Polynomial approximate solutions to the Boussinesq equation

We provide closed-form approximate solutions to models of horizontal infiltration described by the Boussinesq equation in a semi-infinite aquifer that is initially dry. The approximations preserve such important qualitative properties as scaling and wetting fronts. They are applicable to four types of boundary conditions, two on head and two on flux, enumerated in the paper. All the considered problems admit self-similar variables that allow reduction to boundary value problems for a nonlinear ordinary differential equation. This work extends recent results by Lockington et al. [Lockington DA, Parlange J-Y, Parlange MB, Selker J. Similarity solution of the Boussinesq equation. Adv Water Resour 2000;23(7):725–9] and Telyakovskiy et al. [Telyakovskiy AS, Braga GA, Furtado F. Approximate similarity solutions to the Boussinesq equation. Adv Water Resour 2002;25(2):191–4], with new approximations developed for two of the four cases and a new extension of a previously existing method for a third case. Numerical results extending the work of Shampine [Shampine LF. Some singular concentration dependent diffusion problems. ZAMM 1973;53:421–2] provide a basis for assessing the accuracy of the new methods.     

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.     

Similarity solution of axisymmetric flow in porous media

Applications of the axisymmetric Boussinesq equation to groundwater hydrology and reservoir engineering have long been recognised. An archetypal example is invasion by drilling fluid into a permeable bed where there is initially no such fluid present, a circumstance of some importance in the oil industry. It is well known that the governing Boussinesq model can be reduced to a nonlinear ordinary differential equation using a similarity variable, a transformation that is valid for a certain time-dependent flux at the origin. Here, a new analytical approximation is obtained for this case. The new solution,, which has a simple form, is demonstrated to be highly accurate.     

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.     

The Boussinesq and anelastic liquid approximations for convection in the Earth’s core

Convection in the Earth’s core is usually studied in the Boussinesq approximation in which the compressibility of the liquid is ignored. The density of the Earth’s core varies from ICB to CMB by approximately 20%. The question of whether we need to take this variation into account in core convection and dynamo models is examined. We show that it is in the thermodynamic equations that differences between compressible and Boussinesq models become most apparent. The heat flux conducted down the adiabat is much smaller near the inner core boundary than it is near the core-mantle boundary. In consequence, the heat flux carried by convection is much larger nearer the inner core boundary than it is near the core-mantle boundary. This effect will have an important influence on dynamo models. Boussinesq models also assume implicitly that the rate of working of the gravitational and buoyancy forces, as well as the Ohmic and viscous dissipation, are small compared to the heat flux through the core. These terms are not negligible in the Earth’s core heat budget, and neglecting them makes it difficult to get a thermodynamically consistent picture of core convection. We show that the usual anelastic equations simplify considerably if the anelastic liquid approximation, valid if αT?1, where α is the coefficient of expansion and T a typical core temperature, is used. The resulting set of equations are not significantly more difficult to solve numerically than the usual Boussinesq equations. The relationship of our anelastic liquid equations to the Boussinesq equations is also examined.     

A process‐based transfer function approach to model tile‐drain hydrographs

Tile‐drain response to rainfall events is determined by unsaturated vertical flow to the water table, followed by horizontal saturated water movement. In this study, unsaturated vertical movement from the redistribution of water is modelled using a sharp‐front approximation, and the saturated horizontal flow is modelled by an approximate solution to the Boussinesq equation. The unsaturated flow component models the fast response that is associated with the presence of preferential flow paths. By convoluting the responses of the two components, a transfer function is developed that predicts tile‐drain response to unit amounts of infiltrated water. It is observed that the unsaturated flow component can be cast in a form that is linear in a power function of the infiltrated depth. Since the approach is process based, model parameter definitions are easily identified with soil properties at the field scale. Furthermore, it is demonstrated that the transfer function model parameters can be estimated from moment analysis. Using superposition, the transient tile‐drain response to arbitrary amounts of infiltrated water can be constructed. Comparison with data measured from the Water Quality Field Station show that this approach provides a promising method for generating tile‐drain response to rainfall events. Copyright © 2006 John Wiley & Sons, Ltd.     

