Analytical Solution of a Walters’ Liquid B Flow Over a Linear Stretching Sheet in a Porous Medium

This chapter represents the analytical solution of two-dimensional linear stretching sheet problem involving a non-Newtonian liquid and suction by (a) invoking the boundary layer approximation and (b) using this result to solve the stretching sheet problem without using boundary layer approximation. The basic boundary layer equations for momentum, which are non-linear partial differential equations, are converted into non-linear ordinary differential equations by means of similarity transformation. The results reveal a new analytical procedure for solving the boundary layer equations arising in a linear stretching sheet problem involving a non-Newtonian liquid (Walters’ liquid B). The present study throws light on the analytical solution of a class of boundary layer equations arising in the stretching sheet problem.

Analytical solution of unsteady three-dimensional MHD boundary layer flow and heat transfer due to impulsively stretched plane surface

The unsteady magnetohydrodynamic viscous flow and heat transfer of Newtonian fluids induced by an impulsively stretched plane surface in two lateral directions are studied by using an analytic technique, namely, the homotopy method. The analytic series solution presented here is highly accurate and uniformly valid for all time in the entire region. The effects of the stretching ratio and the magnetic field on the surface shear stresses and heat transfer are studied. The surface shear stresses in x- and y-directions and the surface heat transfer are enchanced by increasing stretching ratio for a fixed value of the magnetic parameter. For a fixed stretching ratio, the surface shear stresses increase with the magnetic parameter, but the heat transfer decreases. The Nusselt number takes longer time to reach the steady state than the skin friction coefficients. There is a smooth transition from the initial unsteady state to the steady state.

Analytical solution of the laminar mean flow wave equation in a lined or unlined two-dimensional rectangular duct

The prediction of the sound attenuation in lined ducts with sheared mean flow has been a topic of research for many years. This involves solving the sheared mean flow wave equation, satisfying the relevant boundary condition. As far as the authors' knowledge goes, this has always been done using numerical techniques. Here, an analytical solution is presented for the wave propagation in two-dimensional rectangular lined ducts with laminar mean flow. The effect of laminar mean flow is studied for both the downstream and the upstream wave propagation. The attenuation values predicted for the laminar mean flow case are compared with those for the case of uniform mean flow. Analytical expressions are derived for the transfer matrices.

Bearing capacity of foundations subjected to groundwater flow

The ultimate bearing capacity of strip foundations subjected to horizontal groundwater flow has been computed by making use of the stress characteristics method which is well known for its capability in solving quite accurately different stability problems in geotechnical engineering. The numerical solution has been generated both for smooth and rough footings placed on frictional soils. A correction factor (fγ) associated with Nγ term, to account for the existence of ground water flow, has been introduced. The variation of fγ has been obtained as a function of hydraulic gradient (i) for different values of soil frictional angle. The magnitude of fγ reduces continuously with an increase in the value of i.

An analytical solution to the optimal power flow

This letter presents an approach for a geometrical solution of an optimal power flow (OPF) problem for a two-bus system (slack and PV busses). The algebraic equations for the calculation of the Lagrange multipliers and for the minimum losses value are obtained. These equations are used to validate the results obtained using an OPF program.

Analytical solution for transient one-dimensional couette flow considering constant and time-dependent pressure gradients

This paperaims to determine the velocity profile, in transient state, for a parallel incompressible flow known as Couette flow. The Navier-Stokes equations were applied upon this flow. Analytical solutions, based in Fourier series and integral transforms, were obtained for the one-dimensional transient Couette flow, taking into account constant and time-dependent pressure gradients acting on the fluid since the same instant when the plate starts it´s movement. Taking advantage of the orthogonality and superposition properties solutions were foundfor both considered cases. Considering a time-dependent pressure gradient, it was found a general solution for the Couette flow for a particular time function. It was found that the solution for a time-dependent pressure gradient includes the solutions for a zero pressure gradient and for a constant pressure gradient.

