Seasonal and event-scale variations in solute chemistry for four Sierra Nevada catchments

Hydrobiogeochemical processes controlling stream water chemistry were examined in four small (<5 km²) catchments having contrasting bedrock lithologies in the western Sierra Nevada foothills of California. The Mediterranean climate with its cool/wet and hot/dry cycle produces strong seasonal patterns in hydrological, biological and geochemical processes. Stream water solutes fall into three general groups according to seasonal fluctuation in concentration: strong, rainy season minimum–dry season maximum (Cl⁻, SO₄²⁻, base cations); weak, rainy season minimum–dry season maximum (Si); and rainy season maximum–dry season minimum (NO₃⁻ and K⁺). Solute dynamics in soil solutions and stream water suggest that mixing of drainage waters from bedrock and soil sources regulate stream water solute concentrations. Patterns are further altered by the leaching of solutes accumulated in the soil over the summer period of desiccation and the temporal discoupling of nutrient cycles that occurs due to differences in the timing between vegetation growth (late spring) and leaching (early winter). Solute concentrations are remarkably similar between watersheds with varying bedrock types, with the exception of nitrate, sulfate and bicarbonate. Three watersheds have nitrogen-bearing metasedimentary bedrock that contributes to elevated nitrate concentrations in stream waters. Watersheds whose bedrock includes mineralized veins of sulfide and carbonate minerals similarly have greater sulfate and bicarbonate concentrations in stream water. Hydrobiogeochemical processes are highly dynamic at the seasonal and storm-event temporal scales and spatially complex at the watershed scale making management of stream water chemical composition, such as nitrate concentrations, very challenging.     

Hydrogeochemical and isotopic evidence of groundwater salinization in a coastal aquifer: a case study in Jeju volcanic island, Korea

In order to identify the origin of saline groundwater in the eastern part of Jeju volcanic island, Korea, a hydrogeochemical and isotopic study has been carried out for 18 observation wells located in east and southeast coastal regions. The total dissolved solid contents of groundwaters are highly variable (77–21,782 mg/l). Oxygen, hydrogen, sulfur, and strontium isotopic data clearly show that the saline water results from mixing of groundwater with seawater. Strontium isotopic compositions and Br/Cl and I/Cl ratios strongly suggest that the source of salinity is modern seawater intrusion. Hydrogeochemical characteristics based on bivariate diagrams of major and minor ions show that changes in the chemical composition of groundwater are mainly controlled by the salinization process followed by cation-exchange reactions. The highly permeable aquifers at the east coastal region are characterized by low hydraulic gradient and discharge rate and high hydraulic conductivity as compared with other regions. These properties enhance the salinization of groundwater observed in the study area. Based on the Cl, Br, and δ¹⁸O data, seawater was determined to have intruded inland some 2.5 km from the coastline. Considering the poor correlation of sampling depth and Cl concentrations observed, the position of seawater-freshwater interface is not uniformly distributed in the study area, due to heterogeneities of the basaltic aquifers.     

Isotopic variations in precipitation at Bangkok and their climatological significance

The stable isotopic composition of precipitation from low to mid latitudes contains information about changes of some climatic factors, such as temperature, precipitation and atmospheric circulation patterns. However, the isotopic variations in the area are very complicated because of the combined influences of these factors. Proper interpretation of the patterns of isotopic variations for palaeoclimate reconstructions in this area requires a detailed understanding of these complex stable isotope controls. The isotopic data (δ¹⁸O and δ²D) in precipitation at the International Atomic Energy Agency–World Meteorological Organization Bangkok station were collected and analysed because of the relatively long and unbroken isotopic records and the particular geographical location. The isotopic variations at Bangkok have strong seasonal patterns owing to distinct source regions in different seasons. In summer, the remote sources of water there can influence the δ¹⁸O values significantly, which is verified by the simple Rayleigh model. In winter, the mixing of isotopically distinct air masses with different origins, i.e. the continental and oceanic air masses, accounts for the isotopic variations. In the transition periods of the Asia–Australia monsoon, namely in May and October, the local vapour contribution may play a role in the isotopic ratios. On the interannual time‐scale, the influences of El Niño–southern oscillation (ENSO) and the Indian Ocean dipole (IOD) on isotopic composition are examined. The indications are that both the ENSO and IOD indices have a significant correlation with the δ¹⁸O ratios, and that the δ¹⁸O ratio in summer rains is significantly more enriched (depleted) during the warm (cold) phase of ENSO/IOD events. All the results suggest that it is useful for us in understanding the water cycling process and may be helpful in palaeoclimate reconstruction in this monsoon region. Copyright © 2006 John Wiley & Sons, Ltd.     

