Spatial dependence of the hydraulic conductivity in a well‐type configuration at the mesoscale

The spatial distribution of the hydraulic conductivity κ is modelled by a power law, and we present a methodological approach to quantify the exponent (crowding index) of such a law as detected within a well‐type flow configuration. Based upon the outcome of several pumping tests conducted into a caisson (mesoscale), we identify the crowding index as function of the volumetric flow rate. Hence, we develop a simple (although approximated) procedure to assess whether the spatial distribution of κ can be characterized by a power law. We demonstrate that, even at the mesoscale, the conductivity κ can not be regarded as a formation's property (nonlocality), in agreement with the recent developments on the theory of flows into radial configurations.     

Analytical findings for power law cross-sections: Uniform flow depth

Analytical results concerning open channel flows are presented, assuming that the cross-section is defined by a power law relationship between the channel width and the channel depth. Explicit equations to compute the normal flow depth are derived by considering the liquid discharge, the channel roughness height and the cross-section geometry (based on knowledge of the power law exponent, the reference width, and the reference depth) as known quantities.Such equations are deduced by writing the physical quantities as a power expansion in the power law exponent and expressing the wetted perimeter using a Gauss hypergeometric function. With the designed procedure, an accurate estimations of the integrals required to invert the uniform flow formula are obtained, at least for cross-sections characterized by aspect ratios of technical interest.Two relationships are proposed between the normal depth and the flow discharge. The first relationship is shown to work well for any discharge, provided that the width to depth ratio is sufficiently large. If this is not the case, the second procedure must be used for non-dimensional discharge larger than a given threshold, while the former procedure remains valid under the threshold.     

Variation of the ionospheric scintillation index with elevation angle of the transmitter

We have analyzed GPS data from 2007–2011 to determine the nature of variation of scintillation index with elevation of the direction of propagation at an observing point Warsaw, Poland, and Hornsund, Svalbard. To compare with the theory, the intensity scintillation index is simulated as a function of elevation angle, azimuth, magnetic field inclination, and shape of irregularities, using the phase screen model of scintillation as formulated by Rino (1979). Data analysis has been done for the seasonal as well as geomagnetic activity dependence of ionospheric scintillation. Scintillation index is a power-law function of the cosecant of the elevation angle. Results show that the power law strongly depends on the form of irregularities, being larger than in isotropic case for irregularities with dimension along the magnetic field direction smaller than those across the magnetic field. The present work also shows the need to use experimentally derived dependence on elevation.     

Estimates of effective permeability for non-Newtonian fluid flow in randomly heterogeneous porous media

An understanding of the interplay between non-Newtonian effects in porous media flow and field-scale domain heterogeneity is of great importance in several engineering and geological applications. Here we present a simplified approach to the derivation of an effective permeability for flow of a purely viscous power–law fluid with flow behavior index n in a randomly heterogeneous porous domain subject to a uniform pressure gradient. A standard form of the flow law generalizing the Darcy’s law to non-Newtonian fluids is adopted, with the permeability coefficient being the only source of randomness. The natural logarithm of the permeability is considered a spatially homogeneous and correlated Gaussian random field. Under the ergodic hypothesis, an effective permeability is first derived for two limit 1-D flow geometries: flow parallel to permeability variation (serial-type layers), and flow transverse to permeability variation (parallel-type layers). The effective permeability of a 2-D or 3-D isotropic domain is conjectured to be a power average of 1-D results, generalizing results valid for Newtonian fluids under the validity of Darcy’s law; the conjecture is validated comparing our results with previous literature findings. The conjecture is then extended, allowing the exponents of the power averaging to be functions of the flow behavior index. For Newtonian flow, novel expressions for the effective permeability reduce to those derived in the past. The effective permeability is shown to be a function of flow dimensionality, domain heterogeneity, and flow behavior index. The impact of heterogeneity is significant, especially for shear-thinning fluids with a low flow behavior index, which tend to exhibit channeling behavior.     

Fitting and goodness-of-fit test of non-truncated and truncated power-law distributions

Recently, Clauset, Shalizi, and Newman have proposed a systematic method to find over which range (if any) a certain distribution behaves as a power law. However, their method has been found to fail, in the sense that true (simulated) power-law tails are not recognized as such in some instances, and then the power-law hypothesis is rejected. Moreover, the method does not work well when extended to power-law distributions with an upper truncation. We explain in detail a similar but alternative procedure, valid for truncated as well as for non-truncated power-law distributions, based in maximum likelihood estimation, the Kolmogorov-Smirnov goodness-of-fit test, and Monte Carlo simulations. An overview of the main concepts as well as a recipe for their practical implementation is provided. The performance of our method is put to test on several empirical data which were previously analyzed with less systematic approaches. We find the functioning of the method very satisfactory.     

