Testing a 1-D analytical salt intrusion model and its predictive equations in Malaysian estuaries

Abstract

Little is known about the salt intrusion behaviour in Malaysian estuaries. Study of salt intrusion generally requires large amounts of data, especially if 2-D or 3-D numerical models are used; thus, in data-poor environments, 1-D analytical models are more appropriate. A fully analytical 1-D salt intrusion model, which is simple to implement and requires minimal data, was tested in six previously unsurveyed Malaysian estuaries (Kurau, Perak, Bernam, Selangor, Muar and Endau). The required data can be collected during a single day of observations. Site measurements were conducted during the dry season (June–August 2012 and February–March 2013) near spring tide. Data on cross-sections (by echo-sounding), water levels (by pressure loggers) and salinity (by moving boat) were collected as model input. A good fit was demonstrated between the simulated and observed salinity distribution for all six estuaries. Additionally, the two calibration parameters (the Van der Burgh coefficient and the boundary condition for the dispersion) were compared with the existing predictive equations. Since gauging stations were only present in some nested catchments in the drainage basins, the river discharge had to be up-scaled to represent the total discharge contribution of the catchments. However, the correspondence between the calibration coefficients and the predictive equations was good, particularly in view of the uncertainty in the river discharge data used. This confirms that the predictive salt intrusion model is valid for the cases studied in Malaysia. The model provides a reliable, predictive tool, which the water authority of Malaysia can use for making decisions on water abstraction or dredging.

Editor D. Koutsoyiannis; Associate editor A. Fiori     

A coupled oscillator model of shelf and ocean tides

The resonances of tides in the coupled open ocean and shelf are modeled by a mechanical analogue consisting of a damped driven larger mass and spring (the open-ocean) connected to a damped smaller mass and spring (the shelf). When both masses are near resonance, the addition of even a very small mass can significantly affect the oscillations of the larger mass. The influence of the shelf is largest if the shelf is resonant with weak friction. In particular, an increase of friction on a near-resonant shelf can, perhaps surprisingly, lead to an increase in ocean tides. On the other hand, a shelf with large friction has little effect on ocean tides. Comparison of the model predictions with results from numerical models of tides during the ice ages, when lower sea levels led to a much reduced areal extent of shelves, suggests that the predicted larger tidal dissipation then is related to the ocean basins being close to resonance. New numerical simulations with a forward global tide model are used to test expectations from the mechanical analogue. Setting friction to unrealistically large values in Hudson Strait yields larger North Atlantic _M2$M_2$ amplitudes, very similar to those seen in a simulation with the Hudson Strait blocked off. Thus, as anticipated, a shelf with very large friction is nearly equivalent in its effect on the open ocean to the removal of the shelf altogether. Setting friction in shallow waters throughout the globe to unrealistically large values yields even larger open ocean tidal amplitudes, similar to those found in simulations of ice-age tides. It thus appears that larger modeled tides during the ice ages can be a consequence of enhanced friction in shallower water on the shelf in glacial times as well as a reduced shelf area then. Single oscillator and coupled oscillator models for global tides show that the maximum extractable power for human use is a fraction of the present dissipation rate, which is itself a fraction of global human power consumption.     

A general solution for groundwater flow in an estuarine's leaky aquifer system after considering aquifer anisotropy

This paper presents an analytical model for describing the tidal effects in a two‐dimensional leaky confined aquifer system in an estuarine delta where ocean and river meet. This system has an unconfined aquifer on top and a confined aquifer on the bottom with an aquitard in between the two. The unconfined and confined aquifers interact with each other through leakage. It was assumed that the aquitard storage was negligible and that the leakage was linearly proportional to the head difference between the unconfined and confined aquifers. This model's solution was based on the separation of variables method. Two existing solutions that deal with the head fluctuation in one‐dimensional or two‐dimensional leaky confined aquifers are shown as special cases in the present solution. Based on this new solution, the dynamic effect of the water table's fluctuations can be clearly explored, as well as the influence of leakage on the behaviour of fluctuations in groundwater levels in the leaky aquifer system. Copyright © 2015 John Wiley & Sons, Ltd.     