Model reduction of stochastic groundwater flow and transport processes

This work presents a comprehensive methodology for the reduction of analytical or numerical stochastic models characterized by uncertain input parameters or boundary conditions. The technique, based on the Polynomial Chaos Expansion (PCE) theory, represents a versatile solution to solve direct or inverse problems related to propagation of uncertainty. The potentiality of the methodology is assessed investigating different applicative contexts related to groundwater flow and transport scenarios, such as global sensitivity analysis, risk analysis and model calibration. This is achieved by implementing a numerical code, developed in the MATLAB environment, presented here in its main features and tested with literature examples. The procedure has been conceived under flexibility and efficiency criteria in order to ensure its adaptability to different fields of engineering; it has been applied to different case studies related to flow and transport in porous media. Each application is associated with innovative elements such as (i) new analytical formulations describing motion and displacement of non-Newtonian fluids in porous media, (ii) application of global sensitivity analysis to a high-complexity numerical model inspired by a real case of risk of radionuclide migration in the subsurface environment, and (iii) development of a novel sensitivity-based strategy for parameter calibration and experiment design in laboratory scale tracer transport.

Application of a coupled ground-surface water flow model to simulate periodic groundwater flow influenced by a sloping boundary, capillarity and vertical flows

Theoretical developments as well as field and laboratory data have shown the influence of the capillary fringe on water table fluctuations to increase with the fluctuation frequency. The numerical solution of a full, partially saturated flow equation can be computationally expensive. In this paper, the influence of the capillary fringe on water table fluctuations is simplified through its parameterisation into the storage coefficient of a fully-saturated groundwater flow model using the complex effective porosity concept [Nielsen, P., Perrochet, P., 2000. Water table dynamics under capillary fringes: experiments and modelling. Advances in Water Resources 23 (1), 503-515; Nielsen, P., Perrochet, P., 2000. ERRATA: water table dynamics under capillary fringes: experiments and modelling (Advances in Water Resources 23 (2000) 503-515). Advances in Water Resources 23, 907-908]. The model is applied to sand flume observations of periodic water table fluctuations induced by simple harmonic forcing across a sloping boundary, analogous to many beach groundwater systems. While not providing information on the moisture distribution within the aquifer, this approach can reasonably predict the water table fluctuations in response to periodic forcing across a sloping boundary. Furthermore, he coupled ground-surface water model accurately predicts the extent of the seepage face formed at the sloping boundary. (C) 2005 Elsevier Ltd. All rights reserved.

Groundwater flow model of the Logan river alluvial aquifer system Josephville, South East Queensland

The study focuses on an alluvial plain situated within a large meander of the Logan River at Josephville near Beaudesert which supports a factory that processes gelatine. The plant draws water from on site bores, as well as the Logan River, for its production processes and produces approximately 1.5 ML per day (Douglas Partners, 2004) of waste water containing high levels of dissolved ions. At present a series of treatment ponds are used to aerate the waste water reducing the level of organic matter; the water is then used to irrigate grazing land around the site. Within the study the hydrogeology is investigated, a conceptual groundwater model is produced and a numerical groundwater flow model is developed from this. On the site are several bores that access groundwater, plus a network of monitoring bores. Assessment of drilling logs shows the area is formed from a mixture of poorly sorted Quaternary alluvial sediments with a laterally continuous aquifer comprised of coarse sands and fine gravels that is in contact with the river. This aquifer occurs at a depth of between 11 and 15 metres and is overlain by a heterogeneous mixture of silts, sands and clays. The study investigates the degree of interaction between the river and the groundwater within the fluvially derived sediments for reasons of both environmental monitoring and sustainability of the potential local groundwater resource. A conceptual hydrogeological model of the site proposes two hydrostratigraphic units, a basal aquifer of coarse-grained materials overlain by a thick semi-confining unit of finer materials. From this, a two-layer groundwater flow model and hydraulic conductivity distribution was developed based on bore monitoring and rainfall data using MODFLOW (McDonald and Harbaugh, 1988) and PEST (Doherty, 2004) based on GMS 6.5 software (EMSI, 2008). A second model was also considered with the alluvium represented as a single hydrogeological unit. Both models were calibrated to steady state conditions and sensitivity analyses of the parameters has demonstrated that both models are very stable for changes in the range of ± 10% for all parameters and still reasonably stable for changes up to ± 20% with RMS errors in the model always less that 10%. The preferred two-layer model was found to give the more realistic representation of the site, where water level variations and the numerical modeling showed that the basal layer of coarse sands and fine gravels is hydraulically connected to the river and the upper layer comprising a poorly sorted mixture of silt-rich clays and sands of very low permeability limits infiltration from the surface to the lower layer. The paucity of historical data has limited the numerical modelling to a steady state one based on groundwater levels during a drought period and forecasts for varying hydrological conditions (e.g. short term as well as prolonged dry and wet conditions) cannot reasonably be made from such a model. If future modelling is to be undertaken it is necessary to establish a regular program of groundwater monitoring and maintain a long term database of water levels to enable a transient model to be developed at a later stage. This will require a valid monitoring network to be designed with additional bores required for adequate coverage of the hydrogeological conditions at the Josephville site. Further investigations would also be enhanced by undertaking pump testing to investigate hydrogeological properties in the aquifer.