Spatial and temporal spectra of the geomagnetic field and their scaling properties

Many natural phenomena show a relationship between their spatial and temporal Fourier spectra. This paper discusses such a connection for the geomagnetic field, when some assumptions are made about the (exponential or power-law) behaviour of the spatial power spectrum of the field itself and that of its time derivative (the spatial spectrum of the secular variation) as estimated from global geomagnetic field models. It is shown that, under either assumption, the temporal spectrum of the geomagnetic field computed at the core–mantle boundary (CMB) would have a power-law behaviour with a negative spectral exponent of about 0.5. At the Earth’s surface, although the temporal spectrum obtained from the power-law spatial model assumes a slightly more complicated form, it can be practically approximated with a power law with a negative exponent of about 3.6. Analysis of magnetic observatory data confirms these results and that the starting hypotheses are reasonable, especially in view of the possibly chaotic state of the dynamical processes underlying the generation and maintenance of the geomagnetic field.     

A Constitutive Scaling Law for Shear Rupture that is Inherently Scale-dependent, and Physical Scaling of Nucleation Time to Critical Point

It is shown that the rupture nucleation length increases up to the critical length with time according to a power law, and that the accelerating phase of nucleation leading up to the critical point is scaled in the framework of fracture mechanics based on slip-dependent constitutive formulation. Geometric irregularity of the rupturing surfaces plays a fundamental role in scaling the accelerating phase of nucleation up to the critical point. A power-law scaling relation between the rupture growth length and the nucleation time to the critical point is derived from theoretical consideration based on laboratory data. This power-law scaling relation has no singularity, and hence it may be useful for the predictive purpose of an imminent, large earthquake.     

Generalized flux law,with an application

A generalized flux law, which can specialize into power‐ and diffusive‐type flux laws, is proposed. When coupled with the continuity equation, a generalized flow equation is the result. The generalized equation specializes into several cases, one of which is the kinematic wave equation. By applying the generalized flow equation and its variants to flow routing in two rivers, the usefulness of the generalized flux law is evaluated. Copyright © 1999 John Wiley & Sons, Ltd.     

Self-similar adiabatic flow headed by a magnetogasdynamic cylindrical shock wave in a rotating non-ideal gas

Similarity solutions are obtained for one-dimensional adiabatic flow behind a magnetogasdynamic cylindrical shock wave propagating in a rotating non-ideal gas in presence of an azimuthal magnetic field. The density of the medium ahead of the shock is assumed to be constant. In order to obtain the similarity solutions the angular velocity of the ambient medium is assumed to be obeying a power law and to be decreasing as the distance from the axis increases. It is found that the similarity solutions exist, in both the cases, when the initial magnetic field is constant or obeying a power law. The effects of an increase in the value of the index for variation of angular velocity of the ambient medium, in the value of the parameter of the non-idealness of the gas and in the strength of the initial magnetic field are obtained.     

Knickpoint recession rate and catchment area: the case of uplifted rivers in Eastern Scotland

Knickpoint behaviour is a key to understanding both the landscape responses to a base‐level fall and the corresponding sediment fluxes from rejuvenated catchments, and must be accommodated in numerical models of large‐scale landscape evolution. Knickpoint recession in streams draining to glacio‐isostatically uplifted shorelines in eastern Scotland is used to assess whether knickpoint recession is a function of discharge (here represented by its surrogate, catchment area). Knickpoints are identified using DS plots (log slope versus log downstream distance). A statistically significant power relationship is found between distance of headward recession and catchment area. Such knickpoint recession data may be used to determine the values of m and n in the stream power law, E = KA^mSⁿ. The data have too many uncertainties, however, to judge definitively whether they are consistent with m = n = 1 (bedrock erosion is proportional to stream power and KPs should be maintained and propagate headwards) or m = 0·3, n = 0·7 (bedrock incision is proportional to shear stress and KPs do not propagate but degrade in place by rotation or replacement). Nonetheless, the E Scotland m and n values point to the dominance of catchment area (discharge) in determining knickpoint retreat rates and are therefore more consistent with the stream power law formulation in which bedrock erosion is proportional to stream power. Copyright © 2005 John Wiley & Sons, Ltd.     

