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Shallow water models of geophysical flows must be adapted to geometric characteristics in the presence of a general bottom topography with non-negligible slopes and curvatures, such as a mountain landscape. In this paper we derive an intrinsic shallow water model from the Navier–Stokes equations defined on a local reference frame anchored on the bottom surface. The equations resulting are characterized by non-autonomous flux functions and source terms embodying only the geometric information. We show that the proposed model is rotational invariant, admits a conserved energy, is well-balanced, and it is formally a second order approximation of the Navier–Stokes equations with respect to a geometry-based order parameter. We then derive a numerical discretization by means of a first order upwind Godunov finite volume scheme intrinsically defined on the bottom surface. We study convergence properties of the resulting scheme both theoretically and numerically. Simulations on several synthetic test cases are used to validate the theoretical results as well as more experimental properties of the solver. The results show the importance of taking into full consideration the bottom geometry even for relatively mild and slowly varying curvatures.

Inspired by the linear Schr\"odinger operator, we consider a generalized $p$-Laplacian operator on discrete graphs and present new results that characterize several spectral properties of this operator with particular attention to the nodal domain count of its eigenfunctions. Just like the one-dimensional continuous $p$-Laplacian, we prove that the variational spectrum of the discrete generalized $p$-Laplacian on forests is the entire spectrum. Moreover, we show how to transfer Weyl's inequalities for the Laplacian operator to the nonlinear case and prove new upper and lower bounds on the number of nodal domains of every eigenfunction of the generalized $p$-Laplacian on generic graphs, including variational eigenpairs. In particular, when applied to the linear case $p=2$, in addition to recovering well-known features, the new results provide novel properties of the linear Schr\"odinger operator.

International audience; Flow in formations having both low-porosity and low-permeability prevalently occurs in fracture networks. Fracture connectivity and permeability are the main features governing flow in these media, and affect the effectiveness of any subsurface operation (e.g. enhanced oil recovery, enhanced geothermal systems). Moreover, the rheology of the fluid is often non-Newtonian due to their complex microstructure, resulting in a non-linear constitutive law at the continuum scale.Modeling flow of complex fluids in a fracture is challenging, when both medium heterogeneity and a realistic fluid rheology, typically shear-thinning, are considered. Full 3-D simulations are computationally expensive and time-consuming; alternatively, a lubrication-based approach can be adopted to implement an efficient 2-D flow solver able to produce vast statistics in reasonable time.We propose a numerical code that solves the 2-D generalized non-linear Reynolds equation for Ellis fluid flow in a single variable aperture fracture, adopting the finite volume method. The inexact Newton-Krylov method solves the associated non-linear system of equations, while a preconditioned conjugate gradient algorithm is used to solve the linear system at each Newton iteration. A parameter continuation strategy is also introduced to handle strongly non-linear cases.A synthetic fracture aperture field w(x) is generated and the flow problem is solved. The numerical scheme is particularly efficient and robust for most input parameters of practical interest. Numerical parameters of the Inexact Newton-Krylov method have been optimized to simulate flow on large mesh (e.g. 1010x1010), considering a range of fracture closure (σw/‹w›) from 0 to 1, a shear-thinning index (n) of the Ellis rheologic model varying from 0.1 (strongly shear-thinning behavior) to 1 (Newtonian case). Pressure gradients (∇P) typical of subsurface flow, either natural or forced, have been considered to study natural phenomena and industrial applications.

Model reduction for fluid flow simulation continues to be of great interest across a number of scientific and engineering fields. In a previous work [arXiv:2104.13962], we explored the use of Neural Ordinary Differential Equations (NODE) as a non-intrusive method for propagating the latent-space dynamics in reduced order models. Here, we investigate employing deep autoencoders for discovering the reduced basis representation, the dynamics of which are then approximated by NODE. The ability of deep autoencoders to represent the latent-space is compared to the traditional proper orthogonal decomposition (POD) approach, again in conjunction with NODE for capturing the dynamics. Additionally, we compare their behavior with two classical non-intrusive methods based on POD and radial basis function interpolation as well as dynamic mode decomposition. The test problems we consider include incompressible flow around a cylinder as well as a real-world application of shallow water hydrodynamics in an estuarine system. Our findings indicate that deep autoencoders can leverage nonlinear manifold learning to achieve a highly efficient compression of spatial information and define a latent-space that appears to be more suitable for capturing the temporal dynamics through the NODE framework., 16 pages, 7 figures, To Appear in the proceedings of the IXth International Conference on Computational Methods for Coupled Problems in Science and Engineering (COUPLED PROBLEMS 2021), 14-16 June, 2021. arXiv admin note: substantial text overlap with arXiv:2104.13962

