Journal articles on the topic 'Rain and rainfall Tasmania Mathematical models'

Recent studies analyze the influence of rainfall on traffic crashes, indicating that precipitation intensity is an important factor, for modeling crashes occurrence. This research presents a relationship between daily-basis traffic crashes and precipitation, from 2014 to 2018, in a rural mountainous Brazilian Highway (BR-376/PR), where field rain gauges were used to obtain precipitation data. Data modeling considered a Negative Binomial regression for precipitation influence in crash frequency. Separate regression models were estimated to account for the rainfall effect in different seasons, and for different vehicle types. All models analyzed presented a positive relationship between daily rainfall intensity and daily crashes number. This can indicate that generally rainfall presence is a hazardous factor. Different critical seasons for rainfall influence were also highlighted, alerting for the possible necessity of distinct road safety policies concerning seasonality. Finally, for the vehicle type analysis, typically, rainfall seemed to have a greater effect in lighter vehicles. Moreover, results are useful for traffic control, in order to increase safety conditions.

It is standard practice in integrated urban water management to apply mathematical models of the total drainage system for calculating the frequency of occurrence of critical states in the receiving water body. The model input for such computations are long term time-series of rainfall data. However, it is inevitable that those rainfall data measurements deviate from reality. This is a result of inaccuracy of the measurement devices, errors in data transmission, local meteorological effects, etc. In this work we investigate the effect of such uncertainty in the rainfall data on the return period of the occurrence of oxygen depletion in the river due to the drainage of storm water. The errors in the rain data measurements are simulated by means of both stochastic and deterministic models. A comparison of the results obtained from the erroneous data series against the reference data reveals the small effect of random deviation in rain measurements. Only a constant and significant offset of the measured data (greater 20%) has an equally significant effect on the modeling result.

AbstractA point process model based on a class of Cox processes is developed to analyse precipitation data at a point location. The model is constructed using state-dependent exponential pulses that are governed by an unobserved underlying Markov chain. The mathematical formulation of the model where both the arrival rate of the rain cells and the initial pulse depth are determined by the Markov chain is presented. Second-order properties of the rainfall depth process are derived and utilised in model assessment. A method of moment estimation is employed in model fitting. The proposed model is used to analyse 69 years of sub-hourly rainfall data from Germany and 15 years of English rainfall data. The results of the analysis using variants of the proposed model with fixed pulse lifetime and variable pulse duration are presented. The performance of the proposed model, in reproducing second-moment characteristics of the rainfall, is compared with that of two stochastic models where one has exponential pulses and the other has rectangular pulses. The proposed model is found to capture most of the empirical rainfall properties well and outperform the two alternative models considered in our analysis.

With regard to reconstructing the gamma background dose rate, existing models are either empirical with limited applicability or have many unknown input parameters, which complicates their application in practice. Due to this, there is a need to search for a new approach and build a convenient, easily applicable and universal model. The paper proposes a mathematical model for reconstructing the temporal evolution of the ambient equivalent γ-radiation dose rate during rain episodes, depending on the density of radon flux from the soil surface, as well as the duration and intensity of rain. The efficiency of the model is confirmed by the high coefficient of determination (R2 = 0.81–0.99) between the measured and reconstructed ambient equivalent dose rate during periods of rain, the simulation of which was performed using Wolfram Mathematica. An algorithm was developed for restoring the dynamics of the ambient equivalent γ-radiation dose rate during rainfall. Based on the results obtained, assumptions were made where the washout of radionuclides originates. The influence of the radionuclides ratio on the increase in the total γ-radiation dose rate was investigated.

Abstract:Soil processes are essential in enabling forest regeneration in disturbed landscapes. Little is known about whether litterfall from dominating pioneer species in secondary rain forest is functionally equivalent to that of mixed rain-forest litter in terms of contribution to soil processes. This study used the litterbag technique to quantify the decomposition and nutrient dynamics of leaf litter characteristic of three wet tropical forest communities in the Paluma Range National Park, Queensland, Australia over 511 d. These were: undisturbed primary rain forest (mixed rain-forest species), selectively logged secondary rain forest (pioneer Alphitonia petriei) and tall open eucalypt forest (Eucalyptus grandis). Mass loss, total N, total P, K, Ca and Mg dynamics of the decaying leaves were determined, and different mathematical models were used to explain the mass loss data. Rainfall and temperature data were also collected from each site. The leaves of A. petriei and E. grandis both decomposed significantly slower in situ than the mixed rain-forest species (39%, 38% and 29% ash-free dry mass remaining respectively). Nitrogen and phosphorus were immobilized, with 182% N and 134% P remaining in E. grandis, 127% N and 132% P remaining in A. petriei and 168% N and 121% P remaining in the mixed rain-forest species. The initial lignin:P ratio and initial lignin:N ratio exerted significant controls on decomposition rates. The exceptionally slow decomposition of the pioneer species is likely to limit soil processes at disturbed tropical rain-forest sites in Australia.

