Dimensional analysis of two-phase flow including a rate-dependent capillary pressure–saturation relationship

The macroscopic modelling of two-phase flow processes in subsurface hydrosystems or industrial applications on the Darcy scale usually requires a constitutive relationship between capillary pressure and saturation, the P_c(S_w) relationship. Traditionally, it is assumed that a unique relation between P_c and S_w exists independently of the flow conditions as long as hysteretic effects can be neglected. Recently, this assumption has been questioned and alternative formulations have been suggested. For example, the extended P_c(S_w) relationship by Hassanizadeh and Gray [Hassanizadeh SM, Gray WG. Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries. Adv Water Resources 1990;13(4):169–86] proposes that the difference between the phase pressures to the equilibrium capillary pressure is a linear function of the rate of change of saturation, thereby introducing a constant of proportionality, the coefficient τ. It is desirable to identify cases where the extended relationship needs to be considered. Consequently, a dimensional analysis is performed on the basis of the two-phase balance equations. In addition to the well-known capillary and gravitational number, the dimensional analysis yields a new dimensionless number. The dynamic number Dy quantifies the ratio of dynamic capillary to viscous forces. Relating the dynamic to the capillary as well as the gravitational number gives the new numbers DyC and DyG, respectively. For given sets of fluid and porous medium parameters, the dimensionless numbers Dy and DyC are interpreted as functions of the characteristic length and flow velocity. The simulation of an imbibition process provides insight into the interpretation of the characteristic length scale. The most promising choice for this length scale seems to be the front width. We conclude that consideration of the extended P_c(S_w) relationship may be important for porous media with high permeability, small entry pressure and high coefficient τ when systems with a small characteristic length (e.g. steep front) and small characteristic time scale are under investigation.     

Effects of fractional wettability on capillary pressure–saturation–relative permeability relations of two-fluid systems

Capillary pressure (P_c)–saturation (S)–relative permeability (k_r) relationships must be quantified to accurately predict non-aqueous phase liquid (NAPL) distribution in the subsurface. Several experimental techniques are presented here for two-fluid P_c–S–k_r relationships for various saturation paths to better define the effect of fractional wettability on these relationships. During the primary drainage path of the P_c–S curves, the air–water system showed no distinct trend as a function of the fraction of sand treated by organosilane (S) to render it non-water wetting. In a NAPL–water system, however, a consistent decrease of capillary pressure with increase of the fraction of non-water wetting sands was observed. The much lower contact angle for air–water (a–w) system may result in the observed insensitivity of the a–w P_c–S curves to fractional wettability, at least for the PD pathway. For the main imbibition path of NAPL–water system, capillary pressure decreased as the fraction of the S component increased, requiring forced imbibition (negative capillary pressures) for a certain range of saturations. Systems with an increasing percentage of the S component also exhibited a higher water k_r and lower NAPL or air k_r at a given saturation for the primary drainage and main imbibition paths in both air–water and NAPL–water systems. The increase of water k_r with increase of the fraction of the S component can be explained by the ability of water to occupy larger and highly conductive pores in such a system. Experimental k_r–S data for the primary drainage path of NAPL–water system presented here were used to test the Bradford et al. [Bradford SA, Abriola LM, Leij FJ. Wettability effects on two- and three-fluid relative permeabilities. J Contam Hydrol 1997;28:171–91] model and the modified Mualem model for estimating the k_r–S curves from measured P_c–S data as a function of fractional wettability. Both models predicted significantly less variation in the k_r–S curves than measured indicating that they did not adequately represent the system under investigation.     

The influence of unavoidable saturation averaging on the experimental measurement of dynamic capillary effects: A numerical simulation study

Many studies over the past four decades have observed that capillary pressure–saturation (P_c–S_w) relationships are often different when measured dynamically under rapidly changing pressure inputs. This phenomenon has been referred to as a dynamic capillary effect, and its magnitude is often quantified by the dynamic capillary coefficient, τ. Experimentally-reported values of τ have varied by orders of magnitude, even for seemingly similar experimental systems. The purpose of the present work is to numerically explore the likely impact of fluid properties on the calculation of τ from experimental measurements. Specifically, the emphasis is on understanding how spatial averaging of the saturation profiles resulting from different fluid combinations contributes to the apparent magnitude of τ derived from experimental measurements.Simulations of dynamic drainage in a packed sand column were conducted using the CompSim multiphase flow simulator. Four nonwetting phase fluids with viscosities spanning four orders of magnitude were studied. Comparison between local and spatially-averaged rates of saturation change show significant differences, with the magnitude of the difference increasing with increasing viscosity to interfacial tension ratio and increasing drainage rate. Results show that at averaging scales likely to be experienced during experimental saturation measurements, this effect is likely to produce significant differences in the ultimate magnitude of the calculated τ values for different fluid systems and drainage rates. This result means that conventional flow phenomena may produce an inherent systematic bias in experimental measurements of τ, amplifying measured values for high viscosity or low interfacial tension systems and for experiments where higher drainage rates are used.     

