1,721,009 research outputs found
The mathematics of early diagenesis: from worms to waves
No full textThe changes that sediments undergo after deposition are collectively known as diagenesis. Diagenesis is not widely recognized as a source for mathematical ideas; however, the myriad processes responsible for these changes lead to a wide variety of mathematical models. In fact, most of the classical models and methods of applied mathematics emerge naturally from quantification of diagenesis. For example, small-scale sediment mixing by bottom-dwelling animals can be described by the diffusion equation; the dissolution of biogenic opal in sediments leads to sets of coupled, nonlinear, ordinary differential equations; and modeling organisms that eat at depth in the sediment and defecate at the surface suggests the one-dimensional wave equation, while the effect of waves on pore waters is governed by the two- or three-dimensional wave equation. Diagenetic modeling, however, is not restricted to classical methods. Diagenetic problems of concern to modern mathematics exist in abundance; these include free-boundary problems that predict the depth of biological mixing or the penetration of O2 into sediments, algebraic-differential equations that result from the fast-reversible reactions that regulate pH in pore waters, inverse calculations of input functions (histories), and the determination of the optimum choice in a hierarchy of possible diagenetic models. This review highlights and explores these topics with the hope of encouraging further modeling and analysis of diagenetic phenomena
Slow growth of an isolated disk-shaped bubble of constant eccentricity in the presence of a distributed gas source
No full textIn this paper we consider the diffusion-controlled (small Péclet number) growth of an isolated, oblate-spheroidal (disk-shaped) bubble of constant eccentricity (aspect ratio) in a medium that actively produces the volatile substance via a distributed source, but does not itself offer significant resistance to growth. Oblate spheroidal bubbles are predicted to grow faster than spherical ones, due to the higher surface area to volume ratio; yet, bubbles of all eccentricities grow proportionally to the square root of time, as expected for a diffusive process. In the presence of a distributed source, however, the growth time becomes dependent on the square-root of the source strength, in the limit as the boundary forcing, i.e., the degree of super-saturation, becomes negligible. Furthermore, we demonstrate that the previously known spherical solution is contained within the more general spheroidal solution. In addition, we produced new expression to describe the growth of a disk in terms of the evolution of the radius of a volume-equivalent sphere and another simple expression relating the growth time of a disk to that of a sphere
Rate of growth of isolated bubbles in sediments with a diagenetic source of methane. (errata)
No full textObservation of bubbles in estuarine and coastal sediments indicates that bubbles at or below 10 cm depth grow on seasonal time scales (May-October). In order to determine the controls on this growth rate, we have constructed a diffusion-reaction model that accounts for the dynamics of methane formation, its diffusion through pore waters, its incorporation into a bubble, and the consequent growth of the bubble. The model produces an explicit equation for the radius of a growing bubble, R(t), with time using mean parameter values and under the assumption that the mechanics of the sediment response to growth can be neglected: [equation]. where w is the porosity, D is the tortuosity-corrected diffusivity, cg is the concentration of gas in the bubble, S is the rate of methanogenesis near the bubble, R1 is the half-separation distance between bubbles (R1 k R), c1 is the ambient CH4 concentration, c0 is the pore-water CH4 concentration at R, t is time, and R0 is the initial bubble radius, if not zero. The effects of the source S and supersaturation (c1 2 c0), thus, appear as separate contributing terms, and this formula can then be applied even in those cases where apparently c1 ≈ c0. The model is applied to three sediments where bubbles have been previously studied, i.e., Cape Lookout Bight (USA), White Oak River (USA) and Eckernfoörde Bay (Germany). In all three cases, using the site-specific time-averaged parameter values, the model predicts seasonal growth rates, consistent with the observations. Furthermore, the source term dominates the rate of growth at the first of these two sites, whereas diffusion from the ambient supersaturation dominates at the German location. Real bubbles may follow a more complicated growth history than predicted by the above equation because of the mechanical properties of sediments; nevertheless, the overall growth times are concordant with ultimate diffusion control. The effects of rectified diffusion, that is, the pumping of gas into a bubble by pressure oscillations, e.g., from waves and tides, were also examined. Existing models for that process suggest that it is negligible, due to the low frequency of these types of oscillations
Mechanical response of sediments to bubble growth
