Environmental Fluid Mechanics - Chapter 11 ppsx

A phreatic aquifer has an impermeable bottomand a free surface, and a leaky aquifer is an aquifer whose boundaries areleaky, i.e., there is flow across its boundaries.. assumption is that

be modeled and simulated by methods applied to inviscid flows, where thespecific discharge originates from a potential function In the present chapter

we explore some further applications of fluid mechanics principles with regard

to groundwater flow, its contamination, and its preservation

Groundwater is always associated with the concept of an aquifer Anaquifer comprises a layer of soil that may store and convey groundwater.Therefore an aquifer is a layer of soil whose effective porosity and permeability(or hydraulic conductivity) are comparatively high There are various types ofaquifers, as illustrated in Fig 11.1: (1) the confined aquifer, (2) the phreatic

aquifer, and (3) the leaky aquifer It should be noted that in addition to this

classification, there are other properties of aquifers that are of interest, such

as the presence and effects of fractures, etc However, regarding the aquifersshown in Fig 11.1, a confined aquifer is an aquifer whose top and bottomconsist of impermeable layers A phreatic aquifer has an impermeable bottomand a free surface, and a leaky aquifer is an aquifer whose boundaries areleaky, i.e., there is flow across its boundaries Figure 11.1c shows a leakyphreatic aquifer, for example

Considering length scales of aquifer flows, the thickness of the aquifer

is usually quite small, of the order of several tens of meters, whereas thehorizontal extent is of the order of kilometers Therefore it can be assumedthat in many cases the groundwater flow is approximately in the horizontal

direction Such an assumption leads to the Dupuit approximation, introduced

Figure 11.1 Typical aquifers: (a) confined aquifer; (b) phreatic aquifer; and (c) leaky phreatic aquifer.

in Chap 4 The essence of this type of approximation is represented in thefollowing section

11.2 THE APPROXIMATION OF DUPUIT

The Dupuit approximation is based on several simplifying assumptions,which are generally quite well satisfied in groundwater systems The major

assumption is that streamlines in groundwater flows are almost horizontal.Such an assumption with regard to a free surface (phreatic) aquifer greatlysimplifies the boundary condition at the free surface The free surface of thephreatic aquifer, by definition, represents a streamline on which the pressure

is equal to atmospheric pressure A boundary condition of prescribed pressure,according to Bernoulli’s equation, is a nonlinear boundary for the calculation

of the potential function Also, the exact location of the streamline of the freesurface is not known prior to the performance of the calculations Both of thesedifficulties are resolved by the employment of the Dupuit approximation

Figure 11.2shows the basic differences between the presentation of thegroundwater flow according to potential flow theory and the modification ofthat presentation by the employment of the Dupuit approximation

Basically, the Dupuit approximation does not consider the exact shape

of the streamlines The conservation of mass is considered with no reference

to the stream function The vertical component of the specific discharge isignored, but the horizontal component of the specific discharge varies along thelongitudinal x coordinate It is assumed that due to the small curvature of thestreamlines, the elevation of the free surface represents the piezometric head,which is constant along vertical lines, instead of along lines perpendicular tothe free surface of the groundwater Therefore the specific discharge vector isapproximated as

of the aquifer is horizontal Therefore the thickness of the flowing water layer

is adopted to represent the value of h As shown in the following paragraph,such an adoption of h forFig 11.2 may provide a complete linearization ofthe equation of flow and the surface boundary condition

The assumption of vertical lines of constant piezometric head impliesthat the specific discharge is uniformly distributed in a vertical cross section

of the aquifer Therefore the total discharge per unit width flowing throughany vertical cross section of the aquifer, shown inFig 11.2b,is given by

Figure 11.2 Differences between the potential flow theory and the Dupuit mation: (a) the potential flow presentation; and (b) the Dupuit approximation.

approxi-the x coordinate, and approxi-the solution of Eq (11.2.2) is given by

By applying two measured values of h at two arbitrary points, we canidentify the value of the two constant coefficients of Eq (11.2.3), namely Qand C Consider that the measured value of h at x D 0 is h0, and at x D L thevalue of h is hL Introducing these values into Eq (11.2.3), we obtain

