Beach water table fluctuations due to spring–neap tides: moving boundary effects

Tidal water table fluctuations in a coastal aquifer are driven by tides on a moving boundary that varies with the beach slope. One-dimensional models based on the Boussinesq equation are often used to analyse tidal signals in coastal aquifers. The moving boundary condition hinders analytical solutions to even the linearised Boussinesq equation. This paper presents a new perturbation approach to the problem that maintains the simplicity of the linearised one-dimensional Boussinesq model. Our method involves transforming the Boussinesq equation to an ADE (advection–diffusion equation) with an oscillating velocity. The perturbation method is applied to the propagation of spring–neap tides (a bichromatic tidal system with the fundamental frequencies ω₁andω₂) in the aquifer. The results demonstrate analytically, for the first time, that the moving boundary induces interactions between the two primary tidal oscillations, generating a slowly damped water table fluctuation of frequency ω₁−ω₂, i.e., the spring–neap tidal water table fluctuation. The analytical predictions are found to be consistent with recently published field observations.     

Two-dimensional approximation for tide-induced watertable fluctuations in a sloping sandy beach

The prediction of watertable fluctuations in a coastal aquifer is important for coastal management. However, most previous approaches have based on the one-dimensional Boussinesq equation, neglecting variations in the coastline and beach slope. In this paper, a closed-form analytical solution for a two-dimensional unconfined coastal aquifer bounded by a rhythmic coastline is derived. In the new model, the effect of beach slope is also included, a feature that has not been considered in previous two-dimensional approximations. Three small parameters, the shallow water parameter (ε), the amplitude parameter (α) and coastline parameter (β) are used in the perturbation approximation. The numerical results demonstrate the significant influence of both the coastline shape and beach slopes on tide-driven coastal groundwater fluctuations.     

A new analytical solution of tidal water table fluctuations in a coastal unconfined aquifer

This paper presents a new perturbation solution of the non-linear Boussinesq equation for one-dimensional tidal groundwater flow in a coastal unconfined aquifer. Built upon the work of Parlange et al. [Parlange, J.-Y., Stagnitti, F., Starr, J.L., Braddock, R.D., 1984. Free-surface flow in porous media and periodic solution of the shallow-flow approximation, J. Hydrol., 70, 251–263], the solution adopts a new perturbation parameter that is by definition less than unit, and thus is applicable to a wider range of physical conditions within the constraint of the Boussinesq approximation. This approach avoids a secular term in the third-order perturbation equation of Parlange et al. (1984), enabling the derivation of the third- and higher-order solutions. In comparison with a numerical (“exact”) solution, the new perturbation solution is shown to be slightly more accurate than that of Parlange et al. (1984) with the second-order approximation. The obtained third-order solution exhibits considerable improvement in accuracy. In relatively simple analytical forms, the present perturbation solution will help to understand better the non-linear characteristics of tidal water table fluctuations in as modeled by the non-linear Boussinesq equation coastal unconfined aquifers.     

New approximation for free surface flow of groundwater: capillarity correction

An existing capillarity correction for free surface groundwater flow as modelled by the Boussinesq equation is re-investigated. Existing solutions, based on the shallow flow expansion, have considered only the zeroth-order approximation. Here, a second-order capillarity correction to tide-induced watertable fluctuations in a coastal aquifer adjacent to a sloping beach is derived. A new definition of the capillarity correction is proposed for small capillary fringes, and a simplified solution is derived. Comparisons of the two models show that the simplified model can be used in most cases. The significant effects of higher-order capillarity corrections on tidal fluctuations in a sloping beach are also demonstrated.     