Spatial and temporal variations of overland flow during rainfall events and in relation to catchment conditions

Improved knowledge on overland flow (OF) generation and its dynamics (i.e. spatial and temporal variations) is essential to understand catchment hydrology, a prerequisite for better water resources and soil management. In this study, our main objective was to quantify the dynamics of OF during rainfall events and to assess its main factors of control. The research study was undertaken in an agricultural 23‐ha catchment of a communal pasture in KwaZulu‐Natal (South Africa) experiencing Mediterranean climate and with variations of soil, topography and vegetation conditions. The dynamics of OF was evaluated during three rainfall seasons (2007 to 2010) by using 1 × 1‐m² microplots (n = 15) located at five landscape positions. At each location, a microplot was equipped with an automatic tipping bucket linked to a logger to estimate the delay between the start of the rain and the start of OF [i.e. the time to runoff initiation (TRI)]. Multivariate analysis was applied to the total OF and TRI data and the information on selected environmental factors (rainfall characteristics; soil type; soil clay content, Clay; proportion of the soil surface covered by vegetation, Cov; proportion of the soil surface covered by crusting, Crust; mean slope gradient, S; soil bulk density, ρ_b; soil water tension at different depths, SWT). The average OF rate over the 3‐year study period varied 2.3‐fold across the catchment (from 15% footslope to 35% backslope), whereas the average TRI varied by a 10.6‐fold factor (between 0.6 min at bottomland and 6.4 min at footslope). TRI temporal variations correlated the most with event duration (r = 0.8) and cumulative amount of rainfall since the onset of the rainy season (r = ?0.47), whereas TRI spatial variations were controlled the most by Crust (?0.97 < r < ?0.77). Ultimately, TRI spatial variations were modelled and mapped in an attempt to model OF dynamics over the entire microcatchment. Copyright © 2012 John Wiley & Sons, Ltd.     

Isotope hydrology and water sources in a heavily urbanized stream

Complex networks of both natural and engineered flow paths control the hydrology of streams in major cities through spatio-temporal variations in connection and disconnection of diverse water sources. We used spatially extensive and temporally intensive sampling of water stable isotopes to disentangle the hydrological sources of the heavily urbanized Panke catchment (~220 km²) in the north of Berlin, Germany. The isotopic data enabled us to partition stream water sources across the catchment using a Bayesian mixing analysis. The upper part of the catchment streamflow is dominated by groundwater (~75%) from gravel aquifers. In dry summer periods, streamflow becomes intermittent in the upper catchment, possibly as a result of local groundwater abstractions. Storm drainage dominates the responses to precipitation events. Although such events can dramatically change the isotopic composition of the upper stream network, storm drainage only accounts for 10%–15% of annual streamflow. Moving downstream, subtle changes in sources and isotope signatures occur as catchment characteristics vary and the stream is affected by different tributaries. However, effluents from a wastewater treatment plant (WWTP), serving 700,000 people, dominate stream flow in the lower catchment (~90% of annual runoff) where urbanization effects are more dramatic. The associated increase in sealed surfaces downstream also reduces the relative contribution of groundwater to streamflow. The volume and isotopic composition of storm runoff is again dominated by urban drainage, though in the lower catchment, still only about 10% of annual runoff comes from storm drains. The study shows the potential of stable water isotopes as inexpensive tracers in urban catchments that can provide a more integrated understanding of the complex hydrology of major cities. This offers an important evidence base for guiding the plans to develop and re-develop urban catchments to protect, restore, and enhance their ecological and amenity value.     

