## Book of abstracts

###### Giorgio Cassiani (Università degli Studi di Padova)

**The “true” meaning of Hydrogeophysics: integration of geophysical data with hydrological modeling**

The use of geophysical methods for the characterization of the shallow subsurface for hydrological/hydrogeological purposes has a long history. Examples go back to 1970’s, with mixed success. In particular, the use of electrical and electromagnetic methods has found extensive application even since, even though some early ambitious goals have not (and cannot) be attained, in particular when correlation is attempted directly between geophysical and hydrological medium properties (typically, electrical conductivity and hydraulic conductivity, or, in more recent years, pore medium polarization properties with hydraulic conductivity). A major step forward, though, has been achieved since the early 1990’s when a new school of thoughts has been developing and the word “hydrogeophysics” has been coined. The main change in the minds of the early developers was to join more closely the results of the (geo)physical measurements with the physical signal produced by hydrological processes. This, in practice means that the geophysical results shall be used, as much as possible, as data (albeit to some extent indirect) to be used to calibrate hydrological models, e.g. using data assimilation techniques, but not only. At the heart of the idea lie two main requirements: (a) that a link is established between geophysically-measured parameters (e.g. electrical resistivity) and the hydrological state variables (moisture content, water salinity etc.), rather than towards hydraulic parameters such as hydraulic conductivity; and (b) that time-lapse geophysical measurements are conducted in order to samples in time, as well as in space of course, the state variables. The approach has had tremendous impacts in understanding of complex hydrological systems, at a variety of scales, supplying hydrological models with an unprecedented amount of 4D data, that the models desperately need for a proper calibration – in this respect mimicking what is successfully done (and can be done even better) in the petroleum industry using 4D seismics. As many success stories, however, the one of “hydrogeophysics” has also some downside. In this case, the misuse of the term has been widespread. Nowadays a variety of studies that are no more than the classical, ancient, use of geophysics for the characterization of the shallow subsurface, often for hydrological purpose, call themselves “hydrogeophysical” studies, even though no attempt is made to link geophysical data with hydrological models. This contribution wishes to state a word of caution in this direction, and make a small step towards a badly needed restoration of the original goal and meaning of a still very innovative and promising approach, not always fully exploited in practice.

###### Vincenzo Casulli (Università di Trento)

**A coupled surface-subsurface model for hydrostatic flows under saturated and variably saturated conditions**

Assuming the validity of the hydrostatic approximation, coupled surface-subsurface flows governed by the Richards and by the Navier-Stokes equations are solved simultaneously by using semi-implicit finite difference equations for velocities and a finite volume approximation for the vertically integrated continuity equation. The resulting three-dimensional algorithm is relatively simple, extremely efficient, and very accurate. Stability, convergence, and exact mass conservation are assured throughout also in presence of wetting and drying, in variable saturated conditions, and during flow transition through the soil interface.

###### Salvatore Cuomo (Università degli Studi di Napoli Federico II)

**Physics-Informed Neural Networks for solving Groundwater Flow Equation**

In recent years, Scientific Machine Learning (SciML) methods for solving partial differential equations (PDEs) have gained wide popularity. Within such a paradigm, Physics Informed Neural Networks (PINNs) are novel deep learning frameworks for solving forward and inverse problems with non-linear PDEs. Recently, PINNs have shown promising results in different application domains. In this paper, we approach the groundwater flow equations numerically by searching for the unknown hydraulic head. Since singular terms in differential equations are very challenging from a numerical point of view, we approximate the Dirac distribution by different regularization terms. Furthermore, from a computational point of view, this study investigate how a PINN can solve higher-dimensional flow equations. In particular, we analyze the approximation error for one and two-dimensional cases in a statistical learning framework. The numerical experiments discussed include one and two-dimensional cases of a single or multiple pumping well in an infinite aquifer, demonstrating the effectiveness of this approach in the hydrology application domain.

###### Vittorio Di Federico (Università di Bologna)

**Flow in fractured media: from deterministic to random approaches**

Non-Newtonian fluid flow in fractured media is relevant for subsurface operations aimed at resources recovery, land remediation, and geothermal exploitation. In these contexts, complex fluids (e.g., polymer solutions, foams, muds) are employed as carriers for suspended nanoparticles and fracking proppant, and potentially, as heat-exchange working fluids in enhanced geothermal systems. The interplay between the heterogeneity of fractured geological media, from the pore- to field scale, and the non-linear rheology of working fluids strongly influence flow localization, medium transmissivity, and transport phenomena across the formation. The quantitative characterization of flow and transport features is fundamental to achieve a desired performance of subsurface industrial activities to mitigate the environmental impact and increase their cost effectiveness. We present results from several of our works dealing with complex fluid flow in single fractures, from analytical conceptual models and analog experiments to numerical approaches in a stochastic framework.

