Time series and stochastic analyses to study the hydrodynamic characteristics of karstic aquifers

The regional study of hydrodynamic characteristics of karstic aquifers is challenging because of the great variety of lithology and the structural complexity found in carbonate formations. In order to improve this situation, a combined approach of time series and stochastic analyses was adopted to assess the hydrodynamic behaviour of the karstic aquifers. To achieve this, daily flow rates of 20 springs were taken from the 11 most significant aquifer units of the Basque Country. The results demonstrate the presence of memory effects, which modulated the input rainfall for short‐, medium‐ and long‐term storage capacity, resulting in hydrodynamic properties such as system memory, response time and mean delay between input and output. They reflect the storage and the manner in which these are filled and emptied, thus indicating the karstification of the aquifer. Likewise, the hydrodynamic and hydraulic classification obtained from the stochastic analysis provides a complementary approach to characterize the hydraulic behaviour of the studied karstic aquifers. The discussed examples indicate that this approach provides an excellent method to research hydrological karst systems. It is also shown that the use of hydrologic time series, alone, does not lead to a satisfactory classification of the hydrodynamic characteristics. Therefore, the general approach to hydrological regionalization in karst areas should take into account the structural complexity, heterogeneity of the lithology and the degree of karstification. Only in this case will the regionalization be physically founded, leading to a regional understanding of the hydrodynamic characteristics and flow conditions in a karst aquifer. Copyright © 2009 John Wiley & Sons, Ltd.     

Brackish karstic springs model: application to Almiros spring in Crete

A mathematical model is proposed to simulate brackish karstic springs. Rainfall data constitutes model input information while output information is the discharge and the chloride concentration of the water versus time. The model was constructed by considering the mass and mechanical energy balance on the hydrodynamic analog, which includes three reservoirs outflowing in a tube that lies adjacent to the spring. Two reservoirs emulate the karstic system, and the third one emulates the sea. The discharge of the spring is given by the sum of the discharge of the reservoirs, and the chloride concentration by the solution of the mixing problem between the fresh and the salty water, which exists in the tube leading to the spring. The model is applied to the spring of Almiros at Heraklion, Crete, Greece. The agreement between model values and field measurements is very good for depletion periods and satisfactory for recharge periods.     

A stochastic model of low flows

A model is developed for annual low flow hydrographs. Its two primary components reflect the fact that hydrologic processes during streamflow rise (function of water input) and recession (function of basin storage) are different. Durations of periods of rise (wet intervals) and recession (dry intervals) are modelled by discrete probability distributions — negative binomial for dry intervals and negative binomial or modified logarithmic series for wet intervals depending on goodness of fit. During wet intervals, the total inflow is modelled by the lognormal distribution and daily amounts are allocated according to a pattern-averaged model. During dry intervals, the flow recedes according to a deterministic-stochastic recession model. The model was applied to three Canadian basins with drainage area ranging from 2210 to 22000 km² to generate 50 realizations of low flow hydrographs. The resulting two standard-error confidence band for the simulated probability distribution of annual minimum 7-day flows enclosed the probability distribution estimated from the observed record. A sensitivity analysis for the three basins revealed that in addition to the recession submodel, the most important submodel is that describing seasonality. The state of the basin at the beginning of the low flow period is of marginal importance and the daily distribution of input is unimportant.     

A stochastic model of water-table elevation

A second order stochastic differential equation is used for modeling of water-table elevation. The data were sampled at the Borden Aquifer as a part of a tracer experiment. The purpose of the water-table data collection was to determine presence of a water flow. We argue that the water-table surface is a simple plane oscillating up and down in time according to an equation for a stochastic oscillator. We derive the model, estimate its parameters and provide arguments for goodness-of-fit of the model.     

A stochastic model of water-table elevation

A second order stochastic differential equation is used for modeling of water-table elevation. The data were sampled at the Borden Aquifer as a part of a tracer experiment. The purpose of the water-table data collection was to determine presence of a water flow. We argue that the water-table surface is a simple plane oscillating up and down in time according to an equation for a stochastic oscillator. We derive the model, estimate its parameters and provide arguments for goodness-of-fit of the model.     

