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Streambed biogeochemical processes strongly influence riverine water quality and gaseous emissions. These processes depend largely on flow paths through the hyporheic zone (HZ), the streambed volume saturated with stream water. Boulders and other macroroughness elements are known to induce hyporheic flows in gravel‐bed streams. However, data quantifying the impact of these elements on hyporheic chemistry are lacking. We demonstrate that, in gravel‐bed rivers, the amount of dissolved oxygen (DO) in the bed depends chiefly on changes in bed shape, or morphology, such as the formation of scour and depositional areas, caused by the boulders, among other factors. The study was conducted by comparing DO distributions across different bed states and hydraulic conditions. Our experimental facility replicates conditions observed in natural gravel‐bed streams. We instrumented a section in the bed with DO sensors. Results generally indicate that boulder placement on planar beds has some effects, which are significant at high base flows, on increasing hyporheic oxygen amount compared to the planar case without boulders. Conversely, boulder‐induced morphological changes noticeably and significantly increase the amount of oxygen in the HZ, with the increase depending on sediment inputs during flood flows able to mobilize the sediment. Therefore, streambeds of natural, plane‐bed streams may have deeper oxic zones than previously thought because the presence of boulders and the occurrence of flood flows with varying sediment inputs induce streambed variations among these elements.

Hyperbolic balance laws with uncertain (random) parameters and inputs are ubiquitous in science and engineering. Quantification of uncertainty in predictions derived from such laws, and reduction of predictive uncertainty via data assimilation, remain an open challenge. That is due to nonlinearity of governing equations, whose solutions are highly non-Gaussian and often discontinuous. To ameliorate these issues in a computationally efficient way, we use the method of distributions, which here takes the form of a deterministic equation for spatio-temporal evolution of the cumulative distribution function (CDF) of the random system state, as a means of forward uncertainty propagation. Uncertainty reduction is achieved by recasting the standard loss function, i.e. discrepancy between observations and model predictions, in distributional terms. This step exploits the equivalence between minimization of the square error discrepancy and the Kullback–Leibler divergence. The loss function is regularized by adding a Lagrangian constraint enforcing fulfilment of the CDF equation. Minimization is performed sequentially, progressively updating the parameters of the CDF equation as more measurements are assimilated.

Steady-state distributions of water potential and salt concentration in coastal aquifers are typically modelled by the Henry problem, which consists of a fully coupled system of flow and transport equations. Coupling arises from the dependence of water density on salt concentration. The physical behaviour of the system is fully described by two dimensionless groups: (i) the coupling parameter $\\alpha$, which encapsulates the relative importance of buoyancy and viscous forces, and (ii) the Péclet number $\\mbox{\\textit{Pe}}$, which quantifies the relative importance of purely convective and dispersive transport mechanisms. We provide a systematic analytical analysis of the Henry problem for a full range of the Péclet number. For moderate $\\mbox{\\textit{Pe}}$, analytical solutions are obtained through perturbation expansions in $\\alpha$. This allows us to elucidate the onset of density-driven vertical flux components and the dependence of the local hydraulic head gradients on the coupling parameter. The perturbation solution identifies the regions where salt concentration is most pronounced and relates their spatial extent to the development of a convection cell. Next, we compare our solution to a solution of the pseudo-coupled model, wherein flow and transport are coupled only via the boundary conditions. This enables us to isolate the effects caused by density-dependent processes from those induced by external forcings (boundary conditions). For small $\\mbox{\\textit{Pe}}$, we develop a perturbation expansion around the exact solution corresponding to $\\mbox{\\textit{Pe}}\\,{=}\\,0$, which sheds new light on the interpretation of processes observed in diffusion experiments with variable-density flows in porous media. The limiting case of infinite Péclet numbers is solved exactly for the pseudo-coupled model and compared to numerical simulations of the fully coupled problem for large $\\mbox{\\textit{Pe}}$. The proposed perturbation approach is applicable to a wide range of variable-density flows in porous media, including seawater intrusion into coastal aquifers and temperature or pressure-driven density flows in deep aquifers.

Advective–diffusive transport of passive or reactive scalars in confined environments (e.g. tubes and channels) is often accompanied by diffusive losses/gains through the confining walls. We present analytical solutions for transport of a reactive solute in a tube, whose walls are impermeable to flow but allow for solute diffusion into the surrounding medium. The solute undergoes advection, diffusion and first-order chemical reaction inside the tube, while diffusing and being consumed in the surrounding medium. These solutions represent a leading-order (in the radius-to-length ratio) approximation, which neglects the longitudinal variability of solute concentration in the surrounding medium. A numerical solution of the full problem is used to demonstrate the accuracy of this approximation for a physically relevant range of model parameters. Our analysis indicates that the solute delivery rate can be quantified by a dimensionless parameter, the ratio of a solute’s residence time in a tube to the rate of diffusive losses through the tube’s wall.