Stable isotopes in the source waters of the Yamuna and its tributaries: seasonal and altitudinal variations and relation to major cations

Water samples from the Yamuna and its tributaries, one of the major river systems draining the Himalaya, have been analysed for their stable oxygen and hydrogen isotopes during three seasons (summer, monsoon and post‐monsoon). The data show clear seasonal and altitudinal variations; waters from higher altitudes and those collected during monsoon season are characterized by relatively depleted isotopic composition. Regression analysis of δD–δ¹⁸O data of samples collected during summer and monsoon seasons shows that the slope of the best‐fit lines are nearly identical to those of precipitation at New Delhi for the same period. The similarity in their slopes suggests that the isotopic composition of precipitation contributing water to these rivers are reasonably well preserved in both monsoon and non‐monsoon seasons, however, during the non‐monsoon period both rainfall and river waters carry signatures of evaporation. The ‘deuterium excess’ in river waters during the three seasons though overlap with each other, the values during October are higher. This can be understood in terms of recycled moisture contributions to precipitation. The ‘altitude effect’ for δ¹⁸O in these waters is determined to be 0·11‰ per 100 m, a factor of about two less than that reported for the Ganga source waters from similar altitudinal range. The variability in altitude effects in rivers draining the Himalaya seems to be controlled by the ‘amount effect’ associated with the monsoon. The significant spatial variability in altitude effect in these river basins, which are a few hundred kilometers apart, suggests that reconstruction of palaeoelevation in the Himalaya, based on δ¹⁸O‐altitude gradients, would depend critically on its proper assessment in the region. This study has established a relationship between total cation abundance and δ¹⁸O in waters of the Yamuna mainstream; total cations (corrected for cyclic components) double for a 1·4 km decrease in altitude as the Yamuna flows downstream. Copyright © 2002 John Wiley & Sons, Ltd.     

Contribution of glacier meltwater to streamflow in the Wind River Range,Wyoming, inferred via a Bayesian mixing model applied to isotopic measurements

The Wind River Range (WRR) of Wyoming has the largest concentration of alpine glaciers in the American Rockies and contributes to several major river systems in the western United States. Declines in the areal extent and volume of these glaciers are well documented, and eventual loss of alpine glaciers will reduce the amount of water available for agricultural and domestic use. The contribution of glacial melt to streamflow remains largely unquantified in Wyoming. We used isotope measurements and Bayesian modeling to estimate the fractional contribution of glacier meltwater to Dinwoody Creek (DC) in the WRR on bi‐weekly and seasonal (spring, summer, and fall) time scales over 2 years. In 2007 and 2008, we made temporally intensive measurements of the stable isotope composition of water from the DC watershed. Samples of the primary sources of streamflow (snowmelt, glacier melt, rain, and baseflow) were collected during field campaigns, and automated collection of stream samples occurred over the melt season. Isotope data (D and ¹⁸O) were analyzed within a hierarchical Bayesian framework that incorporated temporal and spatial correlations. Glacial melt contributed a significant proportion (～53–59%) to streamflow in a low‐flow year (2007) or when streamflow was low during a high‐flow year (2008). In 2008, a large and persistent snowpack contributed significantly (～0·42–51%) to streamflow in mid‐summer. The large contribution of glacial melt to streamflow suggests that the loss of glaciers may impact riparian ecosystems and human water supplies in the late summer and in years with low snowpack. Copyright © 2011 John Wiley & Sons, Ltd.     

Local topography and erosion rate control regolith thickness along a ridgeline in the Sierra Nevada,California

The ridgelines of mountain ranges are a source of geomorphic information unadulterated by the arrival of sediment from upslope. Studies along ridgecrests, therefore, can help identify and isolate the controls on important regolith properties such as thickness and texture. A 1.5 km section of ridgeline in the Sierra Nevada (CA) with a tenfold decrease in erosion rate (inferred from ridgetop convexity) provided an opportunity to conduct a high‐resolution survey of regolith properties and investigate their controls. We found that regolith along the most quickly eroding section of the ridge was the rockiest and had the lowest clay concentrations. Furthermore, a general increase in regolith thickness with a slowing of erosion rate was accompanied by an increase in biomass, changes in vegetation community, broader ridgeline profiles, and an apparent increase in total available moisture. The greatest source of variation in regolith thickness at the 10–100 m scale, however, was the local topography along the ridgeline, with the deepest regolith in the saddles and the thinnest on the knobs. Because regolith in the saddles had higher surface soil moisture than the knobs, we conclude that the hydrological conditions primarily driven by local topography (i.e. rapid vs. slow drainage and water‐storage potential) provide the fundamental controls on regolith thickness through feedbacks incorporating physical weathering by the biota and chemical weathering. Moreover, because the ridgeline saddles are the uppermost extensions of first‐order valleys, we propose that the fluvial network affects regolith properties in the furthest reaches of the watershed. Copyright © 2015 John Wiley & Sons, Ltd.     