Feedback between residual circulations and sediment distribution in highly turbid estuaries: An analytical model

Motivated by field studies of the Ems estuary which show longitudinal gradients in bottom sediment concentration as high as O(0.01 kg/m⁴), we develop an analytical model for estuarine residual circulation based on currents from salinity gradients, turbidity gradients, and freshwater discharge. Salinity is assumed to be vertically well mixed, while the vertical concentration profile is assumed to result from a balance between a constant settling velocity and turbulent diffusive flux. Width and depth of the model estuary are held constant. Model results show that turbidity gradients enhance tidally averaged circulation upstream of the estuarine turbidity maximum (ETM), but significantly reduce residual circulation downstream, where salinity and turbidity gradients oppose each other. We apply the condition of morphodynamic equilibrium (vanishing sediment transport) and develop an analytical solution for the position of the turbidity maximum and the distribution of suspended sediment concentration (SSC) along a longitudinal axis. A sensitivity study shows great variability in the longitudinal distribution of suspended sediment with the applied salinity gradient and six model parameters: settling velocity, vertical mixing, horizontal dispersion, total sediment supply, fresh water flow, and water depth. Increasing depth and settling velocity move the ETM upstream, while increasing freshwater discharge and vertical mixing move the ETM downstream. Moreover, the longitudinal distribution of SSC is inherently asymmetric around the ETM, and depends on spatial variations in the residual current structure and the vertical profile of SSC.     

Initial growth of phytoplankton in turbid estuaries: A simple model

An idealised model is presented and analysed to gain more fundamental understanding about the dynamics of phytoplankton blooms in well-mixed, suspended sediment dominated estuaries. The model describes the behaviour of subtidal currents, suspended sediments, nutrients and phytoplankton in a channel geometry. The initial growth of phytoplankton and its spatial distribution is calculated by solving an eigenvalue problem. The growth rates depend on the position in the estuary due to along-estuary variations in nutrient concentration and suspended sediment concentration. The model yields an insight into how the onset of blooms in the model depends on physical and biological processes (turbulent mixing, fresh water discharge, light attenuation, imposed nutrient concentrations at the river and sea side). In particular, the model demonstrates that the joint action of spatial variations in turbidity and in nutrients causes the maximum phytoplankton concentrations to occur seaward of the estuarine turbidity maximum.     

A simple model of hillslope response for overland flow generation

This paper deals with the derivation of the hydrological response of a hillslope on the assumption of quick runoff by surface runoff generation. By using the simple non‐linear storage based model, first proposed by Horton, an analytical solution of the overland flow equations over a plane hillslope was derived. This solution establishes a generalization for different flow regimes of Horton's original solution, which is valid for the transitional flow regime only. The solution proposed was compared successfully with that of Horton and, for the turbulent flow regime, to the one derived from kinematic wave theory. This solution can be applied easily to both stationary and non‐stationary rainfall excess events. An analytical solution for the instantaneous response function (IRF) was also derived. Finally, simple expressions to compute peak and time to peak of IRF are proposed. Copyright © 2001 John Wiley & Sons, Ltd.     

Application of analytical solution for stream depletion due to groundwater pumping in Gapcheon watershed,South Korea

A number of groundwater wells for agricultural activity, including rice farming and greenhouses, have been developed near streams over the past 20 years in South Korea. The result of a stream depletion calculation using an analytical solution of complimentary error function shows that groundwater pumping at 1949 wells drilled in the Gapcheon watershed can produce stream depletion. This amount is estimated at about 7% of annual baseflow and reaches as high as 18% of monthly baseflow during the maximum agricultural water consumption period in May. Agricultural wells have a larger effect on stream depletion than domestic wells because of their higher pumping rate. Stream depletion from agricultural wells located within 200 m from a stream represents 65% of the total depletion rate. Agricultural water policy for water use at nearby streams should be changed to reduce stream depletion and thereby maintain sustainable water development in South Korea. Copyright © 2010 John Wiley & Sons, Ltd.     

