Determination of Optimal Diffusion Coefficients in Lake Zirahuén through a Local Inverse Problem

In this study, optimal diffusion coefficients for Lake Zirahuén, Mexico, were found under particular conditions based on images taken with a drone of a dye release experiment. First, the dye patch concentration was discretized using image processing tools, and it was then approximated by an ellipse, finding the optimal major and minor axes. The inverse problem was implemented by comparing these observational data with the concentration obtained numerically from the 2D advection–diffusion equation, varying the diffusion tensor. When the tensor was isotropic, values of K11=K22≈0.003 m2/s were found; when nonequal coefficients were considered, it was found that K11≈0.005 m2/s and K22≈0.002 m2/s, and the cross-term K12 influenced the results of the orientation of the ellipse. It is important to mention that, with this simple technique, the parameter estimation had consequences of great importance as the value for the diffusion coefficient was bounded significantly under particular conditions for this site of study.

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Assessing lateral flows and solute transport during floods in a conduit-flow-dominated karst system using the inverse problem for the advection–diffusion equation

Hydrology and Earth System Sciences ◽

10.5194/hess-21-3635-2017 ◽

2017 ◽

Vol 21 (7) ◽

pp. 3635-3653 ◽

Cited By ~ 15

Author(s):

Cybèle Cholet ◽

Jean-Baptiste Charlier ◽

Roger Moussa ◽

Marc Steinmann ◽

Sophie Denimal

Keyword(s):

Inverse Problem ◽

Solute Transport ◽

Unsaturated Zone ◽

Flood Routing ◽

Saturated Zone ◽

Karst System ◽

Advection Diffusion Equation ◽

Advection Diffusion ◽

Karst Conduit ◽

Conduit Network

Abstract. The aim of this study is to present a framework that provides new ways to characterize the spatio-temporal variability of lateral exchanges for water flow and solute transport in a karst conduit network during flood events, treating both the diffusive wave equation and the advection–diffusion equation with the same mathematical approach, assuming uniform lateral flow and solute transport. A solution to the inverse problem for the advection–diffusion equations is then applied to data from two successive gauging stations to simulate flows and solute exchange dynamics after recharge. The study site is the karst conduit network of the Fourbanne aquifer in the French Jura Mountains, which includes two reaches characterizing the network from sinkhole to cave stream to the spring. The model is applied, after separation of the base from the flood components, on discharge and total dissolved solids (TDSs) in order to assess lateral flows and solute concentrations and compare them to help identify water origin. The results showed various lateral contributions in space – between the two reaches located in the unsaturated zone (R1), and in the zone that is both unsaturated and saturated (R2) – as well as in time, according to hydrological conditions. Globally, the two reaches show a distinct response to flood routing, with important lateral inflows on R1 and large outflows on R2. By combining these results with solute exchanges and the analysis of flood routing parameters distribution, we showed that lateral inflows on R1 are the addition of diffuse infiltration (observed whatever the hydrological conditions) and localized infiltration in the secondary conduit network (tributaries) in the unsaturated zone, except in extreme dry periods. On R2, despite inflows on the base component, lateral outflows are observed during floods. This pattern was attributed to the concept of reversal flows of conduit–matrix exchanges, inducing a complex water mixing effect in the saturated zone. From our results we build the functional scheme of the karst system. It demonstrates the impact of the saturated zone on matrix–conduit exchanges in this shallow phreatic aquifer and highlights the important role of the unsaturated zone on storage and transfer functions of the system.

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Identification of a space- and time-dependent source in a variable coefficient advection-diffusion equation from Dirichlet and Neumann boundary measured outputs

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2020-0087 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Cristiana Sebu

Keyword(s):

Inverse Problem ◽

Diffusion Equation ◽

Variable Coefficient ◽

Source Function ◽

Practical Interest ◽

Time Dependent ◽

Input Output ◽

Space And Time ◽

Advection Diffusion Equation ◽

Advection Diffusion

AbstractThis paper considers the inverse problem of identifying an unknown space- and time-dependent source function F(x,t) in the variable coefficient advection-diffusion equationu_{t}=(D(x)u_{x})_{x}-(V(x)u)_{x}+F(x,t)from the Dirichlet \nu(t):=u(\ell,t) and Neumann f(t):=-D(0)u_{x}(0,t), t\in(0,T], boundary measured outputs. This problem was motivated by several important real-world applications in the field of contaminant hydrogeology, and the novel analysis presented here is highly relevant to problems of practical interest. The input-output operators corresponding to the Dirichlet and Neumann measured boundary data are introduced. The inverse problem is then formulated as a system of operator equations consisting of these operators and the measured outputs. The compactness and Lipschitz continuity of the input-output operators are proved in the relevant classes of admissible source functions ℱ and \mathcal{F}_{r}. These results together with the derived trace estimates allow us to show the existence of a quasi-solution of the inverse source problem as a minimum of the Tikhonov functional, under minimal regularity assumptions with respect to the source function and other inputs. An explicit gradient formula for the Fréchet gradient of the Tikhonov functional is also derived by means of an appropriate adjoint problem.

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Determination of the pollutant far-fields dilution using variable turbulent diffusion coefficients

Water Science & Technology ◽

10.2166/wst.1995.0196 ◽

1995 ◽

Vol 32 (7) ◽

pp. 41-46 ◽

Cited By ~ 2

Author(s):

T. V. Gardanov

Keyword(s):

Diffusion Coefficient ◽

Turbulent Diffusion ◽

Diffusion Coefficients ◽

Concentration Distribution ◽

Far Field ◽

Wastewater Discharge ◽

Main Subject ◽

Turbulent Diffusion Coefficient ◽

Advection Diffusion

The main subject of the paper is deriving appropriate relationships to calculate the horizontal turbulent diffusion coefficient, the size of the far-field polluted plume and the concentration distribution in the case of the continuous steady state wastewater discharge into receiving marine environment. The simplified advection diffusion-diffusion equation with variable (depending on the distance from the outfalls) coefficients of turbulent diffusion has been solved for this purpose.

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