An analytical solution for non-Darcian flow in a confined aquifer using the power law function

We have developed a new method to analyze the power law based non-Darcian flow toward a well in a confined aquifer with and without wellbore storage. This method is based on a combination of the linearization approximation of the non-Darcian flow equation and the Laplace transform. Analytical solutions of steady-state and late time drawdowns are obtained. Semi-analytical solutions of the drawdowns at any distance and time are computed by using the Stehfest numerical inverse Laplace transform. The results of this study agree perfectly with previous Theis solution for an infinitesimal well and with the Papadopulos and Cooper’s solution for a finite-diameter well under the special case of Darcian flow. The Boltzmann transform, which is commonly employed for solving non-Darcian flow problems before, is problematic for studying radial non-Darcian flow. Comparison of drawdowns obtained by our proposed method and the Boltzmann transform method suggests that the Boltzmann transform method differs from the linearization method at early and moderate times, and it yields similar results as the linearization method at late times. If the power index n and the quasi hydraulic conductivity k get larger, drawdowns at late times will become less, regardless of the wellbore storage. When n is larger, flow approaches steady state earlier. The drawdown at steady state is approximately proportional to r¹⁻ⁿ, where r is the radial distance from the pumping well. The late time drawdown is a superposition of the steady-state solution and a negative time-dependent term that is proportional to t^{(1−n)/(3−n)}, where t is the time.     

State of stress in the lithosphere: Inferences from the flow laws of olivine

The experimental flow data for rocks and minerals are reviewed and found to fit a law of the form $$\dot \varepsilon = A'\left[ {sinh (\alpha \sigma )} \right]^n \exp \left[ {{{ - (E * + PV * )} \mathord{\left/ {\vphantom {{ - (E * + PV * )} {RT}}} \right. \kern-\nulldelimiterspace} {RT}}} \right]$$ where \(\dot \varepsilon \) This law reduces to the familiar power-law stress dependency at low stress and to an exponential stress dependency at high stress. Using the material flow law parameters for olivine, stress profiles with depth and strain rate are computed for a representative range of temperature distributions in the lithosphere. The results show that the upper 15 to 25 km of the oceanic lithosphere must behave elastically or fail by fracture and that the remainder deforms by exponential law flow at intermediate depths and by power-law flow in the rest. A model computation of the gravitational sliding of a lithospheric plate using olivine rheology exhibits a very sharp decoupling zone which is a consequence of the combined effects of increasing stress and temperature on the flow law, which is a very sensitive function of both.     

On a power series solution to the Boussinesq equation

For certain initial and boundary conditions the Boussinesq equation, a nonlinear partial differential equation describing the flow of water in unconfined aquifers, can be reduced to a boundary value problem for a nonlinear ordinary differential equation. Using Song et al.'s (2007) [7] approach, we show that for zero head initial condition and power-law flux boundary condition at the inlet boundary, the solution in the form of power series can be obtained with Barenblatt's (1990) [2] rescaling procedure applied to the power series solution obtained in Song et al. (2007) [7] for the power-law head boundary condition. Polynomial approximations can then be obtained by taking terms from the power series. Although for a small number of terms the newly obtained approximations may be worse than polynomial approximations obtained by other techniques, any desired accuracy can be achieved by taking more terms from the power series.     

Probabilistic Regularities in Unfavorable Hydrochemical Phenomena

The balance equation for a substance washed out in a river basin is analyzed under the assumption that the runoff of this substance and its reserves in the watershed are directly proportional. The proportionality factor is perturbed by a random component, which accounts for the effect of atmospheric precipitation. The balance equation is transformed into a stochastic differential equation with a multiplicative white noise, which is used to construct a Fokker-Plank equation for the probability density of chemical flow. A stationary solution containing a power function is found for this equation. Because of the proportionality of the concentration and chemical flow, the concentration distribution also obeys the power law. Statistical treatment of empirical data on some water quality characteristics and water flow showed that the power law adequately describes the probability of unfavorable hydrochemical events. The parameters of this law for turbidity, color index, permanganate oxidability, and ammonia concentration are evaluated.__________Translated from Vodnye Resursy, Vol. 32, No. 4, 2005, pp. 452–458.Original Russian Text Copyright © 2005 by Dolgonosov, Korchagin.     

