For base year 2010, anthropogenic activities created ~210 (190 to 230) TgN of reactive nitrogen Nr from N2. This human-caused creation of reactive nitrogen in 2010 is at least 2 times larger than the rate of natural terrestrial creation of ~58 TgN (50 to 100 TgN yr−1) (Table 6.9, Section 1a). Note that the estimate of natural terrestrial biological fixation (58 TgN yr−1) is lower than former estimates (100 TgN yr−1, Galloway et al., 2004), but the ranges overlap, 50 to 100 TgN yr−1 vs. 90 to 120 TgN yr−1, respectively). Of this created reactive nitrogen, NOx and NH3 emissions from anthropogenic sources are about fourfold greater than natural emissions (Table 6.9, Section 1b). A greater portion of the NH3 emissions is deposited to the continents rather than to the oceans, relative to the deposition of NOy, due to the longer atmospheric residence time of the latter. These deposition estimates are lower limits, as they do not include organic nitrogen species. New model and measurement information (Kanakidou et al., 2012) suggests that incomplete inclusion of emissions and atmospheric chemistry of reduced and oxidized organic nitrogen components in current models may lead to systematic underestimates of total global reactive nitrogen deposition by up to 35% (Table 6.9, Section 1c). Discharge of reactive nitrogen to the coastal oceans is ~45 TgN yr−1 (Table 6.9, Section 1d). Denitrification converts Nr back to atmospheric N2. The current estimate for the production of atmospheric N2 is 110 TgN yr−1 (Bouwman et al., 2013).

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Carbon and Other Biogeochemical Cycles

Responding to the latest developments in rock physics research, this popular reference book has been thoroughly updated while retaining its comprehensive coverage of the fundamental theory, concepts, and laboratory results. It brings together the vast literature from the field to address the relationships between geophysical observations and the underlying physical properties of Earth materials - including water, hydrocarbons, gases, minerals, rocks, ice, magma and methane hydrates. This third edition includes expanded coverage of topics such as effective medium models, viscoelasticity, attenuation, anisotropy, electrical-elastic cross relations, and highlights applications in unconventional reservoirs. Appendices have been enhanced with new materials and properties, while worked examples (supplemented by online datasets and MATLAB® codes) enable readers to implement the workflows and models in practice. This significantly revised edition will continue to be the go-to reference for students and researchers interested in rock physics, near-surface geophysics, seismology, and professionals in the oil and gas industries.

The Rock Physics Handbook

The concept of tortuosity is used to characterize the structure of porous media, to estimate their electrical and hydraulic conductivity, and to study the travel time and length for tracer dispersion, but different types of tortuosity—geometric, hydraulic, electrical, and diffusive—have been used essentially interchangeably in the literature. Here, we critically review the tortuosity models developed empirically, analytically, and numerically for flow in both saturated and unsaturated porous media. We emphasize that the proposed tortuosity models are distinct and thus may not be used interchangeably. Given the variety of models that have been developed, and the sharp differences between some of them, no consensus has emerged unifying the models in a coherent way. Related treatments of tortuosity are found in the literature on porous catalysts. In such materials, nonlinear reactions ordinarily accompany transport, and the effective diffusivity within the pore space in the presence of the reactions is distinct from the one in their absence. Thus, because tortuosity may be defined as the ratio of the effective diffusivities in the bulk material and within the pore space, a careful treatment of tortuosity may need to distinguish between transport with and without reactions. This complication is ultimately relevant to soils as well, because bioremediation and biodegradation in soils are always accompanied by nonlinear reactions. Common models of tortuosity include both logarithmic functions and power laws. In many cases, the differences between the logarithmic and power-law phenomenologies are not great, but power laws can usually be reconciled with percolation concepts. Invoking percolation theory provides both insight into the origin of the power functions and a framework for understanding differences between tortuosity models.