保安湖浮游物和颗粒有机碎屑现存量及季节变动与渔业生产

1987年3月至1988年3月对保安湖所属大湖和桥墩湖浮游物和颗粒有机碎屑现存量作了测定。浮游物平均干重、无灰干重和颗粒有机碎屑量大湖分别为337.6、127.7和96.9t;桥墩湖分别为100.5、50.3和40.2t。浮游物和颗粒有机碎屑现存量具有明显的季节变动。黄丝草、聚草和苦草的碎屑分解对浮游物的形成起着重要的作用。用浮游物碳量估算鲢鳙鱼产潜力大湖为12.3kg/亩;桥墩湖为11kg/亩。     

Using high resolution tracer data to constrain water storage,flux and age estimates in a spatially distributed rainfall‐runoff model

Models simulating stream flow and conservative tracers can provide a representation of flow paths, storage distributions and mixing processes that is advantageous for many predictive purposes. Compared with models that only simulate stream flow, tracer data can be used to investigate the internal consistency of model behaviour and to gain insight into model performance. Here, we examine the strengths and weaknesses of a data‐driven, spatially distributed tracer‐aided rainfall‐runoff model. The model structure allowed us to assess the influence of landscape characteristics on the routing and mixing of water and tracers. The model was applied to a site in the Scottish Highlands with a unique tracer data set; ~4 years of daily isotope ratios in stream water and precipitation were available, as well as 2 years of weekly soil and ground water isotopes. The model structure was based on an empirically based, lumped tracer‐aided model previously developed for the catchment. The best model runs were selected from Monte Carlo simulations based on dual calibration criteria using objective functions for both stream isotopes and discharge at the outlet. Model performance for these criteria was reasonable (Nash–Sutcliffe efficiencies for discharge and isotope ratios were ~0.4–0.6). The model could generally reproduce the variable isotope signals in the soils of the steeper hill slopes where storage was low, and damped isotope responses in valley bottom cells with high storage. The model also allowed us to estimate the age distributions of internal stores, water fluxes and stream flow. Average stream water age was ~1.6 years, integrating older groundwater in the valley bottom and dynamic younger soil waters. By tracking water ages and simulating isotopes, the model captured the changes in connectivity driven by distributed storage dynamics. This has substantially improved the representation of spatio‐temporal process dynamics and gives a more robust framework for projecting environmental change impacts. Copyright © 2016 The Authors Hydrological Processes Published by John Wiley & Sons Ltd.     

Using lumped conceptual rainfall-runoff models to simulate daily isotope variability with fractionation in a nested mesoscale catchment

This paper presents 19 months of stable isotope (δ²H and δ¹⁸O) data to enhance understanding of water and solute transport at two spatial scales (2.3 km² and 122 km²) in the agricultural Lunan catchment, Scotland. Daily precipitation and stream isotope data, weekly lake and spring isotope data and monthly groundwater isotope data revealed important insights into flow pathways and mixing of water at both scales. In particular, a deeper groundwater flow path significantly contributes to total streamflow (25-50%). Upstream lake isotope dynamics, susceptible to evaporative fractionation, also appeared to have an important influence on the downstream isotope composition. This unique tracer data set facilitated the conceptualization of a lumped catchment-scale flow-tracer model. The incorporation of hydrological, mixing and fractionation processes based on these data improved simulations of the stream δ²H isotope response at the catchment outlet from 0.37 to 0.56 for the Nash-Sutcliffe statistic. The stable isotope data successfully aided model conceptualization and calibration in the quest for a simple water and solute transport model with improved representation of process dynamics.     

Peculiarities of long-term variations in the temperature and heat content of the troposphere and their relation to solar activity

The results of an analysis of the spatial-temporal variations in the tropospheric temperature regime in the Northern and Southern hemispheres in 1948–2006 are presented. The possible effect of solar activity on the tropospheric temperature regime is discussed in the scope of the proposed mechanism by which the heliogeophysical factors affect the climatic characteristics and atmospheric circulation in the high-latitude troposphere through atmospheric electricity.     