Analysis of the nonlinear storage–discharge relation for hillslopes through 2D numerical modelling

Storage–discharge curves are widely used in several hydrological applications concerning flow and solute transport in small catchments. This article analyzes the relation Q(S) (where Q is the discharge and S is the saturated storage in the hillslope), as a function of some simple structural parameters. The relation Q(S) is evaluated through two‐dimensional numerical simulations and makes use of dimensionless quantities. The method lies in between simple analytical approaches, like those based on the Boussinesq formulation, and more complex distributed models. After the numerical solution of the dimensionless Richards equation, simple analytical relations for Q(S) are determined in dimensionless form, as a function of a few relevant physical parameters. It was found that the storage–discharge curve can be well approximated by a power law function Q/(LK_s) = a(S/(L²(? ? θ_r)))^b, where L is the length of the hillslope, K_s the saturated conductivity, ? ? θ_r the effective porosity, and a, b two coefficients which mainly depend on the slope. The results confirm the validity of the widely used power law assumption for Q(S). Similar relations can be obtained by performing a standard recession curve analysis. Although simplified, the results obtained in the present work may serve as a preliminary tool for assessing the storage–discharge relation in hillslopes. Copyright © 2012 John Wiley & Sons, Ltd.     

Comparison of relative permeability–fluid saturation–capillary pressure relations in the modelling of non-aqueous phase liquid infiltration in variably saturated,layered media

The characteristic functions relating relative permeabilities and capillary pressures to fluid saturations (k_r–S–P_c models) are of great importance for the modelling of sub-surface multi-phase flow and transport. In order to test their performance and to identify their important parameters, four well-known three-phase k_r–S–P_c models have been tested against published experimental data on non-aqueous phase liquid (NAPL) migration in the unsaturated zone. Both homogenous systems and systems with embedded heterogeneities have been analysed.     

Prediction of capillary air-liquid interfacial area vs. saturation function from relationship between capillary pressure and water saturation

This short communication investigates if the capillary air-liquid interfacial area vs. saturation relationship A_lv(S) can be predicted from the capillary pressure vs. saturation relationship S(h), using the theoretical sample scale model of Diamantopoulos and Durner (2013, 2015). We selected three published experimental datasets, where S(h) and A_lv(S) relationships had been measured for the same porous media. The sample scale model was fitted to the retention curve S(h) of each porous medium and then used to predict the air-liquid interfacial area A_lv(S). We also included in the analysis the thermodynamic models of Leverett (1941) and Grant and Gerhard (2007). For two sandy materials and especially for high saturation values, the model predicted the capillary A_lv(S) successfully, which was in one case given by a pore-network model simulation (Kibbey and Chen, 2012) and in the other case experimentally determined (Brusseau et al., 2006). For glass bead experiments, the contact angle needed to be fitted to properly describe the experimental A_lv(S) curve.     

Coda Q Estimates in the Koyna Region, India

—The coda Q, Q _c ?, have been estimated for the Koyna region of India. The coda waves of 76 seismograms from thirteen local earthquakes, recorded digitally in the region during July–August, 1996, have been analyzed for this purpose at nine central frequencies viz., 1.5, 2.0, 3.0, 4.0, 6.0, 8.0, 12.0, 16.0 and 24.0 Hz using a single backscattering model. All events with magnitude less than 3 fall in the epicentral distances less than 60 km and have focal depths which range from 0.86 to 9.43 km. For the 30 sec coda window length the estimated Q _c values vary from 81 to 261 at 1.5 Hz and 2088 to 3234 at 24 Hz, whereas the mean values of Q _c with the standard error vary from 148 ± 13.5 at 1.5 Hz to 2703 ± 38.8 at 24 Hz. Both the estimated Q _c values and their mean values exhibit the clear dependence on frequency in the region and a frequency dependence average attenuation relationship, Q _c = 96f ^1.09, has been obtained for the region, covering an approximate area of 11500 km² with the surfacial extent of about 120 km and depth of 60 km.¶Lapse time dependence of Q _c has also been studied for the region, with the coda waves analyzed at five lapse time windows from 20 to 60 sec duration with the difference of 10 sec. The frequency dependence average Q _c relationships obtained at these window lengths Q _c = 66f ^1.16 (20 sec), Q _c = 96f ^1.09 (30 sec), Q _c =131f ^1.04 (40 sec), Q _c = 148f ^1.04 (50 sec), Q _c = 182f ^1.02 (60 sec) show that the frequency dependence (exponentn) remains mostly stationary at all the lapse time window lengths, while the change in Q ₀ value is significant. Lapse time dependence of Q _c in the region is also interpreted as the function of depth.     