Tortuosity in Porous Media: A Critical Review

Diffusion and Reactions in Fractals and Disordered Systems

Spontaneous imbibition of wetting liquids in porous media is a ubiquitous natural phenomenon which has received much attention in a wide variety of fields over several decades. Many traditional and recently presented capillary-driven flow models are derived based on Hagen–Poiseuille (H–P) flow in cylindrical capillaries. However, some limitations of these models have motivated modifications by taking into account different geometrical factors. In this work, a more generalized spontaneous imbibition model is developed by considering the different sizes and shapes of pores, the tortuosity of imbibition streamlines in random porous media, and the initial wetting-phase saturation. The interrelationships of accumulated imbibition weight, imbibition rate and gas recovery and the properties of the porous media, wetting liquids, and their interactions are derived analytically. A theoretical analysis and comparison denote that the presented equations can generalize several traditional and newly developed models fr...

https://eps.utk.edu/faculty/perfect/Papers/Cai%20et%20al.%20(2014).pdf

Generalized Modeling of Spontaneous Imbibition Based on Hagen− Poiseuille Flow in Tortuous Capillaries with Variably Shaped Apertures

We study numerically the tortuosity-porosity relation in a microscopic model of a porous medium arranged as a collection of freely overlapping squares. It is demonstrated that the finite-size, slow relaxation and discretization errors, which were ignored in previous studies, may cause significant underestimation of tortuosity. The simple tortuosity calculation method proposed here eliminates the need for using complicated, weighted averages. The numerical results presented here are in good agreement with an empirical relation between tortuosity (T) and porosity (varphi) given by T-1 proportional, variantlnvarphi , that was found by others experimentally in granule packings and sediments. This relation can be also written as T-1 proportional, variantRSvarphi with R and S denoting the hydraulic radius of granules and the specific surface area, respectively. Applicability of these relations appears to be restricted to porous systems of randomly distributed obstacles of equal shape and size.

Tortuosity-porosity relation in porous media flow

Tortuosity (T) is a parameter describing an average elongation of fluid streamlines in a porous medium as compared to free flow. In this paper several methods of calculating this quantity from lengths of individual streamlines are compared and their weak and strong features are discussed. An alternative method is proposed, which enables one to calculate T directly from the fluid velocity field, without the need of determining streamlines, which greatly simplifies determination of tortuosity in complex geometries, including those found in experiments or three-dimensional computer models. Based on numerical results obtained with this method, (a) a relation between the hydraulic tortuosity of an isotropic fibrous medium and the porosity is proposed, (b) a relation between the divergence rate of T with the system size at percolation porosity and the scaling of the most probable traveling length at bond percolation is found, and (c) a range of porosities for which the shape factor is constant is identified.

Hydraulic tortuosity in arbitrary porous media flow.

Nitrogen (N) loss rates were determined in permeable sediments of the Wadden Sea using a combination of stable N isotope incubation experiments and model simulation approaches during three seasons. Three different incubation methods that employed the isotope pairing technique were used: intact core incubations simulating either (1) diffusive or (2) advective transport conditions and (3) slurry incubations. N loss rates from core incubations under simulated advective transport conditions exceeded those rates measured under diffusive transport conditions by 1–2 orders of magnitude, but were comparable to those observed in slurry incubations. N loss rates generally showed little seasonal and spatial variation (207 6 30 mmol m22 h21) in autumn 2006 and spring and summer 2007. Utilizing an extensive time series of nutrient concentrations and current velocities obtained from a continuous monitoring station, nitrate and nitrite (i.e., NO { ) flux into the sediment was modeled over a full annual cycle. Fluxes were sufficient to support the experimentally derived N loss rates. Combining the measured rates with the modeled results, an annual N removal rate of 745 6 109 mmol N m22 yr21 was estimated for permeable sediments of the Wadden Sea. This rate agrees well with previous N loss estimates for the Wadden Sea based on N budget calculations. Permeable sediments, accounting for 58–70% of the continental shelf area, are an important N sink and their contribution to the global N loss budget should be reevaluated.

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Intensive and extensive nitrogen loss from intertidal permeable sediments of the Wadden Sea

Tortuosity is one of the key parameters describing the geometry and transport properties of porous media. It is defined either as an average elongation of fluid paths or as a retardation factor that measures the resistance of a porous medium to the flow. However, in contrast to a retardation factor, an average fluid path elongation is difficult to compute numerically and, in general, is not measurable directly in experiments. We review some recent achievements in bridging the gap between the two formulations of tortuosity and discuss possible method of numerical and an experimental measurements of the tortuosity directly from the fluid velocity field.

Maciej Matyka

Papers

Tortuosity-porosity relation in porous media flow

Hydraulic tortuosity in arbitrary porous media flow.

Intensive and extensive nitrogen loss from intertidal permeable sediments of the Wadden Sea

How to calculate tortuosity easily

How to Calculate Tortuosity Easily