Q D Kh

2
h2 L

As previously noted, the Dupuit approximation basically neglects thecomponent of the specific discharge in the vertical direction Therefore, in themost general case, that approximation allows two horizontal components ofthe specific discharge, namely,

This expression may look similar to Laplace’s equation, but it refers only

to steady-state conditions In the case of a phreatic aquifer, some quantities

of percolating runoff, called accretion, penetrate into the aquifer through its

free surface Under such conditions, the free surface of the aquifer is not astreamline However, this case also can be completely linearized by the Dupuitapproximation

In cases of flow through a confined aquifer, the boundary conditions ofthe domain are linear, but their shape may lead to some difficulties in solvingLaplace’s equation In such cases, the Dupuit approximation simplifies thecalculations, as it leads to an assumption of unidirectional flow

If the domain of Fig 11.2 is subject to unsteady conditions, then thegroundwater free surface is subject to variations, as shown inFig 11.3.Then,consideration of Eq (11.2.2) and the mass conservation for the elementaryvolume of unit width shown inFig 11.3yields

Figure 11.3 Variation of groundwater surface in a phreatic aquifer.

In the most general case of applying the Dupuit approximation, instead of

If the aquifer is confined, then the Dupuit approximation is useful to

simplify the equations based on Darcy’s law and mass conservation In the

case of a confined aquifer, the parameter h in Eq (11.2.9) is still considered

as the piezometric head with regard to calculation of the specific discharge,but it is replaced by the thickness, B, of the confined aquifer, with regard tothe calculation of the flow rate per unit width of the aquifer, as indicated in

Eq (11.2.6) In a confined aquifer, contrary to a phreatic aquifer, the ness of the region of flowing groundwater is kept almost constant However,variations of flow in the aquifer are accompanied by some compression ofthe water phase, as well as restructuring of the solid skeleton and porosity of

thick-the porous medium Such changes are characterized by thick-the strativity or

coef-ficient of storage, S Therefore in the case of a confined aquifer, Eq (11.2.9)

A common approach is to linearize Eq (11.2.9) by considering that also

in the case of a phreatic aquifer, the transmissivity can be defined by

where hav is an average value of the aquifer thickness By introducing

Eq (11.2.12) into Eq (11.2.9), we obtain

of the soil but cannot be useful for the prediction of the response of the scale aquifer to various types of flow conditions With regard to confinedand phreatic aquifers, the analysis of field tests yields the storativity andtransmissivity of the aquifer Contemporary methods are sometimes used tocharacterize phreatic aquifers by reference to more sophisticated parametricanalysis Sometimes such approaches are needed, mainly in cases of largevariations of the phreatic aquifer thickness With regard to leaky aquifers, anadditional parameter, the leakage factor, is required for the complete para-metric presentation of the aquifer characteristics Other types of aquifers, likefractured aquifers, require definitions of some other characteristic quantities.However, the topic of well hydraulics is based on practical uses of the Dupuitapproximation, as exemplified by Eqs (11.2.10) and (11.2.13), to characterizethe capability of aquifers to supply required quantities of water to water supplysystems

large-The Dupuit approximation is very useful in obtaining simplified imate solutions of groundwater flow problems It simplifies problems of flowbetween impermeable layers, like confined aquifers, by simplifying the format

approx-of space-dependent coefficients In cases approx-of free surface flows, it linearizesthe nonlinear surface boundary condition In cases of immiscible fluid flows,

it linearizes the nonlinear boundary of the interface between the immisciblefluids In coastal aquifers, it is common to assume that the sea saltwater andfreshwater of the aquifer are immiscible fluids Then the Dupuit approxima-tion can be useful for the calculation of the movement and location of theinterface between salt and fresh waters

It will be shown in the following sections of this chapter that the Dupuitapproximation can often be useful for the solution of environmental problemsassociated with groundwater contamination and reclamation Such topics areoften defined as “contaminant hydrology.” Such a definition is suggested toseparate topics of pure hydraulics, which refer only to flow through porousmedia, from topics related to the quality of groundwater