Abstract The research issue focuses on potential interception, which is the maximum amount of water that can be stored on plant surface. Tests under controlled conditions remain the best way to enhance knowledge on interception determinants in forest communities. Such tests can provide data for identification of mathematical models based on ecological criteria. The study presented in this paper concerned tree interception under simulated rain in a range from 2 to 11 mm/h. To perform the experiment a set of sprinklers was designed and built. The study included two deciduous species: beech (Fagus sylvatica L.) and oak (Quercus robur L.). Descriptive characteristic and nonlinear estimation were suggested for the obtained data. Interdependence of potential interception, the intensity of rain and the size of raindrops were described using exponential equation. The intensity and drop size of simulated rainfall significantly influence the obtained values of potential interception. Data analysis shows a decrease of interception value with an increase of intensity of simulated rainfall for both analysed species. Every run of the experiment that differed in the intensity and size of raindrops reached an individual level of potential interception and time needed to realize it. The formation of ability of plants to intercept water depends both on the dynamics and the time of spraying.

One of the major problems in trying to design wastewater treatment schemes to protect bathing water for coastal communities with combined sewerage systems, is to ensure that discharges of storm water do not prejudice compliance with the requirements of the EC Bathing Water Directive. In order to develop an appropriate storm water management strategy for the Fylde coast it was necessary to integrate a number of mathematical models simulating the hydraulic behaviour of the sewerage system and the dispersion of discharges in the receiving waters. From the sewerage system modelling it was apparent that frequent discharges of storm water to the bathing waters could only be avoided by the provision of considerable additional storage in the system. By means of a suitably calibrated simplified sewer model it was possible to investigate the volumes of storm water generated by a 15 year record of local rainfall when different amounts of extra storage and different pumping regimes were employed. The results from these investigations were used to determine the probable concentrations of faecal bacteria in the coastal waters for each of the 15 bathing seasons and determine the percentage of time for which faecal coliform concentrations exceeded the Bathing Water directive standards for the model grid cells representing the identified bathing waters. As a result of the extensive integrated modelling programme for the Fylde coast it has been possible to design a base flow and storm water management system which should maximize the flow passed forward for treatment whilst also ensuring that there is just sufficient storage to ensure protection of the towns from flooding and the compliance of the beaches with the Bathing Water Directive standards.

Infiltration estimation is made by tests such as concentric cylinders, which are prone to errors, such as the lateral movement under the ring. Several possibilities have been developed over the last decades to compensate these errors, which are based on physical, electronic, and mathematical principles. In this research, two approaches are proposed to measure the water infiltration rate in a silty loam soil by means of the mass values of a lysimeter weighing under rainfall conditions and different moisture contents. Based on the fact that with the lysimeter it is possible to determine acting soil flows very precisely, then with the help of mass conservation and assuming a downward vertical movement, 12 rain events were analyzed. In addition, it was possible to monitor the behavior of soil moisture and to establish the content at field capacity from the values of the weighing lysimeter, from which both approach are based. The infiltration rate of these events showed a variable rate at the beginning of the rainfall until reaching a maximum, to descend to a stable or basic rate. This basic infiltration rate was 1.49 ± 0.36 mm/h, and this is because soils with fine textures have reported low infiltration capacity. Four empirical or semi-empirical models of infiltration were calibrated with the values obtained with our approaches, showing a better fit with the Horton’s model.

AbstractThis paper examines the influence of water on the ecology of the eggs of Nematodirus battus, with a view to estimating the importance of including rainfall in mathematical models of parasite abundance. The literature suggests that, under pasture conditions, the availability of moisture is unlikely to be limiting for egg development, while eggs and infective larvae are highly resistant to desiccation. In the presented experiment, eggs that had been kept in salt sludges at 95% and 70% RH and were subsequently put at 15°C produced only a mildly accelerated, but not a mass, hatch, in the first few days after return to water. Eggs kept at higher osmotic pressures died. Mass hatching of infective larvae, described at pasture when spells of rain follow periods of drought, is unlikely to occur as the result of a sudden water influx into eggs. Since water is not necessary for migration of infective larvae from the soil on to grass, such peaks in larval abundance are more likely to arise from the effects of temperature on hatching of eggs.