Linking snowmelt‐derived fluxes and groundwater flow in a high elevation meadow system,Sierra Nevada Mountains,California

Quantifying snowmelt‐derived fluxes at the watershed scale within hillslope environments is critical for investigating local meadow scale groundwater dynamics in high elevation riparian ecosystems. In this article, we investigate the impact of snowmelt‐derived groundwater flux from the surrounding hillslopes on water table dynamics in Tuolumne Meadows, which is located in the Sierra Nevada Mountains of California, USA. Results show water levels within the meadow are controlled by a combination of fluxes at the hillslope boundaries, snowmelt within the meadow and changes in the stream stage. Observed water level fluctuations at the boundaries of the meadow show the hydrologic connection and subsequent disconnection between the hillslope and meadow aquifers. Timing of groundwater flux entering the meadow as a result of spring snowmelt can vary over 20 days based on the location, aspect, and local geology of the contributing area within the larger watershed. Identifying this temporal and spatial variability in flux entering the meadow is critical for simulating changes in water levels within the meadow. Model results can vary significantly based on the temporal and spatial scales at which watershed processes are linked to local processes within the meadow causing errors when boundary fluxes are lumped in time or space. Without a clear understanding of the surrounding hillslope hydrology, it is difficult to simulate groundwater dynamics within high elevation riparian ecosystems with the accuracy necessary for understanding ecosystem response. Copyright © 2010 John Wiley & Sons, Ltd.     

Using stable isotopes to identify major flow pathways in a permafrost influenced alpine meadow hillslope during summer rainfall period

Global warming has leaded to permafrost degradation, with potential impacts on the runoff generation processes of permafrost influenced alpine meadow hillslope. Stable isotopes have the potential to trace the complex runoff generation processes. In this study, precipitation, hillslope surface and subsurface runoff, stream water, and mobile soil water (MSW) at different hillslope positions and depths were collected during the summer rainfall period to analyse the major flow pathway based on stable isotopic signatures. The results indicated that (a) compared with precipitation, the δ²H values of MSW showed little temporal variation but strong heterogeneity with enriched isotopic ratios at lower hillslope positions and in deeper soil layers. (b) The δ²H values of middle-slope surface runoff and shallow subsurface flow were similar to those of precipitation and MSW of the same soil layer, respectively. (c) Middle-slope shallow subsurface flow was the major flow pathway of the permafrost influenced alpine meadow hillslope, which turned into surface runoff at the riparian zone before contributing to the streamflow. (d) The slight variation of δ²H values in stream water was shown to be related to mixing processes of new water (precipitation, 2%) and old water (middle-slope shallow subsurface flow, 98%) in the highly transmissive shallow thawed soil layers. It was inferred that supra-permafrost water levels would be lowered to a less conductive, deeper soil layer under further warming and thawing permafrost, which would result in a declined streamflow and delayed runoff peak. This study explained the “rapid mobilization of old water” paradox in permafrost influenced alpine meadow hillslope and improved our understanding of permafrost hillslope hydrology in alpine regions.     

The effects of soil organic matter on soil water retention and plant water use in a meadow of the Sierra Nevada,CA

Tuolumne Meadows is a groundwater dependent ecosystem in the Sierra Nevada of California, USA, that is threatened by hydrologic impacts that may lead to a substantial loss of organic matter in the soil. In order to provide a scientific basis for management of this type of ecosystem, this paper quantifies the effect of soil organic content on soil water retention and water use by plants. First, we show a substantial dependence of soil water retention on soil organic content by correlating Van Genuchten soil water retention parameters with soil organic content, independent of soil texture. Then, we demonstrate the impact of organic content on plants by simulating the degree to which root water uptake is affected by soil water retention with the use of a physically based numerical model of variably saturated groundwater flow. Our results indicate that the increased water retention by soil organic matter contributes as much as 8.8 cm to transpiration, or 35 additional water‐stress free days, during the dry summer when plants experience increased water stress.     

Trace metals distribution pattern in floodplain sediments of a lowland river in relation to contemporary valley bottom morphodynamics

The distribution of trace metals in alluvial sediments depends on their natural background concentrations, and on the dynamics of contemporary depositional and erosional (mainly flood‐induced) processes. Geological and geochemical investigations were carried out in the valley of Vistula River near Magnuszew (central Poland). Sediment samples were collected from a depth of 35 cm and comprise sediments of all defined geomorphological features. Identification and geological interpretation of the morphodynamic sediment features was supported by aerial photographs and high‐resolution satellite images. These studies revealed that the distribution of trace metals is closely linked to the morphogenesis of the alluvial floodplain. The highest concentrations of Cu, Co, Zn, V, Cr and Ni were observed in crevasse‐splays deposits. By contrast, Sr, Pb and As were concentrated in deposits which fill oxbow lakes (partly infilled with organic deposits). The lowest concentrations of trace metals were detected in flood sediments deposited within erosional troughs. The geomorphological and sedimentological history of the fluvial features explains the pattern of heavy metal distribution on the current floodplain surface. Copyright © 2014 John Wiley & Sons, Ltd.     