A stochastic model of low flows

A model is developed for annual low flow hydrographs. Its two primary components reflect the fact that hydrologic processes during streamflow rise (function of water input) and recession (function of basin storage) are different. Durations of periods of rise (wet intervals) and recession (dry intervals) are modelled by discrete probability distributions — negative binomial for dry intervals and negative binomial or modified logarithmic series for wet intervals depending on goodness of fit. During wet intervals, the total inflow is modelled by the lognormal distribution and daily amounts are allocated according to a pattern-averaged model. During dry intervals, the flow recedes according to a deterministic-stochastic recession model. The model was applied to three Canadian basins with drainage area ranging from 2210 to 22000 km² to generate 50 realizations of low flow hydrographs. The resulting two standard-error confidence band for the simulated probability distribution of annual minimum 7-day flows enclosed the probability distribution estimated from the observed record. A sensitivity analysis for the three basins revealed that in addition to the recession submodel, the most important submodel is that describing seasonality. The state of the basin at the beginning of the low flow period is of marginal importance and the daily distribution of input is unimportant.     

Simple procedure to simulate karstic aquifers

A procedure to simulate karstic aquifers is presented. It is based on a simulation of spring discharge using precipitation and, where necessary, temperature as input data. The karstic aquifer system is considered to be divided into three zones: the surface zone, the unsaturated zone (UZ) and the saturated zone (SZ). Each of these is described by a transfer function that determines the water supplied from the overlying zone. Water loss through evapotranspiration is calculated empirically and subtracted from the total precipitation in order to obtain the effective infiltration into the UZ. The transfer function characterizing the UZ can be expressed as a convolution function. The UZ acts as a buffer, delaying effective infiltration into the SZ. Water discharge from the SZ is described by the recession function of the spring, and this becomes the transfer function that characterizes the emergence of water from the SZ. The model permits the simulation of the influence of pumped abstractions from the system by a simple modification of the transfer functions involved. Copyright © 2007 John Wiley & Sons, Ltd.     

A stochastic approach to site-response component in seismic ground motion coherency model

In order to account for the site-response part of the seismic ground motion coherency for sites of interest, an analytical stochastic methodology is proposed in this paper. By combining the pseudo-excitation method with wave motion finite element simulation techniques, a numerical approach for the computation of the coherency function between observation points is developed firstly. Then the orthogonal expansion method is introduced into this approach to study the effect of the uncertainty in soil properties on the coherency function. Finally some numerical examples are given to show the applicability of the methodology. The computational results demonstrate that the lagged coherency values tend to decrease in the vicinity of the resonant frequencies of the site.     

Elevation effects on rainfall: A stochastic model

The variation in point precipitation with elevation is investigated using an event-based stochastic model of thunderstorm rainfall and empirical data. Parameters of the model correspond to the number of events per unit of time and the depth of rainfall per event. An increase in precipitation with elevation may be due to an increase in the number of events, in the amount of rainfall per event or to some combination of both possibilities. The distribution of the number of events per season is assumed to be a Poisson variate while the distribution of point rainfall depths may be taken as geometric. The summation of a random number of random variables is used to represent seasonal point precipitation. Assuming that the two parameters of the model increase linearly with elevation, then total seasonal rainfall increases as a quadratic polynomial with elevation. The use of the model allows one to obtain the return period of storm rainfall of a given magnitude despite a short historical record. An independent set of data was used to verify the procedure.     

新型随机地震动模型

在研究结构的随机地震反应时，要用大量的符合场地条件的地震记录作为输入数据。但强震历史记录却不是每个地区都有的，因此根据符合场地条件的现有地震记录建立随机地震动模型具有重要意义。本文利用中国抗震规范2001版修正选取的样本波作为目标波，考虑了幅值和频率的双重非平稳性，建立了新型随机地震动模型——改进的时变ARMA模型随机地震动模型。通过使用残差的卡方检验法，对多种非平稳ARMA模型生成的模拟波进行检验；同时又比较丁模拟波与目标波的功率谱密度图和反应谱图。结果证明：此法能够更精确地反映不同场地条件地震动的频谱和幅值的真实内容，从而建立符合目标场地条件的更为有效的模拟地震动，为相关研究与工程设计架起一座桥梁。     