The phenomena of infiltration and the percolation of water in the soil are of fundamental importance for the evaluation of runoff, groundwater recharge, evapotranspiration, soil erosion and transport of chemical substances in surface and groundwater. Within this context, the quantitative determination of the infiltration values is extremely important for the different areas of knowledge, in order to evaluate, mainly the surface runoff. Several types of changes in vegetation cover and topography result in significant changes in the infiltration process, making it necessary to use mathematical models to assess the consequences of these changes. Thus, this article aims to implement the Green-Ampt model using two numerical methods - Newton-Raphson method and W-Lambert function - to determine soil permeability parameters - K and matric potential multiplied by the difference between initial and of saturation - comparing them to the real data obtained in simulations using an automatic rainfall simulator from the Federal University of Goiás - UFG. The Green-Ampt model adjusted well to the data measured from the rain simulator, with a determination coefficient of 0.978 for the Newton-Raphson method and 0.984 for the W-Lambert function.

<p>Objetivou-se estimar a erosividade da chuva, mediante seis modelos matemáticos, de regressão linear avaliando entre estes, qual é mais indicado para as condições climáticas da região de São Mateus-ES. Os dados pluviométricos foram obtidos junto à Agência Nacional das Águas-ANA, sendo de 1947 a 2014 para Itauninhas, de 1971 a 2014 para Barra Nova, de 1981 a 2014 para São João da Cachoeira Grande e de 1993 a 2014 para Boca da Vala. Para estimar a erosividade da chuva, a partir da precipitação anual e do coeficiente de chuva, foram utilizadas diferentes equações utilizadas em outros estados com aplicação ao estado do Espírito Santo ou ajustadas para o próprio estado. Para os modelos matemáticos (II) e (I), os valores médios foram de 6.541,2 MJ ha^-1 mm h^-1 ano^-1 a 936,357 MJ ha^-1 mm h^-1 ano^-1 (Itauninhas), de 6.995,855 MJ ha^-1 mm h^-1 ano^-1 a 1.420,296 MJ ha^-1 mm h^-1 ano^-1 (Barra Nova), de 6.297,272 MJ ha^-1 mm h^-1 ano^-1 a 1.014,815 MJ ha^-1 mm h^-1 ano^-1 (São João da Cachoeira Grande) e de 5.427,659 MJ ha^-1 mm h^-1 ano^-1 a 1.626,489 MJ ha^-1 mm h^-1 ano^-1 (Boca da Vala). Para os municípios de Barra Nova e Boca da Vala a erosividade da chuva foi estimada pela equação EI₃₀ = 6,4492*pi – 391,63 com distribuição leptocúrtica. Para as outras duas localidades, a distribuição foi platicúrtica. A estação climatológica com o maior valor de erosividade média da chuva foi Barra Nova, enquanto Boca da Vala apresentou a menor erosividade, considerando apenas a estimativa da erosividade da chuva pelo modelo matemático II. Os maiores e menores valores de erosividade da chuva foram obtidos com os modelos matemáticos I e II. Para estimar a erosividade da chuva, nas condições climáticos da região de São Mateus-ES, o modelo matemático mais adequado é o II.</p><p align="center"><strong><em>Evaluation of mathematical models to estimate rainfall erosivity in the region of São Mateus-ES</em></strong></p><p><strong>Abstract</strong><strong>: </strong>This study aimed to estimate the rainfall erosivity by six mathematical models, linear regression, and evaluate these, which is more suitable for the climatic conditions of São Mateus-ES region. The rainfall data were obtained from the National Water Agency-ANA, and 1947-2014 for Itauninhas, 1971-2014 to Barra Nova, 1981-2014 for São João da Cachoeira Grande and 1993-2014 for Boca da Vala. To estimate the rainfall erosivity, from the annual precipitation and rainfall coefficient were used different equations used in other states with application to the state of the Holy Spirit or adjusted to the state itself. For mathematical models (II) and (I), the average values were 6541.2 MJ ha^-1 mm h^-1 year^-1 to 936.357 MJ ha^-1 mm h^-1 year^-1 (Itauninhas) of 6995.855 MJ mm ha^-1 h^-1 year^-1 to 1420.296 MJ ha^-1 mm h^-1 year^-1 (Barra nova), to 6297.272 MJ ha^-1 mm h^-1 year^-1 and 1014.815 MJ mm ha^-1 h^-1 year^-1 (São João da Cachoeira Grande) and 5427.659 MJ ha^-1 mm h^-1 year^-1 to 1626.489 MJ ha^-1 mm h^-1 year^-1 (Boca da Vala). For the municipalities of Barra Nova and Boca da Vala the rainfall erosivity was estimated by EI₃₀ = 6.4492*pi - 391.63 with leptokurtic distribution. For the other two locations, the distribution was platykurtic. The climatological station with the highest amount of average rainfall erosivity was Barra Nova, while Boca da Vala had the lowest erosivity, considering only an estimated rainfall erosivity by the mathematical model II. The highest and lowest values erosivity of the rain were obtained with the mathematical models I and II. To estimate the rainfall erosivity in the climatic conditions of São Mateus-ES region, the most suitable mathematical model is II.</p>