The study of the flow in a single fracture is the starting point to understand the complex hydraulic behaviour of geological formations and fractured reservoirs, whose comprehension is of interest in many natural phenomena (e.g., magma intrusion) and the optimization of numerous industrial activities in fractured reservoirs (e.g., Enhanced Oil Recovery, drilling engineering, geothermal energy exploitation). Despite the considerable technical prospects of this topic, the associated mathematical complexity and computational burden have so far mostly discouraged investigations of the combined effects of fracture heterogeneity and of the complex rheology of relevant fluids. Indeed, magmas, foams, muds, and suspensions of natural colloids such as clay particles in water are complex fluids and often present in subsurface applications and natural processes. These fluids are characterized by a shear-thinning behavior, which can be well described by the Ellis model, a continuous three-parameter model that behaves as a power-law fluid at high shear rates and as a Newtonian fluid at low shear rates. The Ellis model parameters are: n the power law exponent, μ0 the low shear rates viscosity, and τ1/2 the shear rate such that μapp(τ1/2)=μ0/2. We use this rheological description in combination with the lubrication theory, which is a depth-averaged formalism permitting us to reduce the full 3-D problem to a 2-D plane formulation. It has been applied to study Newtonian flow in a single fracture for decades and, as far as the aperture gradient remains small (∇d«1), the approximation error introduced by this model is limited. We present here a lubrication-based numerical code aiming at simulating the flow of an Ellis fluid in rough-walled fractures. The code is composed of two modules: a 2D FFT-based fracture aperture field generator and a lubrication-based non-Newtonian flow solver. The former module generates a random aperture field d(x,y) with isotropic spatial correlations, given a mean aperture ⟨d⟩, a coefficient of variation σd/⟨d⟩, a Hurst exponent (H) and a correlation length (lc), reproducing realistic geometries of geological fractures. In the latter module, a 2-D finite volume scheme is adopted to solve the non-linear lubrication equation describing the flow of an Ellis fluid. The equation is discretized on a staggered grid, so that d(x,y) and the pressure field p(x,y) are defined at different locations. Computational efficiency is achieved by means of the inexact Newton algorithm, with the linearized symmetric system of equations solved via variable-fill-in Incomplete Cholesky Preconditioned Conjugate Gradient method (ICPCG), and a parameter-continuation strategy for the cases with strong nonlinearities. The code proves to be stable and robust when solving flow within strongly heterogeneous fractures (e.g., σd/⟨d⟩=1), even on very fine and coarse meshes (e.g., 214×214) and considering a wide range of power-law exponents (e.g., 0.1≤n≤1). The code is validated by comparing the results against analytical solutions (e.g., parallel plates model, sinusoidal profile) and full 3-D CFD simulations, considering different closures.

In this work, we develop Non-Intrusive Reduced Order Models (NIROMs) that combine Proper Orthogonal Decomposition (POD) with a Radial Basis Function (RBF) interpolation method to construct efficient reduced order models for time-dependent problems arising in large scale environmental flow applications. The performance of the POD-RBF NIROM is compared with a traditional nonlinear POD (NPOD) model by evaluating the accuracy and robustness for test problems representative of riverine flows. Different greedy algorithms are studied in order to determine a near-optimal distribution of interpolation points for the RBF approximation. A new power-scaled residual greedy (psr-greedy) algorithm is proposed to address some of the primary drawbacks of the existing greedy approaches. The relative performances of these greedy algorithms are studied with numerical experiments using realistic two-dimensional (2D) shallow water flow applications involving coastal and riverine dynamics.