In the first group of contributions, the fracture is conceptualized with the parallel plate approach and the focus is on a specific phenomenon, the backflow of fracturing fluid from fractures to wellbore in the third phase of hydraulic fracturing, after injection is ceased. We consider an Ostwald-DeWaele (power-law) or Ellis fluid, a planar or radial fracture geometry, and a time-variable aperture depending on the internal fluid pressure and the elastic relaxation of the walls. The relationship between the latter quantities may be linear, akin to a Winkler soil, or nonlinear, due to the progressive softening or stiffening of the boundary associated with the properties of the surrounding rock. The result is an integrodifferential problem that admits a closed-form similarity solution for power-law fluids, albeit implicit for some quantities; for the more realistic, three-parameter Ellis fluid, the solution is partially numerical and depends on dimensionless group N encapsulating the main problem parameters and equal to the ratio between the Cauchy number and the product of Reynolds and Ellis numbers. Results are validated in the laboratory with an original experimental device, an ad hoc replica of a rectangular or circular fracture. The match between theory and experiments is fairly good, with discrepancies of a few percent essentially due to the approximations of the theoretical model, and, for shear-thinning fluids, to the simplified constitutive equation.

The rough walls of geological fractures exhibit a pore-scale spatial variability that can be reproduced as an isotropic self-affine surface. The flow in between is influenced by the aperture variability which induces viscous energy losses and promote the channeling phenomenon. The complex nature of polymer fluids plays an important role as it promotes flow localization in channels of lower apparent viscosity, mitigating energy losses and enhancing fracture-scale transmissivity up to two orders of magnitude higher than simple fluids like water. We investigate this phenomenon implementing a lubrication-based numerical code able to generate synthetic fractures and solve the flow on a large mesh, limiting the computational cost typical of a non-linear scheme. The code is encased in a Monte Carlo framework to produce ensemble statistics through numerous flow simulations over the parameter space, providing new insight on the transition from Darcian to non-linear regime, and quantitatively characterizing transmissivity and flow characteristic length in polymer flow.

###### Nicodemo Di Pasquale (Brunel University London)

**Mathematical modelling and open-source simulation of reverse-osmosis desalination**

The reverse osmosis membrane module is an integral element of a desalination system as it determines the overall performance of the desalination plant. The fraction of clean water that can be recovered via this process is often limited by salt precipitation which plays a critical role in its sustainability. In this work, we present a model to study the complex interplay between flow, transport and precipitation processes in reverse osmosis membranes, which together influence recovery and in turn process sustainability. A reactive porous interface model describes the membrane with a dynamic evolving porosity and permeability to capture the scaling of the membrane. An open-source finite-volume numerical solver is implemented within the OpenFOAM®library and numerical tests are presented here showing the effect of the various parameters of the model and the robustness of the model to describe a wide range of operating conditions.

###### Matthew Farthing (U.S. Army Engineer Research and Development Center)

**Model Reduction and Operator Learning for Environmental Flows**

High-fidelity numerical simulation plays an important role in addressing complex scientific and engineering problems in environmental fluid dynamics. Despite the progress in computational resources and methodologies over the last two decades, high-fidelity models remain computationally expensive and require substantial expertise. Non-intrusive reduced order modeling techniques have become increasingly popular as a way to address these computational challenges because they offer an alternative by leveraging available data and observations without explicit access to the underlying governing equations. Here, we will consider several non-intrusive techniques based on linear and nonlinear dimension reduction as well as recently introduced operator learning frameworks. We will evaluate their performance for riverine, estuarine, and near-shore systems, with an eye to both fast prediction and parameter inference.

###### Massimiliano Ferronato (Università degli Studi di Padova)

**Numerical models for frictional contact mechanics and flow in fractured porous media**

The simultaneous simulation of frictional contact mechanics and fluid flow in fractured geological media is a tightly coupled physical processes and a key component in the design of sustainable technologies for several subsurface applications, such as geothermal energy production, CO2 sequestration and underground gas storage. Typically, the aperture and slippage between the contact surfaces drive the fluid flow in the fractures, while the pressure variation perturbs the stress state in the surrounding medium and influences the contact mechanics itself. This usually produces a stiff non-linear problem associated with a series of generalized saddle-point linear systems, whose solution is often hard to obtain efficiently. In this work, we focus on a blended finite element/finite volume method, where the porous medium is discretized by low-order continuous finite elements with nodal unknowns, cell centered Lagrange multipliers with a stabilization are used to prescribe the contact constraints, and the fluid flow in the fractures is described by a classical two-point flux approximation scheme. A class of scalable preconditioning strategies based on the physically-informed block partitioning of the unknowns and state-of-the-art multigrid techniques is developed for the robust and efficient solution to the resulting sequence of linear systems with the Jacobian matrix. A set of numerical results concerning fractured porous media applications illustrate the robustness of the proposed approach, its algorithmic scalability, and the computational performance on large-size realistic problems.