11.3.1 General Introduction

The basic equations of contaminant transport in any fluid system wereintroduced in Chap 10 The same general approach, using elementary orfinite control volumes, is applicable for modeling transport in porous media.However, in the case of a porous medium, we need to consider that a portion

of the control volume is occupied by the solid matrix, and another portionincorporates the fluid or fluids The elementary volume of reference in aporous medium system also must be much larger than the characteristicpore size Such a representative elementary volume (REV) is much largerthan is usually required, according to continuum mechanics of single-phasematerials In single-phase materials, continuum mechanics requires reference

to an elementary volume significantly larger than the molecular size In

Chap 10, we discussed some topics of dispersion in porous media In thepresent section, we present the basic modeling approach to the analysis andcalculation of contaminant transport in porous media

11.3.2 Basic Equation of Contaminant Transport

Consider a constituent distributed in small concentrations in the water phase.The constituent concentration represents the mass of the constituent per unitvolume of the water phase, consistent with the definition of mass concentra-tion inChap 10.Also as inChap 10,a binary mass system of water and theconstituent is assumed here The total mass of the constituent is assumed to

be very small in comparison to the quantity of water Therefore the tion of the minute quantity of constituent into the water phase does not affect

introduc-the original volume of that water phase The constituent may be present as adissolved material in the water phase, it can be present as a material adsorbed

to the solid skeleton of the porous medium, and it can be added, or taken away,

in different forms to and from the control volume of the porous medium ring to an elementary representative volume of the porous medium, the basicequation of mass conservation of the dissolved constituent in the groundwatercan be obtained as

(2) This term represents the difference between advective fluxes ofcontaminant leaving and entering the elementary control volume of the porousmedium through its surfaces; q is the specific discharge of the flowing waterphase

(3) This term represents the effects of molecular diffusion and namic dispersion on fluxes of contaminant entering and leaving the elementarycontrol volume through its surfaces Fluxes of diffusion and dispersion areproportional to the gradient of the constituent concentration D is a second-order hydrodynamic dispersion tensor, which is represented by a matrix ofthe nine coefficients of dispersion The values of the dispersion coefficientsdepend on the type of the porous medium, its isotropy and homogeneity, andits Peclet number The Peclet number is defined by

dispersion tensor is an isotropic tensor, whose main diagonal components are

smaller than the molecular diffusion coefficient If 0.4 < Pe < 5, then someanisotropy characterizes the hydrodynamic dispersion tensor, and it becomes

a symmetric second-order tensor The principal directions of this tensor areparallel and perpendicular to the velocity vector If Pe > 5, then the effect ofmolecular diffusion is minor, and the dispersion tensor can be represented by

DijD aTjVj C aL
aTViVj

where aTand aLare the transverse and longitudinal dispersivity, respectively.Studies report that the longitudinal dispersivity is between 5 to even 100times larger than the transverse dispersivity The common ratio between thelongitudinal and transverse dispersivity is considered to be between 20 and 40.(4) This term represents phenomena of sorption–desorption A positivevalue of f indicates larger quantities of the constituent adsorbed to the solidskeleton of the porous medium than those desorbed from the solid skeleton It

is common to analyze phenomena of sorption–desorption using linear isothermmodels, such as developed by Langmuir or Freundlich These models provideapproximate linear relationships between the concentration of the constituentdissolved in the water phase and its mass quantity adsorbed to the solidskeleton of the porous medium Such a presentation of the adsorption processleads to incorporation of the fourth term of Eq (11.3.1) with the first term as

∂C

∂t C f D R∂C

where R is called the retardation factor.