PROPOSIÇÃO E VALIDAÇÕES DE EQUAÇÕES PARA ESTIMATIVA DA EROSIVIDADE DE DOIS MUNICÍPIOS DO ESTADO DE SÃO PAULO Reginaldo Barboza da Silva1; Piero Iori2; Francisca Alcivania de Melo Silva11 Universidade Estadual Paulista, Unidade de Registro, Registro, rbsilva@registro.unesp.br2 Departamento de Ciência do Solo, Universidade Federal de Lavras, Lavras, MG 1 RESUMO O uso e extrapolação de equações para localidades cujas características de solo e clima, ainda que parciais, distingam da localidade para a qual tenham sido geradas, ainda permeiam em estudos para estimativa da erosividade (EI30). Este trabalho teve como objetivo propor e validar equações matemáticas para estimativa da erosividade de dois municípios (Sete Barras e Juquiá) do Estado de São Paulo. O ajuste das equações para estimar os valores de erosividade (EI30) em função de valores de coeficiente de chuva (Rc) foi a partir de dados pluviográficos e pluviométricos, respectivamente, utilizando-se de distintas séries históricas. Testes de comparação múltipla e intervalos de confiança foram empregados para comparar médias absolutas de EI30, precipitações (Pp) e Rc. A correlação entre o EI30 e Rc foi verificada pelo coeficiente de correlação de Pearson. O teste da hipótese de igualdade entre as variâncias populacionais foi utilizado para comparar as equações. Dados pluviométricos de uma série histórica diferente das que geraram a equações foram utilizados para validar e avaliar o desempenho das equações obtidas neste estudo e compará-las com outra equação já consolidada pela literatura. Os resultados mostraram que para as condições em que foi realizado o estudo, as equações lineares simples, mostraram ser as mais apropriadas para estimar a erosividade nestes dois municípios. De acordo com o teste da hipótese de igualdade entre as variâncias populacionais. As equações ajustadas para cada município diferiram estatisticamente, de maneira que, a erosividade de cada município deve ser predita por seus modelos respectivos. UNITERMOS: EUPS, erosão, modelagem, Vale do Ribeira. SILVA, R.B.; IORI, P.; SILVA, F. A. de M. PROPOSITION AND COMPARE OF EQUATIONS TO ESTIMATE THE RAINFALL EROSIVITY IN TWO CITIES OF SÃO PAULO STATE 2 ABSTRACT The equations and extrapolation use to localities whose characteristics of soil and climate, even if partial, distinguish the town to which they were generated, still permeate in studies to estimate the rainfall erosivity (EI30). This work has objective to propose and validate mathematical equations to estimate the rainfall erosivity of two cities of Sao Paulo State’s. The adjusted to estimate obtaining and validate data of equations of erosivity (EI30)according to values of coefficient of rain (Rc) were obtained from pluviographic and pluviometric rainfall data, respectively, using of distinct historical rainfall series. Mutiple comparisions test and confidence intervals were performed to compare absolute average of EI30, pluviometric data (Pp), and Rc. The correlation between EI30 and Rc was verified by of Pearson correlation coefficient. Test of the hypothesis of equality between population variance was used to compare the equations. Pluviometrics data of historical series rainfall data different than those that the models were generated were used to validate and to assess the performance of the equations, proposed of this study and compare them with another equation already consolidated in literature. The results show that for the conditions under which the study was conducted, the simple linear equations, shown to be the most appropriate to estimate the rainfall erosivity these two cities. According to the test of the hypothesis of equality variances between populations, the equations adjusted for each city differ statistically so that the rainfall erosivity of each city must be estimated by their respective equation. KEYWORDS: erosion, modeling, USLE, Vale do Ribeira.