Tide-induced variability in beach surface moisture: Observations and modelling

The moisture content w_s of a beach surface strongly controls the availability of sand for aeolian transport. Our predictive capability of the spatiotemporal variability in w_s, which depends to a large extent on water table depth, is, however, limited. Here we show that water table fluctuations and surface moisture content observed during a 10-day period on a medium-grained (365μm) planar (1:30) beach can be predicted well with the nonlinear Boussinesq equation extended to include run-up infiltration and a soil–water retention curve under the assumption of hydrostatic equilibrium. On the intertidal part of the beach the water table is observed and predicted to continuously fall from the moment the beach surface emerges from the falling tide to just before it is submerged by the incoming tide. We find that on the lower 30% of the intertidal beach the water table remains within 0.1–0.2 m from the surface and that the sand is always saturated (w_s≈20%, by mass). Higher up on the intertidal beach, the surface can dry to about 5% when the water table has fallen to 0.4–0.5 m beneath the surface. Above the high-tide level the water table is always too deep (>0.5 m) to affect surface moisture and, without precipitation, the sand is dry (w_s < 5 − 8%). Because the water table depth on the emerged part of the intertidal beach increases with time irrespective of whether the (ocean) tide falls or rises, we find no need to include hysteresis (wetting and drying) effects in the surface-moisture modelling. Model simulations suggest that at the present planar beach only the part well above mean sea level can dry sufficiently (w_s < 10%) for sand to become available for aeolian transport. ©2018 The Authors. Earth Surface Processes and Landforms published by John Wiley & Sons Ltd.     

Modelling of groundwater mound formation resulting from transient recharge

An analytical solution of a linearized Boussinesq equation is obtained to predict water table fluctuations as a result of time varying recharge from a strip basin for any number of recharge cycles. The analytical solution is obtained by using finite Fourier sine transform. Applications of the solution for the prediction of water table fluctuations and sensitivity analysis are demonstrated with the help of example problems. Copyright © 2001 John Wiley & Sons, Ltd.     

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Abstract

This paper develops a new analytical solution for the aquifer system, which comprises an unconfined aquifer on the top, a semi-confined aquifer at the bottom and an aquitard between them. This new solution is derived from the Boussinesq equation for the unconfined aquifer and one-dimensional leaky confined flow equation for the lower aquifer using the perturbation method, considering the water table over-height at the remote boundary. The head fluctuation predicted from this solution is generally greater than the one solved from the linearized Boussinesq equation when the ratio of the tidal amplitude to the thickness of unconfined aquifer is large. It is found that both submarine groundwater discharges from upper and lower aquifers increase with tidal amplitude–aquifer thickness ratio and may be underestimated if the discharge is calculated based on the average head fluctuation. The effects of the aquifer parameters and linearization of the Boussinesq equation on the normalized head fluctuation are also investigated.

Editor D. Koutsoyiannis; Associate editor J. Simunek

Citation Chuang, M.-H., Mahdi, A.-A. and Yeh, H.-D., 2012. A perturbation solution for head fluctuations in a coastal leaky aquifer system considering water table over-height. Hydrological Sciences Journal, 57 (1), 162–172.     

Modelling coastal ground‐ and surface‐water interactions using an integrated approach

Beach water table fluctuations have an impact on the transport of beach sediments and the exchange of solute and mass between coastal aquifer and nearby water bodies. Details are given of the refinement of a dynamically integrated ground‐ and surface‐water model, and its application to study ground‐ and surface‐water interactions in coastal regions. The depth‐integrated shallow‐water equations are used to represent the surface‐water flow, and the extended Darcy's equation is used to represent the groundwater flow, with a hydrostatic pressure distribution being assumed to apply for both these two types of flows. At the intertidal region, the model has two layers, with the surface‐water layer being located on the top of the groundwater layer. The governing equations for these two types of flows are discretized in a similar manner and they are combined to give one set of linear algebraic equations that can be solved efficiently. The model is used to predict water level distributions across sloping beaches, where the water table in the aquifer may or may not decouple from the free water surface. Five cases are used to test the model for simulating beach water table fluctuations induced by tides, with the model predictions being compared with existing analytical solutions and laboratory and field data published in the literature. The numerical model results show that the integrated model is capable of simulating the combined ground‐ and surface‐water flows in coastal areas. Detailed analysis is undertaken to investigate the capability of the model. Copyright © 2009 John Wiley & Sons, Ltd.     