(5) This term refers to the constituent added to the water phase, as aresult of chemical reactions inside the elementary control volume It incor-porates the decay of the constituent mass and possible microbial uptake Thevalue of this term represents the mass of the constituent added (or taken away)

by the internal chemical reactions per unit time, per unit volume of the porousmedium

(6) This term represents the artificial removal of the constituent, whichmay consume water with the constituent The consumed water leaves thesystem with the current concentration level of the constituent

(7) This term represents the artificial recharge of the constituent, whichsupplies water with constituent The constituent concentration of the rechargedwater is CR

11.3.3 Various Issues of Interest

Solutions of Eq (11.3.1) can be developed, provided that the appropriate formsfor each of the terms (4) through (7) are known, values of dispersivities are

given, initial conditions are defined, and boundary conditions of the system arewell presented Various issues of contaminant transport in groundwater are thenquantified by the appropriate solution of Eq (11.3.1), depending on the relativemagnitudes of each of these terms Often, initial and boundary conditions of thesaturated porous system are not very well defined In these cases, it is common

to study the sensitivity of the system to a set of different initial and boundaryconditions After performing a set of such simulations, it is usually possible tochoose a set of initial and boundary conditions in such a way that conservativeresults can be assured Time scales of the different phenomena represented in

Eq (11.3.1) are often very different Therefore considering different time scalephenomena may allow significant simplification of Eq (11.3.1), since someterms may be neglected

Also, it is common to use a conservative approach for the

quantifi-cation of contaminant transport, by considering transport of a conservative

contaminant With this assumption, terms (4) through (7) can be neglected

in most portions of the domain Then contaminant migration in the domain

is affected only by advection and dispersion, and Eq (11.3.1) becomes anadvection–diffusion equation similar to the mass balance equation derived inthe previous chapter,

∂C

At this point, it is important to consider the difference between the timescale of processes associated with pumping of water for water supply purposes,and the time scale of contaminant transport in groundwater Pumping testsusually require several days of pumping During that time period, changes

in groundwater table or piezometric head are measured and evaluated Theeffect of pumping on the groundwater table is quick, and it depends on theavailability of water in the aquifer With regard to contaminant transport in theaquifer, processes are determined by the advection of the contaminant, whichalso is associated with the contaminant dispersion Regarding natural flow in

an aquifer, the magnitude of the hydraulic gradient is usually of order 10
3,the hydraulic conductivity is of order 10 m/d, and the porosity is of order0.2 Therefore the interstitial flow velocity, or the advection velocity, is oforder 5 cm/d Under such conditions, a pollutant discharged into the aquifer

is advected a distance of less than 20 m in a year Therefore contamination inaquifers can persist for many years before there is any indication about such aprocess Close to a pumping well, the hydraulic gradient is large Therefore acontaminant that for many years may have spread only a comparatively smalldistance in the aquifer by natural flow is subject to relatively quick advectionafter its penetration into the region of influence of the well

Some important solutions of the advection–diffusion equation can beapplied to determine basic characteristics of contaminant transport through

porous media As an example, we consider here the Ogata and Banks solutionnamed after the scientists who developed it This problem refers to contaminanttransport in a semi-infinite column, through which the water flows with aconstant velocity V The value of this velocity is, as previously defined, equal

to the specific discharge divided by the porosity of the porous medium At time

t 0, there is no contaminant in the flowing water phase At t > 0, at one end

of the column, where x D 0, the contaminant concentration is kept constant,

at C D C0 This may be the case, for example, when the semi-infinite column

is connected to a large reservoir, in which the contaminant distribution is keptuniform due to mixing Under these conditions, if the dispersion coefficient isconstant, the differential equation (11.3.5) reduces to



11.3.12

where erfc is the complementary error function, defined by

erfc D 1
erf D 1
p2

 0

This solution was developed by Ogata and Banks in 1961, and it isvery useful for the identification of the dispersion coefficient, as explained

in the following paragraphs If Pe is high, then the dispersion coefficient in

Eq (11.3.12) is the longitudinal dispersion, which, according to Eq (11.3.3),

A breakthrough curve is obtained by continuously measuring

contami-nant concentration at a point x D x0, and plotting the results on the C, t plane

An example of a set of breakthrough curves is shown in Fig 11.4, whereeach curve is associated with measurements made at several different loca-tions along the column Breakthrough curves similar to these can be obtained

in field tests, where a tracer is injected into a strip of wells, and inant concentration measurements are performed at different points locateddownstream of the injection

contam-By applying Leibniz’s theorem, we may obtain the rate of change of C

at time t1/2, which is the time at which the contaminant concentration at themeasurement point is half of C0 Thus

DLD C2V2

4t0.5S2 aLD DL

Figure 11.4 Breakthrough curves.

