface libre inconnue de la nappe peut de façon approximative en dépendre. Cette approximation induit une erreur systématique. Ici, la solution optimale des modèles numériques d’élévation (DEMs) est admise comme limite au toit dans les modèles d’écoulement de flux. Suivant l’analyse, l’ondulation de la surface libre de la nappe peut être plus faible que la topographie; c’est pourquoi il existe une borne supérieure aux solutions des DEMs pouvant être utilisée pour représenter la surface de la nappe. Les différentes représentations des solutions spectrales des DEMs ont été comparées et les résultats indiquent que l’ajustement est fortement dépendant du nombre d’harmoniques du spectre solution.

Resumen

El análisis espectral mejora la capacidad para analizar el flujo de agua subterránea en estado estacionario separando las condiciones de borde superior en sus formas periódicas. Específicamente, el análisis espectral permite la comparación del impacto de las escalas individuales espaciales sobre el campo total de flujo. Se presentan las nuevas soluciones espectrales exactas para analizar el flujo de agua subterránea 3D con una forma arbitraria del límite superior. Estas soluciones tienen en cuenta para la declinación en profundidad, la anisotropía y la estratificación de la permeabilidad al mismo tiempo que utilizan el flujo de agua subterránea ó la superficie freática como una condición de borde superior. Bajo ciertas condiciones, el flujo de agua subterránea está controlado por la topografía. En áreas donde el flujo de agua subterránea está controlado por la topografía, el nivel freático desconocido es a menudo aproximado por la topografía. Esta aproximación induce a un error sistemático. Aquí se evalúa la resolución óptima de los modelos digitales de elevación (DEMs) para usar como borde superior en los modelos de flujo de agua subterránea. De acuerdo con el análisis, la ondulación del nivel freático es más suave que la topografía, por lo tanto, existe un límite superior para la resolución de los DEMs que debe ser usado para representar la superficie del agua subterránea. Se compara la capacidad para representar DEMs de varias soluciones espectrales y los resultados indican que el ajuste es fuertemente dependiente del numérico de armónicas en la solución espectral.

摘要

光谱分析可通过将上边界条件分解成周期性的形式以提高分析稳态地下水流的能力,且可比较各个空间单元对总流场的影响。新的精确光谱解可用于分析具有任意形状上边界的三维地下水流。当将地下水流线或潜水面作为上边界条件时,这些精确解与衰减深度、各向异性及分层渗透性有关。在特定的条件下,地下水流由地形控制,这种地区的未知的地下水面可由地形近似。这种近似引起系统误差。对数字高程模型(DEMs)的最佳分辨率进行评价以在地下水流模型中的上边界中进行应用。根据分析,地下水面波动较地形平缓,因此DEMs分辨率的上限用于代表地下水面。对代表各种光谱解的DEMs的能力进行比较,结果表明拟合程度强烈依赖于光谱解中谐波的数目。

Resumo

A análise espectral amplia a capacidade de analisar o escoamento de água subterrânea em regime estacionário, através da separação da sua condição limite superior nas suas formas periódicas. Especificamente, a análise espectral permite comparações do impacte das escalas espaciais individuais no campo de escoamento total. São apresentadas novas soluções espectrais exactas para a análise de escoamentos subterrâneos 3D com um limite superior arbitrário. Estas soluções explicam a anisotropia, a estratificação e a diminuição em profundidade da permeabilidade, enquanto usam o escoamento subterrâneo ou a superfície freática como uma condição limite superior. Sob certas circunstâncias, o escoamento da água subterrânea é controlado pela topografia. Em áreas onde o escoamento da água subterrânea é controlado pela topografia, a superfície freática desconhecida acompanha aproximadamente a topografia. Esta aproximação induz um erro sistemático. Aqui, a resolução óptima do modelo digital de terreno (DEM) é avaliada para ser usada como um limite superior em modelos de escoamento subterrâneo. De acordo com a análise, a ondulação da superfície freática é mais suave que a topografia; consequentemente, há um limite superior para a resolução dos DEM que pode ser usado para a representação da superfície livre da água subterrânea. Foi comparada a capacidade de representar DEMs de várias soluções espectrais, e os resultados indicam que o ajuste é fortemente dependente do número de harmónicos da solução espectral.