Spatial variation in soil resistance to flowing water erosion along a regional transect in the Loess Plateau

The factors influencing soil erosion may vary with scale. It remains unclear whether the spatial variation in soil erosion resistance is controlled by regional variables (e.g. precipitation, temperature, and vegetation zone) or by local specific variables (e.g. soil properties, root traits, land use, and farming operations) when the study area enlarges from a hillslope or catchment to the regional scale. This study was performed to quantify the spatial variations in soil erosion resistance to flowing water under three typical land uses along a regional transect on the Loess Plateau and to identify whether regional or local specific variables are responsible for these changes. The results indicated that the measured soil detachment capacities (D_c) of cropland exhibited an irregular trend along the regional transect. The D_c of grassland increased with mean annual precipitation, except for two sites (Yijun and Erdos). The measured D_c of woodland displayed an inverted ‘U’ shape. The changes in rill erodibility (K_r) of three land uses were similar to D_c, whereas no distinguishable trend was found for critical shear stress (τ_c). No significant correlation was detected between D_c, K_r and τ_c, and the regional variables. The spatial variation in soil erosion resistance could be explained reasonably by changes in soil properties, root traits, land use, and farming operations, rather than regional variables. The adjustment coefficient of K_r for grassland and woodland could be well simulated by soil cohesion and root mass density (R² = 0.70, P < 0.01), and the adjustment coefficient of critical shear stress could be estimated with aggregate stability (R² = 0.57, P < 0.01). The results are helpful for quantifying the spatial variation in soil detachment processes by overland flow and to develop process‐based erosion model at a regional scale. Copyright © 2015 John Wiley & Sons, Ltd.     

Determination of critical shear stress of non‐cohesive soils using submerged jet test and turbulent kinetic energy

Laboratory tests using Jet Erosion Testing (JET) apparatus, impinging normally on a horizontal boundary, were conducted to determine the critical shear stress (τ_c) of non‐cohesive soil samples. A three‐dimensional (3D) SonTek/YSI 16 MHz Micro‐Acoustic Doppler Velocimeter (MicroADV) was used to measure turbulent kinetic energy (TKE) at a radial limit of entrainment in the wall jet zone and the measurements were used to calculate τ_c of the samples. The results showed that TKE increases exponentially with increasing particle size. The τ_c from this study were comparable (R² = 0.8) to the theoretical τ_c from Shields diagram after bed roughness scale ratio (D/k_s), due to the non‐uniform bed conditions, was accounted for. This study demonstrated that JET and TKE can be used to determine τ_c of non‐cohesive soils. The use of JET and TKE was found to be faster and easier when compared to the conventional approach of using flumes. A relationship of TKE at the onset of incipient motion (TKE_c) and samples’ D₅₀ developed in this study can be used to predict τ_c of non‐cohesive soils under similar non‐uniform conditions. Copyright © 2015 John Wiley & Sons, Ltd.     

Assessing the effect of vegetation‐related bank strength on channel morphology and stability in gravel‐bed streams using numerical models

Bank strength due to vegetation dominates the geometry of small stream channels, but has virtually no effect on the geometry of larger ones. The dependence of bank strength on channel scale affects the form of downstream hydraulic geometry relations and the meandering‐braiding threshold. It is also associated with a lateral migration threshold discharge, below which channels do not migrate appreciably across their floodplains. A rational regime model is used to explore these scale effects: it parameterizes vegetation‐related bank strength using a dimensionless effective cohesion, C_r*. The scale effects are explored primarily using an alluvial state space defined by the dimensionless formative discharge, Q*, and channel slope, S, which is analogous to the Q–S diagrams originally used to explore meandering‐braiding thresholds. The analyses show that the effect of vegetation on both downstream hydraulic geometry and the meandering‐braiding threshold is strongest for the smallest streams in a watershed, but that the effect disappears for Q* > 10⁶. The analysis of the migration threshold suggests that the critical discharge ranges from about 5 m³/s to 50 m³/s, depending on the characteristic rooting depth for the vegetation. The analysis also suggests that, where fires frequently affect riparian forests, channels may alternate between laterally stable gravel plane‐bed channels and laterally active riffle‐pool channels. These channels likely do not exhibit the classic dynamic equilibrium associated with alluvial streams, but instead exhibit a cyclical morphologic evolution, oscillating between laterally stable and laterally unstable end‐members with a frequency determined by the forest fire recurrence interval. Copyright © 2008 John Wiley & Sons, Ltd.     