Explicit solution of a free boundary problem for a nonlinear absorption model of mixed saturated-unsaturated flow

In wet soils, zones of saturation naturally develop in the vicinity of impermeable strata, surface ponds and subterranean cavities. Hydrology must be then concerned with transient flow through coexisting unsaturated and saturated zones. The models of advancing saturated zones necessarily involve a nonlinear free boundary problem.A closed-form analytic solution is presented for a nonlinear diffusion model under conditions of ponding at the surface. The soil water diffusivity is restricted to the special functional form D(θ) = a/(b − θ)², where θ is the water content field to be determined and a, b are positive constants. The explicit solution depends on a parameter C (determined by the data of the problem), according to two cases: 1 < C < C₁ or C ≥ C₁, where C₁ is a constant which is obtained as the unique solution of an equation. This result complements the study given in P. Broadbridge, Water Resources Research, 1990, 26, 2435–2443, in order to established when the explicit solution is available. The behavior of the bifurcation parameter C₁ as a function of the driving potential is studied with the corresponding limits for small and large values. Moreover, the sorptivity is proven to be continuously differentiable function of the variable C.     

A finite element solution for the fractional advection–dispersion equation

The fractional advection–dispersion equation (FADE) known as its non-local dispersion, has been proven to be a promising tool to simulate anomalous solute transport in groundwater. We present an unconditionally stable finite element (FEM) approach to solve the one-dimensional FADE based on the Caputo definition of the fractional derivative with considering its singularity at the boundaries. The stability and accuracy of the FEM solution is verified against the analytical solution, and the sensitivity of the FEM solution to the fractional order α and the skewness parameter β is analyzed. We find that the proposed numerical approach converge to the numerical solution of the advection–dispersion equation (ADE) as the fractional order α equals 2. The problem caused by using the first- or third-kind boundary with an integral-order derivative at the inlet is remedied by using the third-kind boundary with a fractional-order derivative there. The problems for concentration estimation at boundaries caused by the singularity of the fractional derivative can be solved by using the concept of transition probability conservation. The FEM solution of this study has smaller numerical dispersion than that of the FD solution by Meerschaert and Tadjeran (J Comput Appl Math 2004). For a given α, the spatial distribution of concentration exhibits a symmetric non-Fickian behavior when β = 0. The spatial distribution of concentration shows a Fickian behavior on the left-hand side of the spatial domain and a notable non-Fickian behavior on the right-hand side of the spatial domain when β = 1, whereas when β = −1 the spatial distribution of concentration is the opposite of that of β = 1. Finally, the numerical approach is applied to simulate the atrazine transport in a saturated soil column and the results indicat that the FEM solution of the FADE could better simulate the atrazine transport process than that of the ADE, especially at the tail of the breakthrough curves.     

Tidal effects on groundwater dynamics in unconfined aquifers

The variation of seawater level resulting from tidal fluctuations is usually neglected in regional groundwater flow studies. Although the tidal oscillation is damped near the shoreline, there is a quasi‐steady‐state rise in the mean water‐table position, which may have an influence on regional groundwater flow. In this paper the effects of tidal fluctuations on groundwater hydraulics are investigated using a variably saturated numerical model that includes the effects of a realistic mild beach slope, seepage face and the unsaturated zone. In particular the impact of these factors on the velocity field in the aquifer is assessed. Simulations show that the tidal fluctuation has substantial consequences for the local velocity field in the vicinity of the exit face, which affects the nearshore migration of contaminant in coastal aquifers. An overheight in the water table as a result of the tidal fluctuation is observed and this has a significant effect on groundwater discharge to the sea when the landward boundary condition is a constant water level. The effect of beach slope is very significant and simplifying the problem by considering a vertical beach face causes serious errors in predicting the water‐table position and the groundwater flux. For media with a high effective capillary fringe, the moisture retained above the water table is important in determining the effects of the tidal fluctuations. Copyright © 2001 John Wiley & Sons, Ltd.     

Higher order analytical solutions of water table fluctuations in coastal aquifers

The groundwater response of coastal aquifers to tidal forcing is described by Laplace's equation coupled with the nonlinear phreatic-free surface boundary condition. Here we describe fluctuations in the water table using two small parameters, extending previous work by proposing an ansatz to compute higher order, semi-analytical solutions. The new solutions are compared with known lower order solutions. The relative difference between the linear solution and higher order solutions can reach up to 30% of the linear solution for shallow beaches. The new solutions exhibit a reduction in the over height of the groundwater fluctuations compared with the lower order solutions. In addition, the super elevation of the water table, both near shore and as the aquifer tends landward, is examined with the inclusion of higher order terms.     