Appendix: Derivation of the decay function, ψ

The decay function is derived for a two layer system, where layer 1 extends from a semi-infinite depth to the layer interface at z  = ε, and layer 2 extends from z  = ε to z  = 0. Solutions for systems containing more layers are derived in the same manner, where two arbitrary constants are added to the solution for each additional layer. The number of continuity conditions also increases by two for each additional layer, enabling the derivation of the decay function regardless of the number of layers.

The decay function in each layer is given by Eq. (20), which provides the following expressions for the individual layers of the two-layer solution:

$$ {\psi_1}(z) = {A_1}{e^{{{r_{{A,1}}}z}}} + {B_1}{e^{{{r_{{B,1}}}z}}} $$

(26)

$$ {\psi_2}(z) = {A_2}{e^{{{r_{{A,2}}}z}}} + {B_2}{e^{{{r_{{B,2}}}z}}} $$

(27)

where:

$$ \begin{gathered} {r_{{A,1}}} = - \frac{{{c_1}}}{2} + \sqrt {{\frac{{{c_1}^2}}{4} + {\alpha_1}\left( {k_x^2 + k_y^2} \right)}} \hfill \\ {r_{{B,1}}} = - \frac{{{c_1}}}{2} - \sqrt {{\frac{{{c_1}^2}}{4} + {\alpha_1}\left( {k_x^2 + k_y^2} \right)}} \hfill \\ {r_{{A,2}}} = - \frac{{{c_2}}}{2} + \sqrt {{\frac{{{c_2}^2}}{4} + {\alpha_2}\left( {k_x^2 + k_y^2} \right)}} \hfill \\ {r_{{B,2}}} = - \frac{{{c_2}}}{2} - \sqrt {{\frac{{{c_2}^2}}{4} + {\alpha_2}\left( {k_x^2 + k_y^2} \right)}} \hfill \\ \end{gathered} $$

Applying the bottom boundary condition Eq. (23) on Eq. (26):

$$ 0 = {A_1}{e^{{{r_{{A,1}}} \cdot \left( { - \infty } \right)}}} + {B_1}{e^{{{r_{{B,1}}} \cdot \left( { - \infty } \right)}}} \Rightarrow {B_1} = 0 \Rightarrow {\psi_1}(z) = {A_1}{e^{{{r_{{A,1}}}z}}} $$

(28)

Applying the potential continuity Eq. (22) on Eqs. (27) and (28):

$$ \begin{gathered} {\psi_1}\left( {x,y,\varepsilon } \right) = {\psi_2}\left( {x,y,\varepsilon } \right) \Leftrightarrow {A_1}{e^{{{r_{{A,1}}}\varepsilon }}} = {A_2}{e^{{{r_{{A,2}}}\varepsilon }}} + {B_2}{e^{{{r_{{B,2}}}\varepsilon }}} \hfill \\ \Rightarrow {A_1} = \left( {{A_2}{e^{{{r_{{A,2}}}\varepsilon }}} + {B_2}{e^{{{r_{{B,2}}}\varepsilon }}}} \right){e^{{ - {r_{{A,1}}}\varepsilon }}} \hfill \\ \end{gathered} $$

(29)