The current climate change could lead to an intensification of extreme weather events, such as sudden floods and fast flowing debris flows. Accordingly, the availability of an early-warning device system, based on hydrological data and on both accurate and very fast running mathematical-numerical models, would be not only desirable, but also necessary in areas of particular hazard. To this purpose, the 2D Riemann–Godunov shallow-water approach, solved in parallel on a Graphical-Processing-Unit (GPU) (able to drastically reduce calculation time) and implemented with the RiverFlow2D code (version 2017), was selected as a possible tool to be applied within the Alpine contexts. Moreover, it was also necessary to identify a prototype of an actual rainfall monitoring network and an actual debris-flow event, beside the acquisition of an accurate numerical description of the topography. The Marderello’s basin (Alps, Turin, Italy), described by a 5 × 5 m Digital Terrain Model (DTM), equipped with five rain-gauges and one hydrometer and the muddy debris flow event that was monitored on 22 July 2016, were identified as a typical test case, well representative of mountain contexts and the phenomena under study. Several parametric analyses, also including selected infiltration modelling, were carried out in order to individuate the best numerical values fitting the measured data. Different rheological options, such as Coulomb-Turbulent-Yield and others, were tested. Moreover, some useful general suggestions, regarding the improvement of the adopted mathematical modelling, were acquired. The rapidity of the computational time due to the application of the GPU and the comparison between experimental data and numerical results, regarding both the arrival time and the height of the debris wave, clearly show that the selected approaches and methodology can be considered suitable and accurate tools to be included in an early-warning system, based at least on simple acoustic and/or light alarms that can allow rapid evacuation, for fast flowing debris flows.

Abstract. The RadAlp experiment aims at improving quantitative precipitation estimation (QPE) in the Alps thanks to X-band polarimetric radars and in situ measurements deployed in the region of Grenoble, France. In this article, we revisit the physics of propagation and attenuation of microwaves in rain. We first derive four attenuation–reflectivity (AZ) algorithms constrained, or not, by path-integrated attenuations (PIAs) estimated from the decrease in the return of selected mountain targets when it rains compared to their dry weather levels (the so-called mountain reference technique – MRT). We also consider one simple polarimetric algorithm based on the profile of the total differential phase shift between the radar and the mountain targets. The central idea of the work is to implement these five algorithms all together in the framework of a generalized sensitivity analysis in order to establish useful parameterizations for attenuation correction. The parameter structure and the inherent mathematical ambiguity of the system of equations makes it necessary to organize the optimization procedure in a nested way. The core of the procedure consists of (i) exploring with classical sampling techniques the space of the parameters allowed to be variable from one target to the other and from one time step to the next, (ii) computing a cost function (CF) quantifying the proximity of the simulated profiles and (iii) selecting parameters sets for which a given CF threshold is exceeded. This core is activated for a series of values of parameters supposed to be fixed, e.g., the radar calibration error for a given event. The sensitivity analysis is performed for a set of three convective events using the 0∘ elevation plan position indicator (PPI) measurements of the Météo-France weather radar located on top of the Moucherotte mountain (altitude of 1901 m a.s.l. – above sea level). It allows the estimation of critical parameters for radar QPE using radar data alone. In addition to the radar calibration error, this includes the time series of radome attenuation and estimations of the coefficients of the power law models relating the specific attenuation and the reflectivity (A–Z relationship) on the one hand and the specific attenuation and the specific differential phase shift (A–Kdp relationship) on the other hand. It is noteworthy that the A–Z and A–Kdp relationships obtained are consistent with those derived from concomitant drop size distribution measurements at ground level, in particular with a slightly non-linear A–Kdp relationship (A=0.28 Kdp1.1). X-Band radome attenuations as high as 15 dB were estimated, leading to the recommendation of avoiding the use of radomes for remote sensing of precipitation at such a frequency.

Abstract. To the authors' knowledge there are relatively few studies that try to answer this question: "Are humans able to add value to computer-generated forecasts and warnings?". Moreover, the answers are not always positive. In particular some postprocessing method is competitive or superior to human forecast. Within the alert system of ARPA Piemonte it is possible to study in an objective manner if the human forecaster is able to add value with respect to computer-generated forecasts. Every day the meteorology group of the Centro Funzionale of Regione Piemonte produces the HQPF (Human Quantitative Precipitation Forecast) in terms of an areal average and maximum value for each of the 13 warning areas, which have been created according to meteo-hydrological criteria. This allows the decision makers to produce an evaluation of the expected effects by comparing these HQPFs with predefined rainfall thresholds. Another important ingredient in this study is the very dense non-GTS (Global Telecommunication System) network of rain gauges available that makes possible a high resolution verification. In this work we compare the performances of the latest three years of QPF derived from the meteorological models COSMO-I7 (the Italian version of the COSMO Model, a mesoscale model developed in the framework of the COSMO Consortium) and IFS (the ECMWF global model) with the HQPF. In this analysis it is possible to introduce the hypothesis test developed by Hamill (1999), in which a confidence interval is calculated with the bootstrap method in order to establish the real difference between the skill scores of two competitive forecasts. It is important to underline that the conclusions refer to the analysis of the Piemonte operational alert system, so they cannot be directly taken as universally true. But we think that some of the main lessons that can be derived from this study could be useful for the meteorological community. In details, the main conclusions are the following: – despite the overall improvement in global scale and the fact that the resolution of the limited area models has increased considerably over recent years, the QPF produced by the meteorological models involved in this study has not improved enough to allow its direct use: the subjective HQPF continues to offer the best performance for the period +24 h/+48 h (i.e. the warning period in the Piemonte system); – in the forecast process, the step where humans have the largest added value with respect to mathematical models, is the communication. In fact the human characterization and communication of the forecast uncertainty to end users cannot be replaced by any computer code; – eventually, although there is no novelty in this study, we would like to show that the correct application of appropriated statistical techniques permits a better definition and quantification of the errors and, mostly important, allows a correct (unbiased) communication between forecasters and decision makers.