A semi‐analytical approach for predicting peak discharge of floods caused by embankment dam failures

This study presents an analytical solution of dam‐break floods in a trapezoidal channel with detailed solution procedure. An approach predicting the peak discharge of floods caused by embankment dam failures was derived from the aforementioned analytical solution with a database of 27 historical dam failures. The prediction performance of this approach has been proved by comparing with other 14 straightforward equations for estimating the peak discharge. The proposed model with a small uncertainty of predicted peak flow rates has a high coefficient of determination and a small standard error, being ranked in the top four of the 15 methods considered in this paper. The robustness and predictive capability of the proposed model are further demonstrated in two case studies, and both were considered in the previous analyses performed by other investigators. This method provides a simple and transparent tool for engineers to predict the peak discharge and is easy to implement for trial and error calculation. Copyright © 2016 John Wiley & Sons, Ltd.     

A two-phase numerical model for suspended-sediment transport in estuaries

This paper presents a full 2-D X/Z numerical model for sediment transport in open channels and estuaries using a two-phase (fluid–solid particle) approach. The physical concept and the mathematical background of the model are given and test-cases have been carried out to validate the proposed model. In order to illustrate its feasibility for a real estuary, the model has been applied to simulate the suspended-sediment transport and the formation of turbidity maximum in the Seine estuary. The numerical results show that the main characteristics of estuarine hydro-sediment dynamics in the Seine estuary are in fact reproduced by the proposed model. A qualitative agreement between the numerical results and the actual observations has been obtained and is presented in this paper.     

A three-dimensional model for tides and salinity in the Bristol Channel

A three-dimensional hydrodynamic numerical model is applied to the simulation of tides, salinity and density currents in the Bristol Channel. The model incorporates a sigma coordinate transformation through the vertical and a grid ☐ system throughout.A suitable formulation is chosen for vertical eddy viscosity by comparing elevations and currents from the model with observations. Two simulations of salinity are made, for February and July 1978. The resulting density current patterns for July 1978 are presented at various levels through the vertical.     

A coupled computational method for multi-solver, multi-domain transient problems in elastodynamics

This work presents an efficient and stable methodology for the coupling of Finite Element Methods (FEM) and Boundary Element Methods (BEM) that is independent of the particular solver and allows for independent temporal discretizations among solvers. The approach satisfies explicitly compatibility conditions and equilibrium of forces at the contact interfaces. Although the proposed approach has been developed in view of the soil-rail-vehicle dynamic interaction problem in High Speed Rail applications, it is expressed in a general form applicable to any multi-domain, multi-phase transient problem. The method development and formulations are presented in detail. Verification and application studies demonstrate the accuracy, efficiency and versatility of the method for the direct time domain solution of dynamic problems including structure-structure interaction and soil-structure interaction. The proposed approach demonstrates high accuracy and efficiency to that of direct coupling solutions and more rigorous methods.     

Analytical model for fully three‐dimensional radial dispersion in a finite‐thickness aquifer

Analytical solutions for contaminant transport in a non‐uniform flow filed are very difficult and relatively rare in subsurface hydrology. The difficulty is because of the fact that velocity vector in the non‐uniform flow field is space‐dependent rather than constant. In this study, an analytical model is presented for describing the three‐dimensional contaminant transport from an area source in a radial flow field which is a simplest case of the non‐uniform flow. The development of the analytical model is achieved by coupling the power series technique, the Laplace transform and the two finite Fourier cosine transform. The developed analytical model is examined by comparing with the Laplace transform finite difference (LTFD) solution. Excellent agreements between the developed analytical model and the numerical model certificate the accuracy of the developed model. The developed model can evaluate solution for Peclet number up to 100. Moreover, the mathematical behaviours of the developed solution are also studied. More specifically, a hypothetical convergent flow tracer test is considered as an illustrative example to demonstrate the three‐dimensional concentration distribution in a radial flow field. The developed model can serve as benchmark to check the more comprehensive three‐dimensional numerical solutions describing non‐uniform flow contaminant transport. Copyright © 2009 John Wiley & Sons, Ltd.     