###### Alessio Fumagalli (Politecnico di Milano)

**A machine learning approach that ensure local mass conservation for single-phase flow in fractured porous media**

Constructing fast solution schemes often involves deciding which errors are acceptable and which approximations can be made for the sake of computational efficiency. Herein, we consider a mixed formulation of Darcy flow and take the perspective that the physical law of mass conservation is significantly more important than the constitutive relationship, i.e. Darcy’s law. Within this point of view, we propose a three-step solution technique that guarantees local mass conservation.

In the first step, an initial flux field is obtained by using a locally conservative method, such as the TPFA Finite Volume Method. Although this scheme is computationally efficient, it lacks consistency and therefore requires a suitable correction. Since this correction is divergence-free, the Helmholtz decomposition ensures that it is given by the curl of a potential field. The second step therefore employs an H(curl)-conforming discretization to compute the correction potential and update the flux field. The pressure field is computed in the final step by using the same TPFA system from the first step.

The procedure guarantees local mass conservation regardless of the quality of the computed correction. Thus, we relax this computation using tools from reduced order modeling. We introduce a reduced basis method that is capable of rapidly producing a potential field for given permeability fields. By applying the curl to this field, we ensure that the correction is divergence-free and mass conservation is not impacted.

Finally, we extend the method to solving Darcy flow in fractured porous media. We rewrite the equations in terms of mixed-dimensional differential operators and identify the problem as a mixed-dimensional Darcy flow system. In turn, the proposed three-step solution procedure directly applies using the mixed-dimensional curl to ensure local mass conservation.

###### Costantino Masciopinto (Consiglio Nazionale delle Ricerche – Istituto di Ricerca Sulle Acque)

**Conceptual Models of Flow and Transport in Fracture-Dominated Aquifers**

Flow and transport mathematical models in fracture-dominated aquifers are characterized by impermeable rock matrix and should consider the simultaneous occurrence of laminar, nonlaminar, and turbulent fluxes in the fractures rather than the laminar flow by the cubic law that has been widely applied in the scientific literature. Some simulations show overestimations up to 75% of the groundwater velocity when non-laminar flows are neglected. Moreover, further model development is needed to address the effect of tortuosity of preferential saturated fluid flow paths in fractures suggesting adjustments to the size of the mean aperture of fractures or tubes.

Model simulations in these aquifers typically present uncertainties owing to the impossibility of determining the exact input data, such as fractures’ spatial positions, orientation and deep angles, lengths, apertures, and fracture densities and intersections. Available data from field reliefs of outcrops or tunnel advancement fronts represent only a fraction of the fracture network properties of an entire geological rock formation of aquifer. Furthermore, the heterogeneities related to the spatial variation of the fracture network properties cannot be “averaged” using equivalent porous models or random analytical solutions.

Among numerous existing conceptual models of a fracture-dominated aquifer, the possible simplest conceptualizations in layered fractured models, tube models and backbones of three-dimensional fracture networks, have been presented. Then we show the mathematical model equations to approximate the pollutant and pathogen flow and transport affected by fast and slow flow pathways in fracture-dominated aquifers.

###### Ilario Mazzieri (Politecnico di Milano)

**A space-time discontinuous Galerkin method for wave propagation problems in coupled poroelastic-elastic domains**

We present a space-time finite element discontinuous Galerkin method on polytopal meshes (XT-PolydG) for the numerical discretization of wave propagation in coupled poroelastic-elastic media. We consider the low-frequency Biot’s equations in the poroelastic medium and the elastodynamics equation for the elastic one. To realize the coupling, suitable transmission conditions on the interface between the two domains are (weakly) embedded in the formulation. The proposed PolydG discretization in space is then coupled with a dG time integration scheme, resulting in a full space-time dG discretization. We present the stability analysis for both the continuous and the semidiscrete formulations, and we derive error estimates for the semidiscrete formulation in a suitable energy norm. The method is applied to a wide set of numerical test cases to verify the theoretical bounds. Examples of physical interest are also presented to investigate the capability of the proposed method in relevant geophysical scenarios.