Some results on the approach of a slowly-varyng gravity wave to a beach

Abstract

Results from the theory of slowly-varying solutions of the non-linear shallow water equations (Varley et al. (1971)) are used to estimate maximum wave height amplifications on a beach of arbitrary shape. Carrier's solution (1966) for a large disturbance out at sea is used as input. Generalizations of the slowly-varying results to non-plane situations are also presented.     

Modelling of water table variation in response to time-varying recharge from multiple basins using the linearised Boussinesq equation

A mathematical model is presented to describe the variations of the water table in an unconfined aquifer due to time-varying recharge applied from four rectangular basins. The model is developed by solving the linearised Boussinesq equation using the extended finite Fourier cosine transform. The time-varying recharge rate is approximated by a number of piecewise linear elements of different lengths and slopes depending on the nature of the variation in recharge rate. Application of this model for the prediction of water table fluctuations and in the sensitivity analysis of various controlling parameters on the aquifer response is demonstrated in an example.     

Orographic controls on rain water isotope distribution in the Mount Lofty Ranges of South Australia

Hydrogen and oxygen isotopes of water are common environmental tracers used to investigate hydrological processes, such as evaporation, vegetation water use, surface water–groundwater interaction, and groundwater recharge. The water isotope signature in surface water and groundwater evolves from the initial rain signature. In mountain terrain, rain water stable isotope composition spatially varies due to complex orographic precipitation processes. Many studies have examined the isotope–elevation relationships, while few have quantitatively investigate the terrain aspect and slope effect on rain isotope distribution. In this paper, we examine the orographic effects more completely, including elevation, terrain slope and aspect, on stable isotope distribution in the Mount Lofty Ranges (MLR) of South Australia, using a multivariate regression model. The regression of precipitation isotope composition suggests that orographic effects are the dominant controls on isotope spatial variability. About 75% of spatial variability in δ¹⁸O and deuterium excess is represented by the regression using solely orography-related variables (elevation, terrain aspect and slope), with about 25% of δ¹⁸O spatial variability attributed to the terrain aspect and slope effect. The lapse rate is about −0.25‰ for every 100 m at both windward and leeward slopes. However, at the same elevation, δ¹⁸O at the leeward slope (eastern MLR) is 0.5‰ larger than that at the windward slope. The difference can be explained by different mechanisms – continuous rain-out processes on the windward side and sub-cloud evaporation on the leeward side. Both δ¹⁸O and deuterium excess maps (1 km resolution) are constructed based on the regression results for the MLR. Both maps are consistent with groundwater of local precipitation origin, and useful to examine groundwater recharge.     

Water–Rock interactions in the basement beneath Long Valley Caldera: an oxygen isotope study of the Long Valley Exploratory Well drill cores