Applying the flux continuity condition Eq. (21) on Eqs. (27) and (28):

$$ \begin{gathered} - {K_{{v,1}}}\left( \varepsilon \right){\left. {\frac{{\partial {\psi_m}}}{{\partial z}}} \right|_{{z = \varepsilon }}} = - {K_{{v,2}}}\left( \varepsilon \right){\left. {\frac{{\partial {\psi_{{m + 1}}}}}{{\partial z}}} \right|_{{z = \varepsilon }}} \hfill \\ \Leftrightarrow \hfill \\ {K_{{v,1}}}\left( \varepsilon \right){A_1}{r_{{A,1}}}{e^{{{r_{{A,1}}}\varepsilon }}} = {K_{{v,2}}}\left( \varepsilon \right)\left( {{A_2}{r_{{A,2}}}{e^{{{r_{{A,2}}}\varepsilon }}} + {B_2}{r_{{B,2}}}{e^{{{r_{{B,2}}}\varepsilon }}}} \right) \hfill \\ \end{gathered} $$

(30)

Combining Eq. (29) and Eq. (30), provide an expression of A ₂:

$$ {A_2} = \frac{{\left( {{K_{{v,2}}}\left( \varepsilon \right){r_{{B,2}}} - {K_{{v,1}}}\left( \varepsilon \right){r_{{A,1}}}} \right){e^{{{r_{{B,2}}}\varepsilon }}}}}{{\left( {{K_{{v,1}}}\left( \varepsilon \right){r_{{A,1}}} - {K_{{v,2}}}\left( \varepsilon \right){r_{{A,2}}}} \right){e^{{{r_{{A,2}}}\varepsilon }}}}}{B_2} $$

(31)

Assuming a constant value of ψ at z = 0 provides a relationship between A ₂ and B ₂. The simplest relationship is obtained by

$$ {\psi_2}(0) = 1 \Rightarrow {A_2} + {B_2} = 1 $$

(32)

Here it is noted that Eq. (32) is not a real top boundary condition. Equation (32) is used to derive the arbitrary constants A ₁, A ₂ and B ₂, providing an exact solution that satisfies Eq. (19) as well as the bottom boundary condition and continuity conditions. The solution for the potential field Eq. (18) is then derived in a second step adjusted to meet the top boundary condition, located at the water table, by selecting the correct Fourier coefficients.

From Eq. (31) + Eq. (32), one derives an expression for B ₂:

$$ {B_2} = \frac{{\left( {{K_{{v,1}}}\left( \varepsilon \right){r_{{A,1}}} - {K_{{v,2}}}\left( \varepsilon \right){r_{{A,2}}}} \right){e^{{{r_{{A,2}}}\varepsilon }}}}}{{\left( {{K_{{v,2}}}\left( \varepsilon \right){r_{{B,2}}} - {K_{{v,1}}}\left( \varepsilon \right){r_{{A,1}}}} \right){e^{{{r_{{B,2}}}\varepsilon }}} - \left( {{K_{{v,2}}}\left( \varepsilon \right){r_{{A,2}}} - {K_{{v,1}}}\left( \varepsilon \right){r_{{A,1}}}} \right){e^{{{r_{{A,2}}}\varepsilon }}}}} $$

(33)

Inserting Eq. (33) in Eq. (31), provides an expression for A ₂:

$$ {A_2} = \frac{{\left( {{K_{{v,2}}}\left( \varepsilon \right){r_{{B,2}}} - {K_{{v,2}}}\left( \varepsilon \right){r_{{A,1}}}} \right){e^{{{r_{{B,2}}}\varepsilon }}}}}{{\left( {{K_{{v,2}}}\left( \varepsilon \right){r_{{B,2}}} - {K_{{v,1}}}\left( \varepsilon \right){r_{{A,1}}}} \right){e^{{{r_{{B,2}}}\varepsilon }}} - \left( {{K_{{v,2}}}\left( \varepsilon \right){r_{{A,2}}} - {K_{{v,1}}}\left( \varepsilon \right){r_{{A,1}}}} \right){e^{{{r_{{A,2}}}\varepsilon }}}}} $$

(34)