###### Sorin Pop (Hasselt University)

**Non-standard models for flow in porous media**

Mathematical models that are commonly used for porous media flows assume that quantities like saturation, phase pressure differences, or relative permeability are related by monotone, algebraic relationships. Under such assumptions, the solutions of the resulting mathematical models satisfy the maximum principle. On the other hand, phenomena like saturation overshoot, or the formation of finger profiles have been observed in experiments. Such results are ruled out by standard models. Moreover, the relationships determined experimentally for the same type of medium differ, depending on the context (e.g. drainage or imbibition) or the dynamics of the flow.

This motivated the development of non-standard models, where dynamic or hysteretic effects are included in the above-mentioned relationships. The resulting models are nonlinear evolution systems of (pseudo-)parabolic and possibly degenerate equations, involving differential inclusions. Here we present briefly aspects related to the existence and uniqueness of weak solutions to such models. We then discuss different numerical schemes, including aspects like the rigorous convergence of the discretization, domain decomposition, and solving the emerging nonlinear time-discrete or fully discrete problems.

###### Amilcare M. Porporato (Princeton University)

**Moisture fluctuations in soil biogeochemical cycles: from the emblematic case of iron-redox cycles to current challenges**

Infiltration of unpredictable rainfall inputs determine complex space time variability in soils, which in turn drives pulsing dynamics of biogeochemical processes and ultimately controls plant productivity and ecosystem carbon storage. After briefly discussing the moisture controls on the variety of soil biogeochemical processes going from aerobic to anaerobic conditions, we focus on the emblematic case of iron-redox ‘cycles’, whereby the alternation of dry and wet conditions promotes the cycling between ferrous oxide reduction during oxic conditions and the biologically mediated ferric oxidation in anoxic conditions. The statistical analysis of soil moisture level crossing, coupled to simplified dynamical system analysis of biogeochemical cycling, unveils the presence of optimal ecohydrologic conditions leading to maximal rates of iron redox cycling, with implication for plant growth and organic matter decomposition. We close by highlighting some important challenges and open questions, especially in relation to the random spatial variability of such processes, which are important for their upscaling.

###### Florin A. Radu (University of Bergen)

**Efficient solvers for Richards’ equation**

Richards’ equation is modelling flow in variably saturated porous media. It is a highly nonlinear, degenerate elliptic-parabolic equation, which is known to be very challenging to solve. In this talk we discuss different solution strategies for Richards’ equation, with the main focus on linearization methods. The Newton method and the L-scheme (a stabilized Picard method) will be considered, as well as a combination of the two. The combined scheme, which is both robust and efficient, is based on an adaptive switching between the two linearization methods. The theoretical properties of the scheme will be discussed. Illustrative numerical examples will be shown.

###### Monica Riva (Politecnico di Milano)

**Nanoscale investigation and Stochastic assessment of calcite dissolution**

We investigate the main traits of the spatial heterogeneity of dissolution rates that are directly observed through Atomic Force Microscopy on the surface of a millimeter-scale calcite sample in contact with deionized water. Our analyses are framed within a stochastic approach and are motivated from the observation that detailed characterizations of mineral dissolution/precipitation rates is critical in the context of a variety of applications including, e.g., aquifer contamination assessment, geologic carbon sequestration, or hydraulic fracturing of hydrocarbon reservoirs. Experimental evidences based on Atomic Force Microscopy (AFM) enable direct observations of the mechanisms taking place across the mineral surface during the reaction and constitute a key information basis for interpretive modeling efforts. In this context, the dissolution process is evidenced to be strongly affected by several sources of variability at the local (i.e., micro-scale) mineral-fluid interface and a marked spatial heterogeneity in the dissolution rate is documented. In this general framework, we collect datasets of surface topography at several observation times from which reaction rate maps are evaluated. The study is aimed at (1) characterizing the statistical behavior of dissolution rates and their spatial increments within a unique and consistent theoretical framework; (2) identifying an appropriate interpretive model for such statistics; and (3) evaluating quantitatively, through observed trends of model parameters, the temporal evolution of the spatial heterogeneity of the dissolution reaction.

###### Gerardo Severino (Università degli Studi di Napoli Federico II)

**Flow and transport in a doublet-type flow configuration**

Transport takes place between an injecting well and a pumping one through to a porous formation. The controlling parameter is the conductivity which, unlike the classical approach, here is regarded, in line with field findings, as spatially variable. This renders the problem at stake extremely difficult to solve. However, a simple solution is achieved by adopting a few simplifying assumptions, which nevertheless resemble most of the existing aquifers, and therefore it is applicable to numerous real world situations. It is shown that the proposed solution finds application in the identification of the aquifer’s parameters as well as the quantification of efficiency of decontamination procedures. Finally, the theoretical framework is applied to a couple of transport experiments, in order to illustrate (and to quantify) how dispersion process develops in the zone delimited by the two wells.