Modeling Valley-Scale Lateral Hyporheic Exchange

J. Allgeier, S. Martin, O. A. Cirpka

Modeling Valley-Scale Lateral Hyporheic Exchange

simple → simpler

Motivation

Modeling Valley-Scale Lateral Hyporheic Exchange

Motivation

Modeling Valley-Scale Lateral Hyporheic Exchange

needed to fit titlefooter

Motivation

Modeling Valley-Scale Lateral Hyporheic Exchange

needed to fit titlefooter

Motivation

Modeling Valley-Scale Lateral Hyporheic Exchange

Motivation

Modeling Valley-Scale Lateral Hyporheic Exchange


Motivation

Modeling Valley-Scale Lateral Hyporheic Exchange


Motivation: System Understanding

Modeling Valley-Scale Lateral Hyporheic Exchange


Motivation: System Understanding

Modeling Valley-Scale Lateral Hyporheic Exchange



configuration

for spacing

Qex

Methods: Model Definition

Methods: Model Definition

\(\displaystyle\frac{\partial^2 h}{\partial x^2} + \frac{T_y}{T_x}\frac{\partial^2 h}{\partial y^2} = 0\)

Results: Semi-Analytical Solution




equations


Results: Semi-Analytical Solution




\( \begin{aligned} h(x,y) &= \textstyle h_1 +\textstyle\frac{h_2-h_1}{L} +\textstyle\sum\limits_{n=1}^{\infty}A_n \sin(cx) \sinh(cy)\\ \Psi(x,y) &= - T_x \left(A_0 + y \cdot \textstyle\frac{h_2-h_1}{L} + \textstyle\sum\limits_{n=1}^\infty \textstyle\frac{A_n}{\kappa} \cos(cx) \cosh(c\kappa y)\right)\\ \mathbf{A}&=\operatorname*{arg\,min}_\mathbf{\hat{A}}\textstyle\sum\limits_{j=1}^M \left(\hat{A}_0 + \textstyle\sum\limits_{n=1}^\infty \textstyle\frac{\hat{A}_n}{\kappa} \cos(cx_j) \cosh(c\kappa f_b(x_j))+ \textstyle\frac{Q(x_j)}{T_x} + \textstyle\frac{h_2-h_1}{L} f_b(x_j) \right)^2\\ c&= \textstyle\frac{n \pi}{L}; \qquad \kappa = \sqrt{\textstyle\frac{T_x}{T_y}}\\ \end{aligned}\)


Results: Semi-Analytical Solution




A = lhs\rhs;


Results: Semi-Analytical Solution




magic


Results: Semi-Analytical Solution




magic



Results: Semi-Analytical Solution


Results: Proxy-Relationships


  • many configurations
  • determine Qex semi-analytically
  • transform

Results: Proxy-Relationships




approximation

Results: Proxy-Relationships




\( Q_\mathrm{ex} \approx \textstyle\frac{h_1-h_2}{L} \cdot T_x \cdot (w_\mathrm{max}-w_\mathrm{min}) \cdot \operatorname*{sech}\left(a_1 \cdot \textstyle\frac{w_\mathrm{mean}}{L}\right) \)


Results: Proxy-Relationships




\( Q_\mathrm{ex} \approx \textstyle\frac{h_1-h_2}{L} \cdot T_x \cdot (w_\mathrm{max}-w_\mathrm{min}) \cdot \operatorname*{sech}\left(a_1 \cdot \textstyle\frac{w_\mathrm{mean}}{L}\right) \)



Results: Proxy-Relationships

\( Q_\mathrm{ex} \approx \textcolor{gray}{\frac{h_1-h_2}{L} \cdot T_x \cdot} (w_\mathrm{max}-w_\mathrm{min}) \cdot \operatorname*{sech}\left(\textcolor{gray}{a_1 \cdot} \frac{w_\mathrm{mean}}{L}\right) \)

Application: Real Systems

  • real floodplain aquifers
  • estimate flux
  • estimate area
  • estimate travel times
  • Ammer & Neckar

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