A 2.5D coupled FE-BE model for the prediction of railway induced vibrations

Ground vibrations induced by railway traffic at grade and in tunnels are often studied by means of two-and-half dimensional (2.5D) models that are based on a Fourier transform of the coordinate in the longitudinal direction of the track. In this paper, the need for 2.5D coupled finite element-boundary element models is demonstrated in two cases where the prediction of railway induced vibrations is considered. A recently proposed novel 2.5D methodology is used where the finite element method is combined with a boundary element method, based on a regularized boundary integral equation. In the formulation of the boundary integral equation, Green's functions of a layered elastic halfspace are used, so that no discretization of the free surface or the layer interfaces is required. In the first case, two alternative models for a ballasted track on an embankment are compared. In the first model, the ballast and the embankment are modelled as a continuum using 2.5D solid elements, whereas a simplified beam representation is adopted in the second model. The free field vibrations predicted by both models are compared to those measured during a passage of the TGVA at a site in Reugny (France). A very large difference is found for the free field response of both models that is due to the fact that the deformation of the cross section of the embankment is disregarded in the simplified representation. In the second case, the track and free field response due to a harmonic load in a tunnel embedded in a layered halfspace are considered. A simplified methodology based on the use of the full space Green's function in the tunnel–soil interaction problem is investigated. It is shown that the rigorous finite element-boundary element method is required when the distance between the tunnel and the free surface and the layer interfaces of the halfspace is small compared to the wavelength in the soil.     

A model for tides and currents in the English Channel and southern North Sea

The amplitude and phase of 11 tidal constituents for the English Channel and southern North Sea are calculated using a frequency domain, finite element model. The governing equations — the shallow water equations — are modifed such that sea level is calculated using an elliptic equation of the Helmholz type followed by a back-calculation of velocity using the primitive momentum equations. Triangular elements with linear basis functions are used. The modified form of the governing equations provides stable solutions with little numerical noise. In this field-scale test problem, the model was able to produce the details of the structure of 11 tidal constituents including O₁, K₁, M₂, S₂, N₂, K₂, M₄, MS₄, MN₄, M₆, and 2MS₆.     

平面SV波在饱和土半空间中圆柱形孔洞周边的散射

在Biot饱和多孔介质动力学理论的基础上,利用Fourier—Bessel级数展开法,得到SV波在饱和土半空间中圆柱形孔洞周边的散射问题的解析解答。与已有相关问题的解析解答进行对比,验证了此解的正确性,并给出算例,分析了入射频率对柱面上的应力集中因子的影响。     

Analytical and numerical solutions for wave propagation in water-saturated porous layered half-space

A fluid-saturated one-layer continuum underlain by a rigid half-space is considered. An exact solution is developed in frequency domain for analyzing disturbance induced by a strip footing located at the surface with vertical harmonic excitation. Since the analytical solution can be used only for very simple conditions, a finite element model has been developed also and compared with the exact solution. It is shown that finite element results are in close agreement with the results which have been obtained by a transformation technique. The proposed finite element scheme can take into account the complex geometry and inhomogeneity for practical problems. Besides this, the analytical results exhibit the overall characteristic of wave propagation in porous media and will provide a representative test problem which can be used for a quantitative evaluation of the accuracy of various numerical solution methods.     