Exploring the feasibility of earthquake early warning using records of the 2008 Wenchuan earthquake and its aftershocks

Earthquake early warning system (EEWS) is one of the effective ways to mitigate earthquake damage and can provide few seconds to tens of seconds of advanced warning time of impending ground motions, allowing for mitigation measures to be taken in the short term. After the devastating Ms8.0 Wenchuan earthquake, hundreds of M4-6 earthquakes occurred with depth range of 2–24 km. We explore a practical approach to earthquake early warning in Wenchuan area by determining a ground-motion period parameter τ_c and a high-pass filtered vertical displacement amplitude parameter Pd from the initial 3 s of the P waveforms of these aftershocks with M≥4.0. The empirical relationships both between τ_c and M, and between Pd and peak ground velocity PGV for the Wenchuan area are presented. The τ_c result shows that it is systematically greater for slow earthquakes, leading to a possible false alarm. The moment rate function is used to handle the fact that the Pd parameter alone miss the M=8.0 mainshock. These two relationships can be used to detect the occurrence of a major earthquake and provide onsite warning in the area around the station where onset of strong ground motions is expected within seconds after the arrival of the P wave. The robustness of onsite early warning can be increased by using multistation data when the station density is high or by combing τ_c and Pd as a single indicator.     

Lattice-Boltzmann simulations of the capillary pressure–saturation–interfacial area relationship for porous media

Hysteresis in the relationship between capillary pressure (P_c), wetting phase saturation (S_w) and nonwetting–wetting interfacial area per volume (a_nw) is investigated using multiphase lattice-Boltzmann simulations of drainage and imbibition in a glass bead porous system. In order to validate the simulations, the P_c–S_w and a_nw–S_w main hysteresis loops were compared to experimental data reported by Culligan et al. [Culligan KA, Wildenschild D, Christensen BS, Gray WG, Rivers ML, Tompson AB. Interfacial area measurements for unsaturated flow through porous media. Water Resour Res 2004;40:W12413]. In general, the comparison shows that the simulations are reliable and capture the important physical processes in the experimental system. P_c–S_w curves, a_nw–S_w curves and phase distributions (within the pores) show good agreement during drainage, but less satisfactory agreement during imbibition. Drainage and imbibition scanning curves were simulated in order to construct P_c–S_w–a_nw surfaces. The root mean squared error (RMSE) and mean absolute error (MAE) between drainage and imbibition surfaces was 0.10 mm⁻¹ and 0.03 mm⁻¹, respectively. This small difference indicates that hysteresis is virtually nonexistent in the P_c–S_w–a_nw relationship for the multiphase system studied here. Additionally, a surface was fit to the main loop (excluding scanning curves) of the drainage and imbibition P_c–S_w–a_nw data and compared to the surface fit to all of the data. The differences between these two surfaces were small (RMSE = 0.05 mm⁻¹ and MAE = 0.01 mm⁻¹) indicating that the P_c–S_w–a_nw surface is adequately represented without the need for the scanning curve data, which greatly reduces the amount of data required to construct the non-hysteretic P_c–S_w–a_nw surface for this data.     

A simple parameterization of bulk canopy resistance from climatic variables for estimating hourly evapotranspiration

