The behavior of many physical systems is described by means of differential equations. These equations are usually derived from balance principles and certain modelling assumptions. For realistic situations, the solution of the associated initial boundary value problems requires the use of some discretization technique, such as finite differences or finite volumes. This research tackles the numerical solution of a 1D differential equation to predict water surface profiles in a river, as well as to estimate the so-called roughness parameter. A very important concern when solving this differential equation is the ability of the numerical model to capture different flow regimes, given that hydraulic jumps are likely to be observed. To approximate the solution, Physics-Informed Neural Networks (PINN) are used. Benchmark cases with different bed profile shapes, which induce different flows types (supercritical, subcritical, and mixed) are tested first. Then a real mountain river morphology, the so-called Step-pool, is studied. PINN models were implemented in Tensor Flow using two neural networks. Different numbers of layers and neurons per hidden layer, as well as different activation functions (AF), were tried. The best performing model for each AF (according to the loss function) was compared with the solution of a standard finite difference discretization of the steady-state 1D model (HEC-RAS model). PINN models show good predictability of water surface profiles for slowly varying flow cases. For a rapid varying flow, the location and length of the hydraulic jump is captured, but it is not identical to the HEC-RAS model. The predictability of the tumbling flow in the Step-pool was good. In addition, the solution of the estimation of the roughness parameter (which is an inverse problem) using PINN shows the potential of this methodology to calibrate this parameter with limited cross-sectional data. PINN has shown potential for its application in open channel studies with complex bed profiles and different flow types, having in mind, however, that emphasis must be given to architecture selection.

Due to the presence of boulders and different morphologies, mountain rivers contain various resistance sources. To correctly simulate river flow using 1-D hydrodynamic models, an accurate estimation of the flow resistance is required. In this article, a comparison between the physical roughness parameter (PRP) and effective roughness coefficient (ERC) is presented for three of the most typical morphological configurations in mountain rivers: cascade, step-pool, and plane-bed. The PRP and its variation were obtained through multiple measurements of field variables and an uncertainty analysis, while the ERC range was derived with a GLUE procedure implemented in HEC-RAS, a 1-D hydrodynamic model. In the GLUE experiments, two modes of the Representative Friction Slope Method (RFSM) between two cross-sections were tested, including the variation in the roughness parameter. The results revealed that the RFSM effect was limited to low flows in cascade and step-pool. Moreover, when HEC-RAS selected the RSFM, only acceptable results were presented for plane-bed. The difference between ERC and PRP depended on the flow magnitude and the morphology, and as shown in this study, when the flow increased, the ERC and PRP ranges approached each other and even overlapped in cascade and step-pool. This research aimed to improve the roughness value selection process in a 1-D model given the importance of this parameter in the predictability of the results. In addition, a comparison was presented between the results obtained with the numerical model and the values calculated with the field measurements

River flow velocity is determined by the energy available for flow motion and the energy fraction lost by flow resistance. We compared the performance of different equations for the Darcy-Weisbach resistance coefficient (f) and empirical equations to predict flow velocity. The set of equations was tested using data from the Quinuas headwater mountain river in the Andean region. The data was collected in three Cascades, two Step-pools, and one Plane-bed covering a wide range of velocity magnitudes. The results reveal that nondimensional hydraulic geometry equations (NDHG) with a Nash-Sutcliffe efficiency index (EF) varying from 0.6–0.85 provide the most accurate velocity prediction. Furthermore, the study proposes a methodology applicable to all morphologies for defining the NDHG parameters using easily measured field data. The results show an improvement in predictability with EF values in the range of 0.81–0.86. Moreover, the methodology was tested against data from the literature, which was not divided into morphologies providing EF values of around 0.9. The authors encourage the application of the presented methodology to other reaches to obtain additional data about the NDHG parameters. Our findings suggest that those parameters could be related to reach characteristics (e.g., certain characteristic grain size), and in that case, the methodology could be useful in ungauged streams.

Hydrodynamic models (HMs) are tools for simulating flow behavior through the solution of conservation equations. These equations can have different degrees of simplification, which influence the model structure. One-dimensional (1D) HMs are still popular due to their simplicity. A crucial parameter for obtaining accurate 1D HM outputs is the effective Manning roughness factor (EMRF). The EMRF reflects additional numerical and dissipative aspects beyond boundary roughness. Although generalized likelihood uncertainty estimation (GLUE) is an important method for uncertainty analysis, it requires the selection of a likelihood function and a cutoff threshold. The goal of this study was to determine the effect of the likelihood function on the EMRF characteristics for mountain river morphologies, considering a certain cutoff threshold. The results show that the error model and the treatment of the residual in the objective function affect the EMRF range and limits in the studied reaches with a cascade or step pool. Furthermore, the analysis shows that these morphologies deviate from the model structure, which may affect the likelihood curve shape. Notably, the EMRF and measured roughness did not intersect in the studied reach with a plane bed, which is attributed to the presence of vegetation on the banks of that reach.