Abstract Variations in salinity and flow rate in the aerial, naturally salty spring of Almyros of Heraklion on Crete were monitored during two hydrological cycles. We describe the functioning of the coastal karstic system of the Almyros and show the influence of the duality of the flow in the karst (conduits and fractured matrix) on the quality of the water resource in the coastal area. A mechanism of saltwater intrusion into this highly heterogeneous system is proposed and validated with a hydraulic mathematical model, which describes the observations remarkably well. Introduction. – Fresh groundwater is a precious resource in many coastal regions, for drinking water supply, either to complement surface water resources, or when such resources are polluted or unavailable in the dry season. But coastal groundwater is fragile, and its exploitation must be made with care to prevent saltwater intrusion as a result of withdrawal, for any aquifer type, porous, fractured or karstic. In karstic zones, the problem is very complex because of the heterogeneous nature of the karst, which makes it difficult to use the concept of representative elementary volume developed for porous or densely fractured systems. The karstic conduits focus the major part of the flow in preferential paths, where the water velocity is high. In coastal systems, these conduits have also an effect on the distribution of the saline intrusion. As was shown e.g. by Moore et al. [1992] and Howard and Mullings [1996], both freshwater and salt-water flow along the fractures and conduits to reach the mixing zone, or the zone where these fluids are superposed in a dynamic equilibrium because of their differences in density ; but the dynamics of such a saltwater intrusion are generally unknown and not represented in models. Such coastal karstic systems are intensely studied at this moment in the Mediterranean region [Gilli, 1999], both as above sea-level or underwater springs, for potential use in areas where this resource would be of great value for economic development. This article discusses the freshwater-saltwater exchange mechanisms in the karstic aquifer of the Almyros of Heraklion aquifer (Crete) and explains the salinity variations observed in the spring. First, the general hydrogeology of the study site is described, then the functioning of the spring : a main conduit drains the freshwater over several kilometres and passes at depth through a zone where seawater is naturally present. The matrix-conduit exchanges are the result of pressure differences between the two media. These processes are represented in a mathematical model that confirms their relevance. General hydrogeology of the studied site. – The karstic coastal system of the Almyros of Heraklion (Crete) covers 300 km2 in the Ida massif whose borders are a main detachment fault, and the Sea of Crete in the north, the Psiloritis massif (highest summit at 2,456 m) in the south and west, and the collapsed basin of Heraklion filled in by mainly neo-geneous marl sediments in the east. The watershed basin consists of the two lower units of characteristic overthrust formations of Crete (fig. 1) : the Cretaceous Plattenkalk and the Cretaceous Tripolitza limestones. The two limestone formations are locally separated by interbedded flysch or phyllade units that form an impervious layer [Bonneau et al., 1977 ; Fassoulas, 1999] and may lead to different flow behaviour within the two karstic formations. Neo-tectonic activity has dissected these formations with large faults and fractures. The present-day climate in Crete is of Mediterranean mountain type, with heavy rain storms and snow on the summits in winter. Rainfall is unevenly distributed over the year, with 80 % of the annual total between October and March and a year-to-year average of 1,370 mm. The flow rate of the spring is high during the whole hydrologic cycle, with a minimum in summer on the order of 3 m3.s−1 and peak flow in winter reaching up to 40 m3.s −1. The water is brackish during low flow, up to a chloride content of 6 g.l−1, i.e. 23 % of seawater, but it is fresh during floods, when the flow rate exceeds 15 m3.s−1. During the 1999–2000 and 2000–2001 hydrologic cycles, the water was fresh during 14 and 31 days, respectively. The water temperature is high and varies very little during the year (see table I). In the areas of Kéri and Tilissos (fig. 1), immediately south of the spring, the city of Heraklion extracts water from the karstic system through a series of 15 wells with depth reaching 50 to 100 m below sea level. Initially, when the wells were drilled, the water was fresh, but nowadays the salinity rises progressively, but unequally from well to well (fig. 2). The relatively constant temperatures and salinities of the wells, during the hydrological cycle, contrast with the large salinity variations at the spring (fig. 2 and table I). They show that the karstic system is complex and comprises different compartments, where each aquifer unit reacts to its individual pressures (pumping, rainfall) according to its own hydrodynamic characteristics [Arfib et al., 2000]. The Almyros spring seems disconnected from the surrounding aquifer and behaves differently from that which feeds the wells (upper Tripolitza limestone). It is recharged by fresh water from the mountains, which descends to depths where it probably acquires its salinity. The spring would thus be the largest resource of the