On the effective hydraulic conductivity and macrodispersivity for density-dependent groundwater flow

In this paper, semi-analytical expressions of the effective hydraulic conductivity ( KE) and macrodispersivity ( αE) for 3D steady-state density-dependent groundwater flow are derived using a stationary spectral method. Based on the derived expressions, we present the dependence of KE and αE on the density of fluid under different dispersivity and spatial correlation scale of hydraulic conductivity. The results show that the horizontal KE and αE are not affected by density-induced flow. However, due to gravitational instability of the fluid induced by density contrasts, both vertical KE and αE are found to be reduced slightly when the density factor ( γ ) is less than 0.01, whereas significant decreases occur when γ exceeds 0.01. Of note, the variation of KE and αE is more significant when local dispersivity is small and the correlation scale of hydraulic conductivity is large.

Topographical, structural and geophysical characterization of fracture zones: implications for groundwater flow and vulnerability

The main objective of this study is to evaluate selected geophysical, structural and topographic methods on regional, local, and tunnel and borehole scales, as indicators of the properties of fracture zones or fractures relevant to groundwater flow. Such information serves, for example, groundwater exploration and prediction of the risk of groundwater inflow in underground construction. This study aims to address how the features detected by these methods link to groundwater flow in qualitative and semi-quantitative terms and how well the methods reveal properties of fracturing affecting groundwater flow in the studied sites. The investigated areas are: (1) the Päijänne Tunnel for water-conveyance whose study serves as a verification of structures identified on regional and local scales; (2) the Oitti fuel spill site, to telescope across scales and compare geometries of structural assessment; and (3) Leppävirta, where fracturing and hydrogeological environment have been studied on the scale of a drilled well. The methods applied in this study include: the interpretation of lineaments from topographic data and their comparison with aeromagnetic data; the analysis of geological structures mapped in the Päijänne Tunnel; borehole video surveying; groundwater inflow measurements; groundwater level observations; and information on the tunnel s deterioration as demonstrated by block falls. The study combined geological and geotechnical information on relevant factors governing groundwater inflow into a tunnel and indicators of fracturing, as well as environmental datasets as overlays for spatial analysis using GIS. Geophysical borehole logging and fluid logging were used in Leppävirta to compare the responses of different methods to fracturing and other geological features on the scale of a drilled well. Results from some of the geophysical measurements of boreholes were affected by the large diameter (gamma radiation) or uneven surface (caliper) of these structures. However, different anomalies indicating more fractured upper part of the bedrock traversed by well HN4 in Leppävirta suggest that several methods can be used for detecting fracturing. Fracture trends appear to align similarly on different scales in the zone of the Päijänne Tunnel. For example, similarities of patterns were found between the regional magnetic trends, correlating with orientations of topographic lineaments interpreted as expressions of fracture zones. The same structural orientations as those of the larger structures on local or regional scales were observed in the tunnel, even though a match could not be made in every case. The size and orientation of the observation space (patch of terrain at the surface, tunnel section, or borehole), the characterization method, with its typical sensitivity, and the characteristics of the location, influence the identification of the fracture pattern. Through due consideration of the influence of the sampling geometry and by utilizing complementary fracture characterization methods in tandem, some of the complexities of the relationship between fracturing and groundwater flow can be addressed. The flow connections demonstrated by the response of the groundwater level in monitoring wells to pressure decrease in the tunnel and the transport of MTBE through fractures in bedrock in Oitti, highlight the importance of protecting the tunnel water from a risk of contamination. In general, the largest values of drawdown occurred in monitoring wells closest to the tunnel and/or close to the topographically interpreted fracture zones. It seems that, to some degree, the rate of inflow shows a positive correlation with the level of reinforcement, as both are connected with the fracturing in the bedrock. The following geological features increased the vulnerability of tunnel sections to pollution, especially when several factors affected the same locations: (1) fractured bedrock, particularly with associated groundwater inflow; (2) thin or permeable overburden above fractured rock; (3) a hydraulically conductive layer underneath the surface soil; and (4) a relatively thin bedrock roof above the tunnel. The observed anisotropy of the geological media should ideally be taken into account in the assessment of vulnerability of tunnel sections and eventually for directing protective measures.