A two‐dimensional analytical solution for groundwater flow in a leaky confined aquifer system near open tidal water

Groundwater in coastal areas is commonly disturbed by tidal fluctuations. A two‐dimensional analytical solution is derived to describe the groundwater fluctuation in a leaky confined aquifer system near open tidal water under the assumption that the groundwater head in the confined aquifer fluctuates in response to sea tide whereas that of the overlying unconfined aquifer remains constant. The analytical solution presented here is an extension of the solution by Sun for two‐dimensional groundwater flow in a confined aquifer and the solution by Jiao and Tang for one‐dimensional groundwater flow in a leaky confined aquifer. The analytical solution is compared with a two‐dimensional finite difference solution. On the basis of the analytical solution, the groundwater head distribution in a leaky confined aquifer in response to tidal boundaries is examined and the influence of leakage on groundwater fluctuation is discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

Effects of relative motion between gas and liquid on 1-dimensional steady flow in silicic volcanic conduits: 1. An analytical method

We investigate the effects of vertical relative motion between gas and liquid on eruption styles by formulating a model for 1-dimensional steady flow in volcanic conduits. As magma ascends and decompresses, volatiles exsolve and volume fraction of gas increases. As a result, magma fragmentation occurs and the flow changes from bubbly flow to gas-pyroclast flow. In our model, a transitional region (‘permeable flow region’) is introduced between the bubbly flow region and the gas-pyroclast flow region. In this region, both the gas and the liquid are continuous phases, allowing the efficient vertical escape of gas through the permeable structure. We describe the features of conduit flow with relative motion of gas and liquid using non-dimensional numbers α, γ and ε. The parameter α represents the ratio of effects of wall friction to gravitational load, and is proportional to magma flow rate. The parameter γ represents the degree of decompression for the gas-pyroclast flow to reach the sound velocity at α = 1, and is proportional to r_c²/μ for given magma temperature and initial volatile content, where r_c is conduit radius and μ is liquid viscosity. The parameter ε is defined as the ratio of liquid–wall friction force to liquid–gas interaction force in the permeable flow region, and represents the efficiency of gas escape from magma. The values of γ and ε are determined only by magma properties and geological conditions such as liquid viscosity, magma permeability and conduit radius. We formulate a 1-dimensional steady-state conduit flow model to find non-dimensional magma flow rate α as a function of magma properties and geological conditions (e.g., γ and ε) under given boundary conditions. When the relative motion is taken into account with the assumption that magma fragmentation occurs when the gas volume fraction reaches some critical values, the pressure at the fragmentation level (P_f) decreases as the magma flow rate (α) decreases or the efficiency of gas escape (ε) increases, because gas escape suppresses the increase in the gas volume fraction accompanying magma ascent. When ε is so large that P_f is below the atmospheric pressure (P_a), the flow reaches the vent before fragmentation at low α. On the other hand, when ε is so small that P_f is greater than P_a, the flow reaches the vent after fragmentation at high α. These steady-state solutions of the flow at low and high α correspond to effusive and explosive eruptions, respectively. We present a graphical method to systematically find α. On the basis of the graphical method, a simple regime map showing the relationship between the assemblage of the solutions of conduit flow and the magma properties or the geological conditions is obtained.     

A convection-conduction model for analysis of the freeze-thaw conditions in the surrounding rock wall of a tunnel in permafrost regions

The results of simulated tidal current field, wave field and storm-induced current field are employed to interpret the depositional dynamic mechanism of formation and evolution of the radial sand ridges on the Yellow Sea door. The anticlockwise rotary tidal wave to the south of Shandong Peninsula meets the following progressive tidal wave from the South Yellow Sea, forming a radial current field outside Jianggang. This current field provides a necessary dynamic condition for the formation and existence of the radial sand ridges on the Yellow Sea seafloor. The results of simulated “old current field (holocene)” show that there existed a convergent-divergent tidal zone just outside the palaeo-Yangtze River estuary where a palaeo-underwater accumulation was developed. The calculated results from wave models indicate that the wave impact on the topography, under the condition of high water level and strong winds, is significant. The storm current induced by typhoons landing in the Yangtze River estuary and turning away to the sea can have an obvious influence, too, on the sand ridges. The depmitional dynamic mechanism of formation and evolution of the radial sand ridges on the Yellow Sea seafloor is “tidal current-induced formation—storm-induced chang—tidal current-induced recovery”. Project supported by the National Natural Science Foundation of China (Grant No. 49236120).