Network routing approaches are widely used to study the evolution in time of self-adapting systems. However, few advances have been made for problems where adaptation is governed by time-dependent inputs. In this work we study a dynamical system where the edge conductivities of a network are regulated by time-varying mass loads injected on nodes. Motivated by empirical observations, we assume that conductivities adapt slowly with respect to the characteristic time of the loads. Furthermore, assuming the loads to be periodic, we derive a dynamics where the evolution of the system is controlled by a matrix obtained with the Fourier coefficients of the input loads. Remarkably, we find a sufficient condition on these coefficients that determines when the resulting network topologies are trees. We show an example of this on the Bordeaux bus network where we tune the input loads to interpolate between loopy and tree topologies. We validate our model on several synthetic networks and provide an expression for long-time solutions of the original conductivities., Comment: 12 pages, 5 figures

Routing optimization is a relevant problem in many contexts. Solving directly this type of optimization problem is often computationally unfeasible. Recent studies suggest that one can instead turn this problem into one of solving a dynamical system of equations, which can instead be solved efficiently using numerical methods. This results in enabling the acquisition of optimal network topologies from a variety of routing problems. However, the actual extraction of the solution in terms of a final network topology relies on numerical details which can prevent an accurate investigation of their topological properties. In this context, theoretical results are fully accessible only to an expert audience and ready-to-use implementations for non-experts are rarely available or insufficiently documented. In particular, in this framework, final graph acquisition is a challenging problem in-and-of-itself. Here we introduce a method to extract networks topologies from dynamical equations related to routing optimization under various parameters' settings. Our method is made of three steps: first, it extracts an optimal trajectory by solving a dynamical system, then it pre-extracts a network and finally, it filters out potential redundancies. Remarkably, we propose a principled model to address the filtering in the last step, and give a quantitative interpretation in terms of a transport-related cost function. This principled filtering can be applied to more general problems such as network extraction from images, thus going beyond the scenarios envisioned in the first step. Overall, this novel algorithm allows practitioners to easily extract optimal network topologies by combining basic tools from numerical methods, optimization and network theory. Thus, we provide an alternative to manual graph extraction which allows a grounded extraction from a large variety of optimal topologies., 17 pages, 7 main Figures, 3 SI figures

A peat deposit close to Venice was monitored both in the field and in the lab (1) to investigate the hydrological response of peat soil to changing meteorological conditions in the frame of land subsidence assessment. The whole area is about 3 meters lower than the sea level and therefore subsidence is a major issue. Predictions highlighted the risk of an almost complete disappearance of the peat layer in this area during the next 50 years, due to the increased frequency of warmer periods. Unfortunately, despite the considerable impacts that are expected to affect peatland worldwide, only a few measured datasets are currently available to assess the response of a peat deposit to enhanced drying due to global warming.The lab measurements were performed both at the pedon and at the core scale. An undisturbed peat monolith of approximately 0.7 m3 was collected, transferred to the lab, and instrumented to monitor matric potential, water content, and total weight. This undisturbed peat lysimeter allows to monitor water content variations (both through the weight monitoring and time domain reflectometry sensors), and matric potential, with drier conditions with respect to the field campaign. A complete cycle of wetting and drainage was performed, raising the water table from the bottom to the top of the sample and down again. Additional measurements of matric potential and water content were collected by testing peat cores on a suction table.A set of water retention curves was experimentally determined. They were derived for a range of matric potential much broader than that experienced in situ . Variations were found, with respect to the field natural conditions, in the relations between the matric potential and the volumetric water content of different horizons as a result of the initial prolonged drying. Also, the hysteresis behaviours in the lab and in the field were different, with much wider loops in the lab conditions because of extended range of potential. Hydraulic non-equilibrium between the water content and water potential could also be a possible cause, but further modelling work is necessary to assess it. The van Genuchten parameters were obtained for both wetting and drying, for modelling purposes.(1) Previati et al. (2019), Hydrological Processes.