On the upper bounding approach to thermal convection at moderate rayleigh numbers

Abstract

Two upper bounding problems for thermal convection in a layer of fluid contained between perfectly conducting stress-free boundaries are treated numerically. Since the Euler equations resulting from this variational approach are simpler than the Navier-Stokes equations, they allow numerical calculations to be carried out economically to fairly large values of the Rayleigh number. The upper bounding problem formulated by Howard (1963), which yields a Nusselt number independent of Prandtl number, diverges from the correct behavior as the Rayleigh number increases. In hopes of coming closer to results of previous investigations of the Boussinesq equations of motion, a more restrictive upper bounding problem is formulated. For large Prandtl numbers the momentum equation is linearized and is used as an explicit side constraint on the variational problem, thereby forcing the solutions to more closely resemble the solutions of the Boussinesq equations. Numerical calculations at values of the Rayleigh number up to 1.5 × 10⁵ indicate that the additional constraint decreases the upper bound on the Nusselt number; it appears that this upper bound differs by only a multiplicative factor from that calculated from solutions of the full equations of motion and may be a reasonable approximation for large Rayleigh numbers.     

Incorporating velocity shear into the magneto-Boussinesq approximation

Motivated by consideration of the solar tachocline, we derive, via an asymptotic procedure, a new set of equations incorporating velocity shear and magnetic buoyancy into the Boussinesq approximation. We demonstrate, by increasing the magnetic field scale height, how these equations are linked to the magneto-Boussinesq equations of Spiegel and Weiss (Magnetic buoyancy and the Boussinesq approximation. Geophys. Astrophys. Fluid Dyn. 1982, 22, 219–234).     

Polynomial-based approximate solutions to the Boussinesq equation near a well

This paper presents a method for constructing polynomial-based approximate solutions to the Boussinesq equation with cylindrical symmetry. This equation models water injection at a single well in an unconfined aquifer; as a sample problem we examine recharge of an initially empty aquifer. For certain injection regimes it is possible to introduce similarity variables, reducing the original problem to a boundary-value problem for an ordinary differential equation. The approximate solutions introduced here incorporate both a singular part to model the behavior near the well and a polynomial part to model the behavior in the far field. Although the nonlinearity of the problem prevents decoupling of the singular and polynomial parts, the paper presents an approach for calculating the solution based on its spatial moments. This approach yields closed-form expressions for the position of the wetting front and for the form of the phreatic surface. Comparison with a highly accurate numerical solution verifies the accuracy of the newly derived approximate solutions.     

Nonlinear aspects of phreatic water filtration

Horizontal gravity filtration of groundwater in soil is considered. Under Boussinesq approximation, the problem is reduced to a one-dimensional nonlinear parabolic equation in phreatic water level. The problem of linearizing the original equation is discussed. The comparison of gravity-filtration problem solutions in the nonlinear and linearized formulations shows considerable discrepancies to exist between the solutions, especially, for boundary problems with mixed boundary conditions, when the value of the function is not fixed on the right boundary. An analytical solution is obtained for steady-state flow from a water body into the soil with subsequent leakage into underlying beds. Two regimes are shown to exist: one with an infinite exponential tail, and another in the form of a finite groundwater mound. A new approach is proposed to the linearization problem—quasilinearization with the use of the Burgers equation.     

Properties of low‐frequency trapped mode in viscous‐fluid waveguides

We derived the velocity and attenuation of a generalized Stoneley wave being a symmetric trapped mode of a layer filled with a Newtonian fluid and embedded into either a poroelastic or a purely elastic rock. The dispersion relation corresponding to a linearized Navier–Stokes equation in a fracture coupling to either Biot or elasticity equations in the rock via proper boundary conditions was rigorously derived. A cubic equation for wavenumber was found that provides a rather precise analytical approximation of the full dispersion relation, in the frequency range of 10^?3 Hz to 10³ Hz and for layer width of less than 10 cm and fluid viscosity below 0.1 Pa· s [100 cP]. We compared our results to earlier results addressing viscous fluid in either porous rocks with a rigid matrix or in a purely elastic rock, and our formulae are found to better match the numerical solution, especially regarding attenuation. The computed attenuation was used to demonstrate detectability of fracture tip reflections at wellbore, for a range of fracture lengths and apertures, pulse frequencies, and fluid viscosity.     