The Long Valley Exploratory Well, at the center of the Resurgent Dome of Long Valley caldera, penetrated pre-caldera basement rocks at a depth of 2101.72–2313.0 m, beneath the caldera-forming Bishop Tuff and post-caldera Early Rhyolite. The basement rocks contain prominent quartzites, with ubiquitous milky white quartz veins (with minor calcite and pyrite) and fractures of varied orientation and geometry. The other members of the basement sequence are very fine-grained quartz-rich graphitic pelites with calcite veins, spotted hornfels, and shallow intrusive rocks. Previous studies established the presence of a post-caldera, paleohydrothermal system (500–100 ka) to a depth of 2000 m that affected the Bishop Tuff and a recent (40 ka to present) hydrothermal system at shallow depth (<1 km). The deeper extent of these hydrothermal activities is established in this paper by a detailed oxygen isotope analysis of the drill core samples. 238 analyses of δ¹⁸O in 50 quartz veins within the 163.57 m depth interval of basement rocks reveal extreme heterogeneity in δ¹⁸O values (8–19.5‰). Majorities of the 84 bulk analyses of quartzites show variation of δ¹⁸O within a narrow range of 14–16‰. However, certain samples of these quartzites near the contacts with veins and fractures exhibit sharp drops in δ¹⁸O. The interbedded pelitic rocks and spotted hornfels have whole-rock δ¹⁸O ranging from 2.2 to 11.8‰. Clear, euhedral vuggy quartz that partially fills earlier open fractures in both the quartzites and quartz veins, has distinctive δ¹⁸O, ranging between −3.2 and +8.4‰. Low values of δ¹⁸O are also found in the hydrothermal minerals and whole rocks adjacent to the thin veins, clearly indicating infiltration of meteoric water. Three distinct observed patterns of fractionation in δ¹⁸O between veins and host quartzites are analyzed with the principles of mass balance, equilibrium oxygen isotope fractionation in closed system, and kinetically controlled oxygen isotope exchange in an open system. This analysis suggests that the early quartz veins formed due to a magmatic-hydrothermal activity with no influx of external water once the system comprising the sedimentary envelope and a magmatic-hydrothermal fluid phase became closed. Two-stage isotopic exchange processes caused fractionation in the δ values that originally formed arrays with slope 1 in a δ_{vein quartz}–δ_{host quartzite} space. Another array in the same space, with near zero slope was also formed due to variation in temperature, initial isotopic compositions of the quartzite sequence and the fluid phase. Variation in temperature was mostly in the range of 300–400°C giving Δ (=δ_{vein quartz}–δ_{host quartzite})≈−2.8 to +2.8. The δ¹⁸O of the fluid could range from −5 to +10; however a narrower range of +5 to +10 can explain the data. This episode of hydrothermal activity could take place either as a single pulse or in multiple pulses but each as a closed system. A later, fracture-controlled, meteoric water (δ¹⁸O−0.46 to −12.13) flow and interaction (at 250°C) is interpreted from the analysis of δ¹⁸O values of the coexisting quartz and calcite pairs and existence of markedly ¹⁸O-depleted pelitic horizons interbedded with ¹⁸O-enriched quartzite layers. Thus, the interpreted earlier magmatic-hydrothermal activity was overprinted by a later meteoric-hydrothermal activity that resulted in steep arrays of δ¹⁸O values in the δ_{vein quartz}–δ_{host quartzite} space. Calculations show that the likely life span of the post-caldera, hydrothermal activity in the depth range of 2.1–2.3 km beneath Long Valley was 0.08–0.12 Ma. Diffusive ±advective transport of oxygen isotopes from fracture-channelized meteoric water to nearly impermeable wall rocks caused a lowering of δ¹⁸O values in the quartz over short distances and in calcites over greater distances. Thus, the hydrothermal activity appears pervasive even though the meteoric water flow was primarily controlled by fractures.     

The effects of alongshore variation in bottom topography on a boundary current—(topographically induced upwelling)

A theory which describes the constant f-plane flow of a steady inviscid baroclinic boundary current over a continental margin with a bathymetry that varies slowly in the alongshore but rapidly in the offshore directions is developed in the parameter regime (L_D/L)² ≤ Ro 1, where L_D is the internal deformation radius, L the horizontal length scale, and Ro the Rossby number. To lowest order in the Rossby number the flow is along isobaths with speed q_o = V_u(h,z)|Vh|/α, where V_u(h,z) is the upstream speed, α the upstream bottom slope at depth h, and Vh the bottom slope downstream at depth h. The lowest order flow produces a variation in the vertical component of relative vorticity along the isobath as the magnitude and direction of Vh vary in the downstream direction. The variation of vorticity requires a vertical as well as a cross-isobath flow at first order in the Rossby number. The first order vertical velocity is computed from the vorticity equation in terms of upstream conditions and downstream variations of the bathymetry. The density, pressure, and cross-isobath flow at first order in the Rossby number are then calculated. It is shown that in the cyclonic region of current (d/dh(V_u/α) > 0), if the isobaths diverge in the downstream direction ((∂/∂s)|Vh| < 0), then upwelling and onshore flow occur. The theory is applied to the northeastern Florida shelf to explain bottom temperature observations.     