We derive deterministic cumulative distribution function (CDF) equations that govern the evolution of CDFs of state variables whose dynamics are described by the first-order hyperbolic conservation laws with uncertain coefficients that parametrize the advective flux and reactive terms. The CDF equations are subjected to uniquely specified boundary conditions in the phase space, thus obviating one of the major challenges encountered by more commonly used probability density function equations. The computational burden of solving CDF equations is insensitive to the magnitude of the correlation lengths of random input parameters. This is in contrast to both Monte Carlo simulations (MCSs) and direct numerical algorithms, whose computational cost increases as correlation lengths of the input parameters decrease. The CDF equations are, however, not exact because they require a closure approximation. To verify the accuracy and robustness of the large-eddy-diffusivity closure, we conduct a set of numerical experiments which compare the CDFs computed with the CDF equations with those obtained via MCSs. This comparison demonstrates that the CDF equations remain accurate over a wide range of statistical properties of the two input parameters, such as their correlation lengths and variance of the coefficient that parametrizes the advective flux.

We study transport of a reactive solute in a chemically heterogeneous porous medium whose chemical properties are uncertain. The dissolved substance undergoes a heterogeneous chemical reaction with a solid phase in the presence of advection caused by extraction/injection from a point source. We present semi-analytical solutions for the probability density function of the solute concentration, which allows us to quantify predictive uncertainty associated with uncertain reaction rate constants for both linear and nonlinear reactions. This enables one to compute probabilities of rare events, which are required for quantitative risk analyses.

Hybrid or multiphysics algorithms provide an efficient computational tool for combining micro- and macroscale descriptions of physical phenomena. Their use becomes imperative when microscale descriptions are too computationally expensive to be conducted in the whole domain, while macroscale descriptions fail in a small portion of the computation domain. We present a hybrid algorithm to model a general class of reaction-diffusion processes in granular porous media, which includes mixing-induced mineral precipitation on, or dissolution of, the porous matrix. These processes cannot be accurately described using continuum (Darcy-scale) models. The pore-scale/Darcy-scale hybrid is constructed by coupling solutions of the reaction-diffusion equations (RDE) at the pore scale with continuum Darcy-level solutions of the averaged RDEs. The resulting hybrid formulation is solved numerically by employing a multiresolution meshless discretization based on the smoothed particle hydrodynamics method. This ensures seamless noniterative coupling of the two components of the hybrid model. Computational examples illustrate the accuracy and efficiency of the hybrid algorithm.

Quantitative descriptions of flow and transport in subsurface environmentsare often hampered by uncertainty in the input parameters. Treatingsuch parameters as random fields represents a useful tool for dealingwith uncertainty. We review the state of the art of stochasticdescription of hydrogeology with an emphasis on statisticallyinhomogeneous (nonstationary) models. Our focus is on composite mediamodels that allow one to estimate uncertainties both in geometricalstructure of geological media consisting of various materials and inphysical properties of these materials.[PUBLICATION ABSTRACT]

The Buckley--Leverett (nonlinear advection) equation is often used to describe two-phase flow in porous media. We develop a new probabilistic method to quantify parametric uncertainty in the Buckley--Leverett model. Our approach is based on the concept of a fine-grained cumulative density function (CDF) and provides a full statistical description of the system states. Hence, it enables one to obtain not only average system response but also the probability of rare events, which is critical for risk assessment. We obtain a closed-form, semianalytical solution for the CDF of the state variable (fluid saturation) and test it against the results from Monte Carlo simulations. [PUBLICATION ABSTRACT]

A probability density function (pdf) formulation is applied to a heterogeneous chemical reaction involving an aqueous solution reacting with a solid phase in a batch. This system is described by a stochastic differential equation with multiplicative noise. Both linear and nonlinear kinetic rate laws are considered. An effective rate constant for the mean field approximation describing the change in mean concentration with time is derived. The effective rate constant decreases with increasing time eventually approaching zero as the system approaches equilibrium. This behavior suggests that a possible explanation for the observed discrepancy between laboratory measured rate constants on uniform grain sizes and field measurements may in part be caused by the heterogeneous distribution of grain sizes in natural systems. [PUBLICATION ABSTRACT]