Using MODFLOW code to approach transient hydraulic head with a sharp‐interface solution

Saltwater intrusion problems have been usually tackled through analytical models because of its simplicity, easy implementation and low computational cost. Most of these models are based on the sharp‐interface approximation and the Ghyben–Herzberg relation, which neglects mixing of fresh water and seawater and implicitly assumes that salt water remains static. This paper provides insight into the validity of a sharp‐interface approximation defined from a steady state solution when applied to transient seawater intrusion problems. The validation tests have been performed on a 3D unconfined synthetic aquifer, which include spatial and temporal distribution of recharge and pumping wells. Using a change of variable, the governing equation of the steady state sharp‐interface problem can be written with the same structure of the steady confined groundwater flow equation as a function of a single potential variable (?). We propose to approach also the transient problem solving a single potential equation (using also the ? variable) with the same structure of the confined groundwater flow equation. It will allow solving the problem by using the classical MODFLOW code. We have used the parameter estimation model PEST to calibrate the parameters of the transient sharp‐interface equation. We show how after the calibration process, the sharp‐interface approach may provide accurate enough results when applied to transient problems and improve the steady state results, thus avoiding the need of implementing a density‐dependent model and reducing the computational cost. This has been proved by comparing results with those obtained using the finite difference numerical code SEAWAT for solving the coupled partial differential equations of flow and density‐dependent transport. The comparison was performed in terms of piezometric heads, seawater penetration, transition zone width and critical pumping rates. Copyright © 2014 John Wiley & Sons, Ltd.     

T-matrix approach to seismic forward modelling in the acoustic approximation

Forward seismic modelling in the acoustic approximation, for variable velocity but constant density, is dealt with. The wave equation and the boundary conditions are represented by a volume integral equation of the Lippmann-Schwinger (LS) or Fredholm type. A T-matrix (or transition operator) approach from quantum mechanical potential scattering theory is used to derive a family of linear and nonlinear approximations (cluster expansions), as well as an exact numerical solution of the LS equation. For models of 4D anomalies involving small or moderate contrasts, the Born approximation gives identical numerical results as the first-order t-matrix approximation, but the predictions of an exact T-matrix solution can be quite different (depending on spatial extention of the perturbations). For models of fluid-saturated cavities involving large or huge contrasts, the first-order t-matrix approximation is much more accurate than the Born approximation, although it does not lead to significantly more time-consuming computations. If the spatial extention of the perturbations is not too large, it is practical to use the exact T-matrix solution which allows for arbitrary contrasts and includes all the effects of multiple scattering.     

NEW TRENDS IN THE NUMERICAL STUDY OF COASTAL SEDIMENTARY PROCESSES

1 INTRODUCTION The conventional method for numerical study of coastal sedimentary processes has been based on the solution of a phase-averaged wave equation, the steady nearshore current equations, the continuity equation for sediment, and an empirical net sediment transport equation. This strategy has advantages in meso-scale problems but it loses accuracy in the vicinity of a structure, where local scour usually occurs. To have a good representation of both meso-scale sediment transport …     

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.     

基于流变体模型的波恩近似方程地震正演模拟

常规τ值法假设应力松弛时间与应变延迟时间近似相等, 造成了常Q模型拟合精度低. 本文利用精确的广义流变体模型Q值计算公式, 研究改进的τ值法求解常Q模型参数. 根据地震波散射理论, 推导了基于广义流变体模型的黏滞性介质一阶波恩近似方程, 结合位移-速度关系得到了含卷积完全匹配层边界条件的黏滞性介质应力-速度方程的一阶波恩近似表达式. 通过数值实验验证并对比了黏滞性介质中全波波动方程、 一阶波恩近似方程以及单程波波动方程的波场特征, 讨论了基于流变体模型的黏滞性介质一阶波恩近似方程对速度扰动和Q扰动的适应性, 以及对旅行时和振幅精度的影响.      

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.     

Stability of the subseismic wave equation for the Earth's fluid core

Abstract

The effects of compressibility on the stability of internal oscillations in the Earth's fluid core are examined in the context of the subseismic approximation for the equations of motion describing a rotating, stratified, self-gravitating, compressible fluid in a thick shell. It is shown that in the case of a bounded fluid the results are closely analogous to those derived under the Boussinesq approximation.     

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.