Drying and dying of a subducted slab: Coupled Li and B isotope variations in Western Anatolia Cenozoic Volcanism

In lavas spanning ~ 10Ma of subduction-related volcanism in Western Anatolia, we observe remarkably similar patterns of δ⁷Li and δ¹¹B variation. In this setting, magmatism records a transition from calc–alkaline to ultrapotassic character, consistent with overall lower mean extents of melting, and a changing mantle source that reflects a fractionating, higher temperature slab input consistent with the gradual cessation of subduction. Subsequent rift-related intraplate magmatism record δ⁷Li signatures within the range observed for MORBs and OIBs, indicating an abrupt transition to a mantle source unmodified by subduction.     

Fractal variations of the Transcarpathians,West Ukraine,seismicity and their potential relation to changing phases of local seismic cycles

The evolution of spatial arrangement of the Transcarpathian earthquakes is studied by means of fractal analysis. Spatial correlation dimension is evaluated for a subset of fixed number of catalogue events consecutively shifted in time by one event forward. Distinctive patterns that may be related to different phases of local seismic cycle/cycles have been identified, corresponding to extreme values of the dimension. Their potential relation to evolution of the seismogenic stress field is discussed. The time it takes for the capacity dimension to reach the so-called percolation threshold is estimated for a subset consecutively extended by adding next catalogue events. Temporal and spatial frames of percolation analysis are discussed in relation to parameters of the future earthquake.     

云南宾川盆地及周边地震波走时的季节性变化和可能的物理机制

基于背景噪声自相关方法, 本文计算了云南宾川盆地及周边区域地震波在四个频段的相对走时变化(dt/t), 发现存在比较显著的年变和半年变特征, 最大变化幅度从0.1~0.2 Hz(7 km左右深度最敏感)的±0.8%逐渐降低至1.0~2.0 Hz(对应1 km以浅介质)的±0.05%.噪声源季节性变化可能是dt/t周期性起伏的一个潜在因素, 但雨水和温度与其有更直接的关联性.其中雨水渗透产生的介质孔隙压变化是年变和半年变的重要贡献因素; 而温度起伏产生的介质热力学形变, 主要对一定深度范围(3 km左右最敏感)的介质有明显的年变影响.不同频段走时变化存在幅度差异和不同步, 可能和各频段敏感核的深度范围以及水与热力学形变从地表向深部渗透/传播过程有关.研究表明, 环境因素对地下介质的地震波走时有较为显著的影响, 区分出它们的贡献, 才能更可靠地评估地震构造应力或人文生态环境的变化.

     

On the relation between excess hydraulic head and flow in a conduit imbedded within a porous matrix

This article provides a partial answer to the question “What is the relation between excess hydraulic head and volume flux of water in a conduit within a porous matrix?”, focusing on the case that the forcing is steady. The conduit is modelled as a horizontal circular cylinder, imbedded within a porous matrix of rectangular cross section, having constant head prescribed on the sidewalls and being confined top and bottom. Laminar flow in the matrix is assumed to obey Darcy's law, while turbulent flow in the conduit is quantified using the Darcy–Weisbach equation. Analysis of the latter equation shows that the length scale of variations in the direction of the conduit is large compared with the scale of lateral and vertical variations. This permits separation of the full three-dimensional non-linear problem into a two-dimensional linear problem for head within the matrix and a one-dimensional non-linear problem for head within the conduit. Analytic solutions are obtained for the distribution of head in the matrix and in a conduit of either infinite or finite length. In both cases, the volume flux of water is proportional to the excess head to the 2/3 power, the conduit radius to the 5/3 power, the matrix permeability to the 1/3 power and gravity to the 1/3 power. The scale of variation of head along the conduit is proportional to the excess head to the ?1/3 power, the conduit radius to the 5/3 power, the matrix permeability to the ?2/3 power and gravity to the 1/3 power.