Power-law velocity profile in turbulent boundary layers: An integral reynolds-number dependent solution

Geophysical flows of practical interest encompass turbulent boundary layer flows. The velocity profile in turbulent flows is generally described by a log- or a power-law applicable to certain zones of the boundary layer, or by wall-wake law for the entire zone of the boundary layer. In this study, a novel theory is proposed from which the power-law velocity profile is obtained for the turbulent boundary layer flow. The new power-law profile is based on the conservation of mass and the skin friction within the boundary layer. From the proposed theory, analytical expressions for the power-law velocity profile are presented, and their Reynolds-number dependency is highlighted. The velocity profile, skin friction coefficient and boundary layer thickness obtained from the proposed theory are validated by the reliable experimental data for zero-pressure gradient turbulent boundary layers. The expressions for Reynolds shear stress and eddy viscosity distributions across the boundary layer are also obtained and validated by the experimental data.     

Wavelet power spectrum analysis of heterogeneities from sonic velocity logs

The continuous wavelet transform (CWT) is used to evaluate local variations in the power-law exponents of sonic log data. The resulting wavelet spectrum can be compared with the corresponding global estimates obtained by conventional Fourier transform methods. In Fourier analysis, the fundamental tool used to characterize a fluctuating velocity distribution is the power spectrum. It represents the energy contained in each wavenumber and thus provides information regarding the importance of each scale of heterogeneity. However, important spatial information regarding the location of events becomes implicit in the phase angle of the Fourier transform. In this paper, it is shown how the square of the amplitude of the wavelet transform is related to the Fourier spectrum and how spatial information can be expressed in an explicit manner. Using the conservation of energy, it is shown that the average wavelet power spectrum over the total depth range is equal to the global power spectrum. A Gaussian wavelet is chosen to realize the wavelet transform. Two synthetic sonic logs with exponential and von Karman correlation functions are used to demonstrate the potential of the suggested analysis. Furthermore, the wavelet transform is applied to the KTB (Continental Deep Drilling Program) sonic log data. The wide range of applications of the CWT shows that this transform is a natural tool for characterizing the structural properties of underground heterogeneities. It offers the possibility of separating the multiscale components of heterogeneities.     

A Simple Analogue Experiment to Account for Power-law and Exponential Decays of Earthquake Sequences

— An event rate decays either exponentially in earthquake swarms, or according to the power law in aftershock sequences, suggesting different physics for these decays. In order to investigate the physics, I measured decays in the rate of acoustic emissions (AE) that occur to relax thermal stress after turning off a home cooking apparatus. Such AE are analogous to an earthquake induced by fluid intrusion because thermo-elasticity was precisely analogous to poro-elasticity. To consider what controlled the decays, different heating rates and durations were specified in several experimental runs for which the mechanical and thermal properties were identical. Temperature decayed exponentially with the same time constant. However, both a power-law decay with a p value roughly equal to two and an exponential decay were observed. To ascertain what was occurring at the AE source, the stress and frictional strength at the AE source were numerically estimated. The stress at the AE source decayed exponentially for both power-law decay and exponential decay. In power-law decay, strength was initially very low and recovered immediately and significantly. However, in exponential decay, the change in strength during the initial stage was not as large as that in power-law decay. It was demonstrated that the time constant of exponential event-rate decay was identical to that of temperature decay if strength is constant.     

On the validity of empirical power laws

Empirical power laws are frequently used to relate parameters in complex hydrological and hydrometeorological processes. The validity of power laws relating two parameters with a common variable may be compromised by spurious influences of the common variable. Theoretical results are presented that allow both the spurious self-correlation coefficient and the slope of a spurious self-correlation to be determineda priori. Raising a common variable to a higher power in either parameter amplifies the spurious effects.Power law regression equations are not single-valued analytical functions and must not be treated as such. Because of the strong influence of a common variable on the correlation coefficient, the transfer of a common variable from one side of a power-law regression equation to another (by cross-multiplying) may severely distort the results. Examples from lake hydrology are presented.     

Quantitative Analysis of Seismicity Before Large Taiwanese Earthquakes Using the G-R Law

Seismicity has been identified as an example of a natural, nonlinear system for which the distribution of frequency and event size follow a power law called the “Gutenberg–Richter (G-R) law.” The parameters of the G-R law, namely b- and a-values, have been widely used in many studies about seismic hazards, earthquake forecasting models, and other related topics. However, the plausibility of the power law model and applicability of parameters were mainly verified by statistical error σ of the b-value, the effectiveness of which is still doubtful. In this research, we used a newly defined p value developed by Clausetet al. (Power-Law Distributions in Empirical Data, SIAM Rev. 51, 661–703, 2009) instead of the statistical error σ of the b-value and verified its effectiveness as a plausibility index of the power-law model. Furthermore, we also verified the effectiveness of K–S statistics as a goodness-of-fit test in estimating the crucial parameter \(M_{\text{c}}\) of the power-law model.