No full textModeling the process of bubble growth in sediments requires an understanding of the physics that controls bubble shape and the interaction of the growing bubble with the sediment. To acquire this understanding we have conducted experiments in which we have injected gas through a fine capillary into natural and surrogate sediment samples and have monitored pressure during bubble growth to provide information about stress and strain. In gas injection studies with natural sediment samples, we have observed two modes of bubble growth behavior. One of these modes, characterized by a saw-tooth record of pressure as the bubble grows, is consistent with fracture of the medium. Observations indicate that bubble growth by fracture should correspond to bubbles that are coin- or disk-shaped. This shape is confirmed in observations of bubbles in natural sediments and in our studies of bubble injection into gelatin, a surrogate sediment material. Interpretation of the stress–strain results for bubble growth also required that we measure Young’s modulus, E. The measurements show E to be near 0.14 MN m2, which differs by more than 4 orders of magnitude from values that have been reported in the literature. Our measurements of E give substantially better estimates of bubble shape than are predicted using the literature values. Our data are interpreted with linear elastic fracture mechanics (LEFM) which predicts that the critical pressure for bubble growth will depend on the bubble volume, V raised to the −1/5 power. While evidence of substantial heterogeneity in sediment properties is apparent in our results, this V−1/5 dependence is confirmed. Through application of LEFM theory, we have determined the critical stress intensity factor, K1c, a material property and the principal determinant of bubble shape and growth by fracture. Our values of K1c range from ∼2.8×10−4 MN m−3/2 to ∼4.9×10−4 MN m−3/2 for our natural sediment samples from Cole Harbor, Nova Scotia. We have also estimated the critical stress intensity factor for Eckernförde Bay samples by analyzing published images of natural bubbles. The K1c obtained in this way is similar to our Cole Harbor results and is ∼5.5×10−4 MN m−3/2
Simulation of potassium feldspar dissolution and illitization in the Statfjord Formation, North Sea
No full textLattice-automaton bioturbation simulator (LABS): implementation for small deposit feeders
No full textA new model for biological activity and its effects in sediments is presented. Sediment is represented as a random 2D collection of solid and water “particles”, distributed on a regular lattice with individually assigned chemical, biological and physical properties, e.g. food versus inert material. Model benthic organisms move through the lattice (the virtual sediment) as programmable entities, i.e., automatons, by displacing or ingesting–defecating particles. Each type of automaton obeys a different set of rules, both deterministic and stochastic, designed to mimic real infauna. In the present version of the model code, the organisms are simple small deposit feeders, resembling capitellids.The results from the model are 2D visualizations of the movement of the animals and the particles with time. The latter provide immediate appreciation of the consequences of animal actions on sediment fabric and composition, including both the mixing, traditionally associated with bioturbation, and the development of biologically-induced heterogeneities, which are observed in real sediments. The output is readily amenable to presentation as computer-generated (QuickTimeTM) movies, for which links are provided to such examples. As a particular case, we present a simulation of the mixing of a sand plug in a muddy sediment which shows that this is process not accomplished by counter-diffusion of sand and mud but by displacement and dilution of the sand with mud that is defecated as feces therein; this mode of mixing appears to be far more favorable to preservation of this sand feature than traditional diffusive models.<br/
Diffusion in a lattice-automaton model of bioturbation by small deposit feeders
Full text linkThe mixing of 210Pb and tagged particles is examined in a lattice-automaton model for bioturbation containing small deposit feeders. The values of the biodiffusion coefficient, DB, calculated using typical biological parameter values, i.e., size, abundance, feeding and locomotion rates, are similar to those expected from marine sediments of a given sedimentation rate. Most biological parameters appear to exert primarily linear effects on DB values, while most nonlinearities seem to be model artifacts or failures of the assumptions in the basic DB model. The model highlights the importance of ingestion-egestion, over simple particle displacement, as an agent of bioturbation. The tagged particles are used to calculate root-mean-squared displacement plots, which are linear over long time spans, indicating diffusive behavior. However, initial trends on such plots are not usually linear, indicating that the calculated DB is time dependent for surprisingly long periods after the beginning of such experiments. The latter constitutes a warning to the interpretation of short-term tracer experiments where tagged-particles are salted onto the sediment-water interface and mixing is dominated by small deposit feeders
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