Numerical Simulation of Groundwater Flow with Gradually Increasing Heterogeneity due to Clogging

Well injection replenishes depleting water levels in a well field. Observation well water levels some distance away from the injection well are the indicators of the success of a well injection program. Simulation of the observation well response, located a few tens of meters from the injection well, is likely to be affected by the effects of nonhomogeneous medium, inclined initial water table, and aquifer clogging. Existing algorithms, such as the U.S. Geological Survey groundwater flow software MODFLOW, are capable of handling the first two conditions, whereas time-dependent clogging effects are yet to be introduced in the groundwater flow models. Elsewhere, aquifer clogging is extensively researched in theory of filtration; scope for its application in a well field is a potential research problem. In the present paper, coupling of one such filtration theory to MODFLOW is introduced. Simulation of clogging effects during “Hansol” well recharge in the parts of western India is found to be encouraging.

Hydraulic Subsurface Measurements and Hydrodynamic Modeling as Indicators for Groundwater Flow Systems in the Rotondo Granite, Central Alps (Switzerland)

Regional groundwater flow in high mountainous terrain is governed by a multitude of factors such as geology, topography, recharge conditions, structural elements such as fracturation and regional fault zones as well as man-made underground structures. By means of a numerical groundwater flow model, we consider the impact of deep underground tunnels and of an idealized major fault zone on the groundwater flow systems within the fractured Rotondo granite. The position of the free groundwater table as response to the above subsurface structures and, in particular, with regard to the influence of spatial distributed groundwater recharge rates is addressed. The model results show significant unsaturated zones below the mountain ridges in the study area with a thickness of up to several hundred metres. The subsurface galleries are shown to have a strong effect on the head distribution in the model domain, causing locally a reversal of natural head gradients. With respect to the position of the catchment areas to the tunnel and the corresponding type of recharge source for the tunnel inflows (i.e. glaciers or recent precipitation), as well as water table elevation, the influence of spatial distributed recharge rates is compared to uniform recharge rates. Water table elevations below the well exposed high-relief mountain ridges are observed to be more sensitive to changes in groundwater recharge rates and permeability than below ridges with less topographic relief. In the conceptual framework of the numerical simulations, the model fault zone has less influence on the groundwater table position, but more importantly acts as fast flow path for recharge from glaciated areas towards the subsurface galleries. This is in agreement with a previous study, where the imprint of glacial recharge was observed in the environmental isotope composition of groundwater sampled in the subsurface galleries. Copyright © 2012 John Wiley & Sons, Ltd.