Internal soil moisture and piezometric responses to rainfall-induced shallow slope failures

Better knowledge regarding internal soil moisture and piezometric responses in the process of rainfall-induced shallow slope failures is the key to an effective prediction of the landslide and/or debris flow initiation. To this end, internal soil moisture and piezometric response of 0.7-m-deep, 1.5-m-wide, 1.7-m-high, and 3.94-m-long semi-infinite sandy slopes rested on a bi-linear impermeable bedrock were explored using a chute test facility with artificial rainfall applications. The internal response time defined by the inflection point of the soil moisture and piezometric response curves obtained along the soil–bedrock interface were closely related to some critical failure states, such as the slope toe failure and extensive slope failures. It was also found that the response times obtained at the point of abrupt bedrock slope decrease can be used as indicators for the initiation of rainfall-induced shallow slope failures. An investigation of spatial distributions of soil water content, ω (or degrees of saturation, S_r), in the slope at critical failure states shows that the 0.2 m – below – surface zone remains unsaturated with S_r 40–60%, regardless of their distances from the toe and the rainfall intensity. Non-uniform distributions of ω (or S_r) along the soil–bedrock interface at critical failure states were always associated with near-saturation states (S_r 80–100%) around the point of bedrock slope change or around the transient ‘toe’ upstream of the slumped mass induced by the retrogressive failure of the slope. These observations suggest the important role of the interflow along the soil–bedrock interface and the high soil water content (or high porewater pressure) around the point of bedrock slope deflection in the rainfall-induced failure of sandy slopes consisting of shallow impermeable bedrocks. The present study proposes an ‘internal response time’ criterion to substantiate the prediction of rainfall-induced shallow slope failures. It is believed that the ‘internal response time’ reflects the overall characteristics of a slope under rainfall infiltration and can be as useful as the conventional meteorology-based threshold times. The ‘internal response time’ theory can be generalized via numerical modeling of slope hydrology, slope geology and slope stability in the future.     

An alternative Boussinesq equation considering the effect of hysteresis on coastal groundwater waves

This paper presents an alternative Boussinesq equation considering hysteresis effect via a third‐order derivative term. By introducing an improved moisture–pressure retention function, this equation describes, with reasonable precision, groundwater propagation in coastal aquifers subject to Dirichlet boundary condition of different oscillation frequencies. Test results confirmed that it is necessary to consider horizontal and vertical flows in unsaturated zone, because of their variable influences on hysteresis. Hysteresis in unsaturated zone can affect the water table wave number of groundwater wave motion, such as wave damping rate and phase lag. Oscillations with different periods exert different hysteresis effect on wave propagation. Truncation/shrinkage of unsaturated zones also affects the strength of hysteresis. These impacts can be reflected in the alternative Boussinesq equation by adjusting the parameter representing the variation rate of moisture associated with pressure change, as opposed to traditional computationally expensive hysteresis algorithms. The present Boussinesq equation is simple to use and can provide feasible basis for future coupling of groundwater and surface water models. Copyright © 2016 John Wiley & Sons, Ltd.     

Eddy energetics in the Faroe-Shetland channel: a model resolution study

Observations indicate that, perhaps as a result of intense shears produced by the topographic steering of the slope current and merging of the warm North Atlantic water with cold Nordic waters, the Faroe-Shetland Channel is a fertile ground for energetic eddies and meanders. How these meso-scale ( 10 km) features are resolved in a three-dimensional numerical model is studied here with grid-nesting, in which the convergence of the simulated energetics is examined as a function of grid sizes Δ =10, 10/3, 10/5 and 10/7 km. Large-scale forcing to the fine-grid nest consists of steady slope-current transport (2 Sv) and north/south climatological density difference only. The mean fields were found to converge at Δ = 10/3 km, but the eddy fields required Δ = 2 km or less. At the finest grids, meso-scale features more in conformity with those observed were found: meanders with amplitudes of O(10 km) and along-slope propagation speeds of about 0.14 m s- 1 cross-channel vacillation of the slope current, with amplitude of about 10 km and periods of 5–10 days, and vertical excursions of isotherms of O(100 m). The meanders were highly ageostrophic (ζ/f> 1, ζ=relative vorticity, f = Coriolis parameter), and developed as a result of conditions favorable to flow instability originated from south of the channel. When adequately resolved, the meander energetics are comparable to those produced by (large-scale) winds, which suggests that a high-resolution grid (Δ < 2 km) will be required for hindcast and forecast studies.