area, if it was possible to prevent its pollution by seawater. A general functioning sketch is proposed (fig. 3), which includes the different geological units of interest. Identification of the functioning of the Almyros spring through monitoring of physical and chemical parameters. – The functioning of the aquifer system of the Almyros spring was analysed by monitoring, over two hydrological cycles, the level of the spring, the discharge, the electric conductivity and the temperature recorded at a 30 min time interval. In the centre of the watershed basin, a meteorological station at an altitude of 800 m measures and records at a 30 min time interval the air temperature, rainfall, relative humidity, wind velocity and direction ; moreover, an automatic rain gauge is installed in the northern part of the basin at an altitude of 500 m. The winter floods follow the rhythm of the rainfall with strong flow-rate variations. In contrast, the summer and autumn are long periods of drought (fig. 7). The flow rate increases a few hours after each rainfall event ; the water salinity decreases in inverse proportion to the flow rate a few hours to a few days later. Observations showed that the water volume discharged at the Almyros spring between the beginning of the flow rate increase and the beginning of the salinity decrease is quite constant, around 770,000 m3 (fig. 4) for any value of the flow rate, of the salinity and also of the initial or final rainfall rates. To determine this constant volume was of the upmost importance when analyzing the functioning of the Almyros spring. The lag illustrates the differences between the pressure wave that moves almost instantaneously through the karst conduit and causes an immediate flow rate increase after rainfall and the movement of the water molecules (transfer of matter) that arrives with a time lag proportionate to the length of the travel distance. The variation of the salinity with the flow rate acts as a tracer and gives a direct indication of the distance between the outlet and the seawater entrance point into the conduit. In the case of the Almyros, the constant volume of expelled water indicates that sea-water intrusion occurs in a portion of the conduit situated several kilometres away from the spring (table II), probably inland, with no subsequent sideways exchange in the part of the gallery leading up to the spring. As the lag between the flow rate and the salinity recorded at the spring is constant, one can correct the salinity value by taking, at each time step, with a given flow rate, the salinity value measured after the expulsion of 770,000 m3 at the spring, which transforms the output of the system so as to put the pressure waves and the matter transfer in phase [Arfib, 2001]. After this correction, the saline flux at the spring, equal to the flow rate multiplied by the corrected salinity, indicates the amount of sea-water in the total flow. This flux varies in inverse proportion to the total flow rate in the high-flow period and the beginning of the low-flow period, thereby demonstrating that the salinity decrease in the spring is not simply a dilution effect (fig. 5). The relationship that exists between flow rate and corrected salinity provides the additional information needed to build the conceptual model of the functioning of the part of the Almyros of Heraklion aquifer that communicates with the spring. Freshwater from the Psiloritis mountains feeds the Almyros spring. It circulates through a main karst conduit that descends deep into the aquifer and crosses a zone naturally invaded by seawater several kilometers from the spring. The seawater enters the conduit and the resulting brackish water is then transported to the spring without any further change in salinity. The conduit-matrix and matrix-conduit exchanges are governed by the head differences in the two media. Mathematical modelling of seawater intrusion into a karst conduit Method. – The functioning pattern exposed above shows that such a system cannot be treated as an equivalent porous medium and highlights the influence of heterogeneous structures such as karst conduits on the quantity and quality of water resources. Our model is called SWIKAC (Salt Water Intrusion in Karst Conduits), written in Matlab®. It is a 1 D mixing-cell type model with an explicit finite-difference calculation. This numerical method has already been used to simulate flow and transport in porous [e.g. Bajracharya and Barry, 1994 ; Van Ommen, 1985] and karst media [e.g. Bauer et al., 1999 ; Liedl and Sauter, 1998 ; Tezcan, 1998]. It reduces the aquifer to a single circular conduit surrounded by a matrix equivalent to a homogeneous porous medium where pressure and salinity conditions are in relation with sea-water. The conduit is fed by freshwater at its upstream end and seawater penetrates through its walls over the length L (fig. 6) at a rate given by an equation based on the Dupuit-Forchheimer solution and the method of images. The model calculates, in each mesh of the conduit and at each time step, the head in conditions of turbulent flow with the Darcy-Weisbach equation. The head loss coefficient λ is calculated by Louis’ formula for turbulent flow of non-parallel liquid streams [Jeannin, 2001 ; Jeannin and Marechal, 1995]. The fitting of the model is intended to simulate the chloride concentration at the spring for a given matrix permeability (K), depth (P) and conduit diameter (D) while varying its length (L) and its relative roughness (kr). The spring flow rates are the measured ones ; at present, the model is not meant to predict the flow rate of the spring but only to explain its salinity variations. Results and discussion. – The simulations of chloride concentrations were made in the period from September 1999 to May 2001. The depth of the horizontal conduit where matrix-conduit exchanges occur was tested down to 800 m below sea level. The diameter of the conduit varied between 10 and 20 m, which is larger than that observed by divers close to the spring but plausible for the seawater intrusion zone. The average hydraulic conductivity of the equivalent continuous matrix was estimated at 10−4 m/s. A higher value (10−3 m/s) was tested and found to be possible since the fractured limestone in the intrusion zone may locally be more permeable but a smaller value (10−5 m/s) produces an unrealistic length (L) of the saline intrusion zone (over 15 km). For each combination of hydraulic conductivity, diameter and depth there is one set of L (length) and kr (relative roughness) calibration parameters. All combinations for a depth of 400 m or more produce practically equivalent results, close to the measured values. When the depth of the conduit is less than 400 m, the simulated salinity is always too high. Figure 7 shows results for a depth of 500 m, a diameter of 15 m and a hydraulic conductivity of 10−4 m/s. The length of the saltwater intrusion zone is then 1,320 m, 4,350 m away from the spring and the relative roughness coefficient is 1.1. All the simulations (table II) need a very high relative roughness coefficient which may be interpreted as an equivalent coefficient that takes into account the heavy head losses by friction and the variations of the conduit dimensions which, locally, cause great head losses. The model simulates very well the general shape of the salinity curve and the succession of high water levels in the Almyros spring but two periods are poorly described due to the simplicity of the model. They are (1) the period following strong freshwater floods, where the model does not account for the expulsion of freshwater outside the conduit and the return of this freshwater which dilutes the tail of the flood and (2) the end of the low-water period when the measured flux of chlorides falls unexpectedly (fig. 5), which might be explained by density stratification phenomena of freshwater-saltwater in the conduit (as observed in the karst gallery of Port-Miou near Cassis, France [Potié and Ricour, 1974]), an aspect that the model does not take into account. Conclusions. – The good results produced by the model confirm the proposed functioning pattern of the spring. The regulation of the saline intrusion occurs over a limited area at depth, through the action of the pressure differences between the fractured limestone continuous matrix with its natural saline intrusion and a karst conduit carrying water that is first fresh then brackish up to the Almyros spring. The depth of the horizontal conduit is more than 400 m. An attempt at raising the water level at the spring, with a concrete dam, made in 1987, which was also modelled, indicates that the real depth is around 500 m but the poor quality of these data requires new tests to be made before any firm conclusions on the exact depth of the conduit can be drawn. The Almyros spring is a particularly favorable for observing the exchanges in the conduit network for which it is the direct outlet but it is not representative of the surrounding area. To sustainably manage the water in this region, it is essential to change the present working of the wells in order to limit the irreversible saline intrusion into the terrain of the upper aquifers. It seems possible to exploit the spring directly if the level of its outlet is raised. This would reduce the salinity in the spring to almost zero in all seasons by increasing the head in the conduit. In its present state of calibration, the model calculates a height on the order of 15 m for obtaining freshwater at the spring throughout the year, but real tests with the existing dam are needed to quantify any flow-rate losses or functional changes when there is continual overpressure in the system. The cause of the development of this karstic conduit at such a great depth could be the lowering of the sea level during the Messinian [Clauzon et al., 1996], or recent tectonic movements.

In order to find the optimal solution for the drainage of rainwater from roads in urban areas, as well as for the evaluation and ranking of conceptual solutions, appropriate mathematical models and software packages were used in this research. For relevant rain episodes, i.e. rainfall of appropriate duration and intensity, runoff coefficients and flows were taken into account and analyzed according to the rational method, all for the purpose of obtaining data on the amount of rainwater entering the sewage system. Through this research, very good correlations and regressions were established between the cross slope of the road and the parameters of rainwater drainage from the road, as well as the correlation and regression relationships of the cross slope of the road and the efficiency of the drain. Likewise, the dependences of the drainage parameters, the efficiency of the drains and the cross slope of the road were determined, expressed through mathematical functions.