Now, inserting Eqs. (33) and (34) in Eq. (29) renders an expression for Eq. (26):

$$ {A_1} = \frac{{\left( {{r_{{B,2}}} - {r_{{A,2}}}} \right){K_{{v,2}}}\left( \varepsilon \right){e^{{\left( {{r_{{A,2}}} + {r_{{B,2}}} - {r_{{A,1}}}} \right)\varepsilon }}}}}{{\left( {{K_{{v,2}}}\left( \varepsilon \right){r_{{B,2}}} - {K_{{v,1}}}\left( \varepsilon \right){r_{{A,1}}}} \right){e^{{{r_{{B,2}}}\varepsilon }}} - \left( {{K_{{v,2}}}\left( \varepsilon \right){r_{{A,2}}} - {K_{{v,1}}}\left( \varepsilon \right){r_{{A,1}}}} \right){e^{{{r_{{A,2}}}\varepsilon }}}}} $$

(35)

All arbitrary constants are now determined and the decay functions Eqs. (26) and (27) become:

$$ {\psi_1}(z) = \frac{{\left( {{r_{{B,2}}} - {r_{{A,2}}}} \right){K_{{v,2}}}\left( \varepsilon \right){e^{{\left( {{r_{{A,2}}} + {r_{{B,2}}} - {r_{{A,1}}}} \right)\varepsilon }}}}}{{\left( {{K_{{v,2}}}\left( \varepsilon \right){r_{{B,2}}} - {K_{{v,1}}}\left( \varepsilon \right){r_{{A,1}}}} \right){e^{{{r_{{B,2}}}\varepsilon }}} - \left( {{K_{{v,2}}}\left( \varepsilon \right){r_{{A,2}}} - {K_{{v,1}}}\left( \varepsilon \right){r_{{A,1}}}} \right){e^{{{r_{{A,2}}}\varepsilon }}}}}{e^{{{r_{{A,1}}}z}}} $$

(36)

$$ {\psi_2}(z) = \frac{{\left( {{K_{{v,2}}}\left( \varepsilon \right){r_{{B,2}}} - {K_{{v,1}}}\left( \varepsilon \right){r_{{A,1}}}} \right){e^{{{r_{{B,2}}}\varepsilon + {r_{{A,2}}}z}}} - \left( {{K_{{v,2}}}\left( \varepsilon \right){r_{{A,2}}} - {K_{{v,1}}}\left( \varepsilon \right){r_{{A,1}}}} \right){e^{{{r_{{A,2}}}\varepsilon + {r_{{B,2}}}z}}}}}{{\left( {{K_{{v,2}}}\left( \varepsilon \right){r_{{B,2}}} - {K_{{v,1}}}\left( \varepsilon \right){r_{{A,1}}}} \right){e^{{{r_{{B,2}}}\varepsilon }}} - \left( {{K_{{v,2}}}\left( \varepsilon \right){r_{{A,2}}} - {K_{{v,1}}}\left( \varepsilon \right){r_{{A,1}}}} \right){e^{{{r_{{A,2}}}\varepsilon }}}}} $$

(37)

If the model does not comprise a layered representation of the subsurface the decay function becomes:

$$ \psi (z) = A{e^{{{r_A}z}}} + B{e^{{{r_B}z}}} $$

(38)

where:

$$ \begin{gathered} {r_A} = - \frac{c}{2} + \sqrt {{\frac{{{c^2}}}{4} + \alpha \left( {k_x^2 + k_y^2} \right)}} \hfill \\ {r_B} = - \frac{c}{2} - \sqrt {{\frac{{{c^2}}}{4} + \alpha \left( {k_x^2 + k_y^2} \right)}} \hfill \\ \end{gathered} $$

The constants A and B are determined for the conditions expressed in Eqs. (28) and (32), which provides the decay function presented in Eq. (25):

$$ \psi (z) = {e^{{{r_A}z}}} = {e^{{\left( { - \frac{c}{2} + \sqrt {{\frac{{{c^2}}}{4} + \alpha \left( {k_x^2 + k_y^2} \right)}} } \right)z}}} $$

(39)