This paper examines a model for estimating canopy resistance r_c and reference evapotranspiration ET_o on an hourly basis. The experimental data refer to grass at two sites in Spain with semiarid and windy conditions in a typical Mediterranean climate. Measured hourly ET_o values were obtained over grass during a 4 year period between 1997 and 2000 using a weighing lysimeter (Zaragoza, northeastern Spain) and an eddy covariance system (Córdoba, southern Spain). The present model is based on the Penman–Monteith (PM) approach, but incorporates a variable canopy resistance r_c as an empirical function of the square root of a climatic resistance r* that depends on climatic variables. Values for the variable r_c were also computed according to two other approaches: with the r_c variable as a straight‐line function of r* (Katerji and Perrier, 1983, Agronomie 3 (6): 513–521) and as a mechanistic function of weather variables as proposed by Todorovic (1999, Journal of Irrigation and Drainage Engineering, ASCE 125 (5): 235–245). In the proposed model, the results show that r_c/r_a (where r_a is the aerodynamic resistance) presents a dependence on the square root of r*/r_a, as the best approach with empirically derived global parameters. When estimating hourly ET_o values, we compared the performance of the PM equation using those estimated variable r_c values with the PM equation as proposed by the Food and Agriculture Organization, with a constant r_c = 70 s m^?1. The results confirmed the relative robustness of the PM method with constant r_c, but also revealed a tendency to underestimate the measured values when ET_o is high. Under the semiarid conditions of the two experimental sites, slightly better estimates of ET_o were obtained when an estimated variable r_c was used. Although the improvement was limited, the best estimates were provided by the Todorovic and the proposed methods. The proposed approach for r_c as a function of the square root of r* may be considered as an alternative for modelling r_c, since the results suggest that the global coefficients of this locally calibrated relationship might be generalized to other climatic regions. It may also be useful to incorporate the effects of variable canopy resistances into other climatic and hydrological models. Copyright © 2005 John Wiley & Sons, Ltd.     

Turbulent transport of magnetic fields. V. Distribution of magnetic energy in a simple α2-dynamo

Abstract

In this paper we analyse the stationary mean energy density tensor T_ij = B_iB_j for the x ²-sphere. This model is one of the simplest possible turbulent dynamos, originally due to Krause and Steenbeck (1967): a conducting sphere of radius R with homogeneous, isotropic and stationary turbulent convection, no differential rotation and negligible resistivity. The stationary solution of the (linear) equation for T_ij is found analytically. Only T_rr , T _θθ and T _φφ are unequal to zero, and we present their dependence on the radial distance r.

The stationary solution depends on two coefficients describing the turbulent state: the diffusion coefficient β≈?u²?τ_c/3 and the vorticity coefficient γ ≈ ?|?×u|²?τ_c/3 where u(r, t) is the turbulent velocity and _c its correlation time. But the solution is independent of the dynamo coefficient α≈??u·?×u?τ_c/3 although α does occur in the equation for T_ij . This result confirms earlier conclusions that helicity is not required for magnetic field generation. In the stationary state, magnetic energy is generated by the vorticity and transported to the boundary, where it escapes at the same rate. The solution presented contains one free parameter that is connected with the distribution of B over spatial scales at the boundary, about which T_ij gives no information. We regard this investigation as a first step towards the analysis of more complicated, solar-type dynamos.     

An alternative approach for the Istanbul earthquake early warning system

Two recent catastrophic earthquakes that struck the Marmara Region on 17 August 1999 (M_w=7.4) and 12 November 1999 (M_w=7.2) caused major concern about future earthquake occurrences in Istanbul and the Marmara Region. As a result of the preparations for an expected earthquake may occur around Istanbul region, an earthquake early warning system has been established in 2002 with a simple and robust algorithm, based on the exceedance of specified thresholds of time domain amplitudes and the cumulative absolute velocity (CAV) levels (Erdik et al., 2003 [1]). In order to improve the capability of Istanbul earthquake early warning system (IEEWS) for giving early warning of a damaging earthquake in the Marmara Region, we explored an alternative approach with the use of a period parameter (τ_c) and a high-pass filtered vertical displacement amplitude parameter (Pd) from the initial 3 s of the P waveforms as proposed by Kanamori (2005) [2] and Wu and Kanamori (2005) 3 and 4. The empirical relationships both between τ_c and moment magnitude (M_w), and between Pd and peak ground velocity (PGV) for the Marmara Region are presented. These relationships can be used to detect a damaging earthquake within seconds after the arrival of P waves, and can provide on-site warning in the Marmara Region.     