11.3.4 Application of the Boundary Layer Approximations

The boundary layer approach can be useful in obtaining approximate solutions

of Eq (11.3.5) For example, consider the one-dimensional contaminant port problem represented by Eq (11.3.6), subject to the boundary conditionsgiven by Eq (11.3.7) By adopting a moving longitudinal coordinate,

Now, referring to the C, x plane shown in Fig 11.5, consider the buildup

of two boundary layers at the moving front of the advected contaminant Inthis figure, it is assumed that advection in the x-direction is the dominanttransport mechanism, so that contaminant dispersion is basically a second-order phenomenon, leading to the development of front and rear boundarylayers We assume that both boundary layers have an identical shape Therefore

Figure 11.5 Boundary layers developed at the moving contaminant front.

the contaminant distribution at the moving front of the contaminant is given by

As the front and rear boundary layers are symmetrical with regard to

x1D 0, we proceed with calculating the development of the front boundarylayer By introducing Eq (11.3.22a) into Eq (11.3.20), applying Leibnitz’stheorem, and integrating the expressions over the boundary layer,

ddt

υ

1 0

As indicated inFig 11.5,the calculation, based on using the boundary layerapproximation, can be applied provided that

As a further example of the application of the boundary layer imation to different problems of contaminant hydrology, consider a quickevaluation of the migration of a NAPL (nonaqueous phase liquid — NAPLs

approx-in groundwater are discussed approx-in more detail approx-in Sec 11.5) NAPL can be eitherlighter or denser than water (in the latter case it is referred to as a DNAPL,

a dense nonaqueous phase liquid), but here we consider light NAPLs NAPLmigration arises due to the dissolution of the NAPL lens, which is createdwhen a NAPL is released into the aquifer Figure 11.6 shows how a NAPL

Figure 11.6 NAPL lens floating on top of the groundwater table.

released at the soil surface arrives at the groundwater table and creates a lens,which floats on top of the groundwater table Due to the contact between theNAPL lens and the flowing groundwater, a region adjacent to the NAPL lens

is contaminated by a dissolved or solubilized NAPL Although a NAPL isconsidered as being immiscible in water, there is a small but finite miscibility,

on the order of 0.1% The advection velocity in the aquifer, V, is uniformlydistributed, and the flow is only in the x-direction Therefore Eq (11.3.6) may

In general, an initial condition may be specified as

However, in the framework of this presentation, a steady state will be assumed,

so that initial conditions are not needed

Under steady-state conditions, and neglecting the first right-hand-sideterm of Eq (11.3.26), we introduce Eq (11.3.27) into Eq (11.3.26), apply

Leibniz’s theorem, and integrate over the boundary layer thickness to obtaind

dx

Vυ

1 0

and

υxD1000 mDp2 ð 0.01 m ð 2 ð 3 ð 1000 m ³ 11 m

Saltwater intrusion into coastal aquifers is a problem of interest in manyplaces around the world It can be quantified and analyzed by various methods.Some methods are based on the assumption that saltwater and freshwater areimmiscible fluids, and there is a sharp interface that separates the two fluids.Such an approach allows a solution to be obtained simply by solving theequations of flow in the two portions of the domain However, saltwater andfreshwater do in fact mix, and several approaches have been developed thattake into account the effect of salt diffusion and dispersion as a perturbation

to the advective transport of the salt Other approaches consider the domainsaturated with groundwater, in which salinity is nonuniformly distributed Theeffect of the salt on the density of the water is taken into account Theseapproaches require the simultaneous solution of the equations of flow and salttransport

It should be noted that saltwater intrusion is not only typical of coastalaquifers It represents an acute issue in many inland aquifers, whose partiallypermeable bottom may convey brine from deep formations into the overlyingfreshwater aquifer, due to natural or artificial causes