A model of cells as practical approach to simulate spring flow in the Itxina karstic aquifer, Basque Country, Spain

The aim of this study is to apply a parsimonious hydrologic model to the Itxina karstic aquifer that can predict changes in discharge resulting from variable inputs (recharge). The Itxina Aquifer was divided into four cells corresponding to different recharge areas. Each cell was treated as a tank to characterize the conditions within the cell. In the model, when the reservoir boundaries coincide with the position of the siphons, the signal simulated is sensitive to input pulses of the recharge. This supports the hypothesis that the siphons are the controlling mechanism in the flow system of the aquifer. The good agreement between predicted and measured discharges demonstrates the ability of the model to simulate the flow in the Itxina Aquifer. These results demonstrated that the hydraulic conductivity increases downstream within the aquifer. The hydraulic conductivities obtained by calibration varied between 4.2 x 10(-3) m/s upstream of the aquifer, 6.0 x 10(-2) m/s in the central region, and 9.5 x 10(-1) m/s in the lower region of the aquifer. These values seem reasonable because the underground features in the principal caves show that the density of caves increases downstream in the Itxina Aquifer. The simple representation of the system produced results comparable to traditional ground water models with fewer data requirements and calibration parameters.     

A multivariate Bernstein copula model for permeability stochastic simulation

This paper introduces a general nonparametric method for joint stochastic simulation of petrophysical properties using the Bernstein copula. This method consists basically in generating stochastic simulations of a given petrophysical property (primary variable) modeling the underlying empirical dependence with other petrophysical properties (secondary variables) while reproducing the spatial dependence of the first one.This multivariate approach provides a very flexible tool to model the complex dependence relationships of petrophysical properties. It has several advantages over other traditional methods, since it is not restricted to the case of linear dependence among variables, it does not require the assumption of normality and/or existence of moments.In this paper this method is applied to simulate rock permeability using Vugular Porosity and Shear Wave Velocity (S-Waves) as covariates in a carbonate double-porosity formation at well log scale. Simulated permeability values show a high degree of accuracy compared to the actual values.     

A doubly stochastic rainfall model with exponentially decaying pulses

We develop a doubly stochastic point process model with exponentially decaying pulses to describe the statistical properties of the rainfall intensity process. Mathematical formulation of the point process model is described along with second-order moment characteristics of the rainfall depth and aggregated processes. The derived second-order properties of the accumulated rainfall at different aggregation levels are used in model assessment. A data analysis using 15 years of sub-hourly rainfall data from England is presented. Models with fixed and variable pulse lifetime are explored. The performance of the model is compared with that of a doubly stochastic rectangular pulse model. The proposed model fits most of the empirical rainfall properties well at sub-hourly, hourly and daily aggregation levels.     

Integrated stochastic modeling of pharmaceuticals in sewage networks

Pharmaceuticals and particularly antibiotics can harm sensitive aquatic species. Their occurrence in urban wastewater systems is the consequence of five successive processes: (i) ingestion of the substance, (ii) accumulation in the urine, (iii) excretion, (iv) degradation in the sewer system and (v) transport to the wastewater treatment plant (WTP). These processes were included in an integrated model that can be used to assess the dynamics of pharmaceuticals at a WTP inlet. First, information on sales data, posology, pharmacokinetics and toilet flushing frequency were combined to create a source model of pharmaceuticals entering a sewer system. This production function was then coupled with a transport/degradation model to simulate concentrations of pharmaceuticals at a WTP inlet. In an example application, the full model was applied to simulate the concentration of the antibiotic ciprofloxacin on an hourly time scale. In this application, the model was calibrated and validated for a case study at a WTP in Lausanne, Switzerland. Validation of the integrated approach was successful despite the high variability evident in the model results. This modeling approach has potential use in pollution management and epidemiology related to wastewater.     

A stochastic distributed model of soil erosion by overland flow

A model has been built to simulate runoff and erosion on slopes of varying form with soil of varying hydraulic conductivity, porosity, storage capacity, and particle size distribution. The model is based on known physical laws and takes into account spatially dependent random variation in the initial conditions of the soil. The rainfall is similarly both random and autocorrelated in space. Detailed results show how runoff is generated as a result of slow conductivity and high water table, how this leads to erosion, the degradation of the land surface, the capture of flow lines, and the development of networks of rills. The last accords well with results from laboratory experiments. Changing the initial conditions produces results that accord with intuition and reality.     