Estimates of fracture parameters of earthquakes

A new estimate of the fracture parameters of earthquakes is provided in this paper. By theMuskhelishvili method (1953) a number of basic relations among fracture-mechanics parameters are derived. A scheme is proposed to evaluate the slip weakening parameters in terms of fault dimension, average slip, and rise time, and the new results are applied to 49 events compiled in the earthquake catalogue ofPurcaru andBerckhemer (1982). The following empirical relations are found in the paper: $$\begin{gathered} \frac{{\tau _B - \tau _f }}{{\tau _\infty - \tau _f }} = 2.339 \hfill \\ {{\omega _c } \mathord{\left/ {\vphantom {{\omega _c } {W = 0.113}}} \right. \kern-\nulldelimiterspace} {W = 0.113}} \hfill \\ \log G_c \left( {{{dyne} \mathord{\left/ {\vphantom {{dyne} {cm}}} \right. \kern-\nulldelimiterspace} {cm}}} \right) = 2 \log L (km) + 6.167 \hfill \\ \log \delta _c (cm) = 2 \log L (km) - 1.652 \hfill \\ \end{gathered} $$ whereG _c is the specific fracture energy,ω _c the size of the slip weakening zone,δ _c the slip weakening displacement,τ _B ?τ _f the drop in strength in the slip weakening zone,τ _∞?τ _f the stress drop,L the fault length, andW the fault width. The investigation of 49 shocks shows that the range of strength dropτ _B ?τ _f is from several doze to several hundred bars at depthh<400 km, but it can be more than 10³ bars ath>500 km; besides, the range of the sizeω _c of the strength degradation zone is from a few tenths of a kilometer to several dozen kilometers, and the range of the slip weakening displacementδ _c is from several to several hundred centimeters. The specific fracture energyG _c is of the order of 10⁸ to 10¹¹ erg cm^?2 when the momentM ₀ is of the order of 10²³ to 10²⁹ dyne cm.     

Estimation of sedimentary layer shear wave velocity using micro-tremor H/V ratio measurements for Bangalore city

For assessing earthquake hazard of metro cities, knowledge of soil amplification, thickness and properties of sedimentary layer are essential. In order to map the soil thickness using microtremor survey method, in Bangalore city, it is required to calibrate the relation between fundamental resonance frequency of the soil layer and its thickness for the region. For this purpose microtremor survey was carried out at 34 locations in the city where borehole log was available. The resonance frequency of the soil is evaluated from the microtremor recordings using the H/V ratio technique. A nonlinear regression relation between the thickness of sedimentary layer h (m), from the borehole logs, and the resonance frequency f_r (Hz), was derived as h=(58.3±8.8)f_r^{−(0.95)±0.1}. Using the model of shear wave velocity increasing with depth at these locations, the derived average shear wave velocity and the corresponding soil thickness were used, to get an empirical relation between V_S (m/s) and depth z(m), as V_s=(174±28)(1+z)^0.16±0.07. This relation also compares reasonably with the fit obtained between simulated V_S and depth from borehole logs for Bangalore city. The calibrated relations can be used at locations in Bangalore city where borehole logs are not available, for finding the thicknesses and shear wave velocities of the local soil layers at the survey locations.     

Spatial scaling properties of annual streamflow in the United States

Abstract

The spatial scaling properties of annual average streamflow is examined using records from 1 433 river basins across the continental United States. The log-linear relationship ln(E[Q^r _i]) = a + b_r ln(A_i) is representative throughout the United States, where E[Q^r _i] represents the expectation of the rth moment of annual streamflow at site i, and A_i represents drainage area. The scaling model parameters a_r and b_r follow nearly perfect linear relationships a_r = rα and b_r = rβ throughout the continental United States. We conclude that the probability distribution of annual streamflow follows simple scaling relationships in all regions of the United States. In temperate regions where climate is relatively homogeneous, scale alone describes most of the variability in the moments of annual streamflow. In the more climatically heterogeneous regions, such as in the Upper Colorado and Missouri river basins, scale alone is a poor predictor of the moments of annual flow.     

Hysteresis properties of titanomagnetites: Grain-size and compositional dependence

Sized fractions of x = 0.6, 0.4, 0.2 and 0.0 titanomagnetites were studied with a vibration magnetometer. In the course particles (d > 150 μm), no compositional dependence of hysteresis parameters was found. H_C was less than 50 Oe, H_R/H_C > 4 and J_R/J_S < 10^?2, reflecting multi-domain behaviour. In contrast, fine particles ($\text{d ? 0.1 μ}\text{m}$) revealed systematic grain-size dependence of parameters with coercive force as high as 2,000 Oe in x = 0.6 titanomagnetite. Grain-size dependence studies revealed broad transition sizes for the onset of true multi-domain behaviour depending upon which factor is chosen. In magnetite it varies from 10 to 20 μm. The experimental critical size for single-domain behaviour for magnetite is about 0.1 μm and for x = 0.6 titanomagnetite 1–2 μm.