A stochastic model for tropical rainfall at a single location

Modelling data that correspond to rainfall accumulated over fixed periods of time presents the challenging problem of dealing with a random variable that has a point mass at zero which corresponds to dry periods that occur with positive probability. One way to overcome this difficulty is to assume that the data correspond to a normal variate w, that has been truncated and transformed. The dry periods correspond to the (unobserved) negative values and the wet periods correspond to some power of the positive ones. The serial structure that is present in rainfall can be modelled by imposing a serial structure to w. We use a dynamic linear model on w using a Fourier representation to allow for the seasonality of the data, which in the case of tropical rainfall is very marked. The model is fitted using a Markov chain Monte Carlo method that uses latent variables to handle both dry periods and missing values. We use the model to estimate and predict both the amount of rainfall and the probability of a dry period. The method is illustrated with data collected in the Venezuelan state of Guárico.     

A stochastic model for air injection into saturated porous media

Air injection into porous media is investigated by laboratory experiments and numerical modelling. Typical applications of air injection into a granular bed are aerated bio-filters and air sparging of aquifers. The first stage of the dynamic process consists of air injection into a fixed or a quasi-fixed water-saturated granular bed. Later stages could include stages of movable beds as well, but are not further investigated here. A series of laboratory experiments were conducted in a two-dimensional box of the size 60 cm × 38 cm × 0.55 cm consisting of glass walls and using glass beads of diameter 0.4–0.6 mm as granular material. The development of the air flow pattern was optically observed and registered using a digital video camera. The resulting transient air flow pattern can be characterized as channelled flow in a fixed porous medium with dynamic tree-like evolution behaviour. Attempts are undertaken to model the air injection process. Multiphase pore-scale modelling is currently disregarded since it is restricted to very small scales. Invasion percolation models taking into account gravity effects are usually restricted to slow processes. On the other hand a continuum-type two-phase flow modelling approach is not able to simulate the observed air flow pattern. Instead a stochastic continuum-type approach is discussed here, which incorporates pore-scale features on a subscale, relevant for the immiscible processes involved. Consequently, the physical process can be modelled in a stochastic manner only, where the single experiment represents one of many possible realizations. However, the present procedure retains realistic water and air saturation patterns and therefore produces similar finger lengths and widths as observed in the experiments. Monte Carlo type modelling leads to ensemble mean water saturation and the related variance.     

2D magnetic resonance tomography applied to karstic conduit imaging

Karstic conduits play a crucial role for water supply in many parts of the world. However, the imaging of such targets is generally a difficult task for most geophysical methods. Magnetic Resonance Sounding (MRS) is a geophysical method designed for imaging of water bearing structures. Initially, MRS was developed for characterizing horizontally stratified aquifers. However, when applying a 1D MRS measuring setup to the imaging of 2D–3D targets, the size of which may be much smaller than the loop, the accuracy and the lateral resolution may not be sufficient. We have studied the possibility of simultaneously processing several MRS aligned along a profile to perform a Magnetic Resonance Tomography (MRT). This work emphasizes the gain of resolution for 2D–3D imagery of MRT versus the interpolation of 1D inversion results of MRS along the same profile. Numerical modelling results show that the MRT response is sensitive to the size and location of the 2D target in the subsurface. Sensitivity studies reveal that by using the coincident transmitting/receiving (TX/RX) setup and shifting the loop around the anomaly area, the depth, section and position of a single karstic conduit with a size smaller than the MRS loop size can be resolved. The accuracy of the results depends on the noise level and signal level, the latter parameter being linked to the depth and volume of the karstic conduit and the water content in the limestone matrix. It was shown that when applying MRT to the localization of 2D anomalies such as karstic conduits, the inclination of the geomagnetic field, the orientation of the MRT profile and the angle of crossover of the conduit by the MRT profile must be taken into account. Otherwise additional errors in interpretation should be expected. A 2D inversion scheme was developed and tested. Both numerical and experimental results confirm the efficiency of the developed approach.