11.4.1 The Sharp Interface Approximation

The sharp interface approximation considers that saltwater and freshwater areimmiscible fluids The sharp interface represents a streamline and a nonlinear

boundary (since its position is not well known before solving the problem)with regard to the velocity vector Such a boundary is similar to the freesurface of a phreatic aquifer As shown hereinafter, by using the Dupuitapproximation, the equation of flow is completely linearized under steady-state conditions Figure 11.7 shows two-dimensional steady-state freshwaterflow in a phreatic coastal aquifer According to the Dupuit approximation,lines of constant potential are vertical Therefore the elevation of a point ofthe groundwater table represents the piezometric head of the freshwater inthe vertical cross section that incorporates that point Considering the smallcontrol volume of the freshwater portion of the aquifer, we obtain

is the depth of the interface below the sea level, N is the rate of accretion perunit width of the aquifer, and x is the longitudinal coordinate

Both sides of the interface are subject to the same pressure Therefore,according toFig 11.7,

Figure 11.7 Saltwater intrusion into a coastal aquifer.

Equation (11.4.4) is a linear ordinary differential equation with respect

to h2f If there is no accretion, then direct integration yields

Qx DK2

The results of the preceding paragraphs show that by a small number ofpiezometric head measurements, it is possible to obtain an estimate of thefreshwater discharge of the aquifer and the location of the interface betweenthe freshwater and the saltwater.

In order to obtain some quantitative information about coastal aquifers,

we consider the following approximate values:

sD 1025 kg/m3

where
fand
sare the density of fresh and saltwater, respectively Introducing

Eq (11.4.10) into Eq (11.4.3),

This result is called the Ghyben–Herzberg relationship after the scientistswho first developed this expression in the beginning of the 20th century byconsidering hydrostatic pressure distribution in the coastal aquifer

As an example, consider a coastal aquifer with an impervious bottomlocated at an elevation of 40 m below sea level Then, according to

Eq (11.4.11), the toe of the interface is located where hfD 1 m If thehydraulic conductivity of that aquifer is 40 m/d, and the toe of the interface islocated at a distance of 1 km from the seashore, then, according to Eq (11.4.8),the aquifer discharge per unit width is



ð 12 D 0.82 m2/d

11.4.12Under unsteady conditions, the interface is subject to movement, andadditional terms representing the variation of the groundwater table and thedisplacement of the interface should be added to the differential Eq (11.4.1).This renders the equation nonlinear Therefore problems of saltwater intrusioninto a coastal aquifer, in which the interface is subject to movement, areusually solved by numerical simulation

11.4.2 Salinity Transport

The sharp interface assumption is often used, at least as a first tion for the evaluation of saltwater intrusion into coastal aquifers It also isused for the evaluation of saltwater intrusion into inland aquifers However,

approxima-as previously noted, freshwater and saltwater are miscible fluids The maindifference between the two fluids is simply the difference in salt concentra-tions Therefore, logically, the appropriate method of simulation of saltwaterintrusion into an aquifer should focus on the water flow associated with advec-tion and dispersion of salt in the domain Such an approach requires that the

flow equation and salt transport equation must be solved simultaneously, asthe increase of the salt concentration increases the density of the water phase.There are various numerical models, some of them in the public domain,that can adequately provide such solutions However, a simplified approachmay apply the sharp interface approximation incorporating the assumption thatthe transition zone between freshwater and saltwater can be represented as aboundary layer Using this approach, it is assumed that the interface repre-sents a boundary of constant salt concentration, as shown in Fig 11.8 Thetransition zone develops along that boundary, and it is similar to a boundarylayer Assuming that the curvature of the interface is small, we adopt a two-dimensional coordinate system x, y, where x is the longitudinal coordinateextended along the interface and y is perpendicular to the interface Due to thesmall curvature of the interface, the equation of salt transport in the proximity

of the interface is given by

where υ is now the thickness of the transition zone, C0 is salt concentration

of the saltwater, and n is a power coefficient We introduce Eq (11.4.14) into

Eq (11.4.13) and integrate over the transition zone to obtain

dυ2

Figure 11.8 Development of the transition zone at the sharp interface.

Under steady-state conditions, and neglecting the first right-hand-sideterm of Eq (11. 3.26), we introduce Eq (11. 3.27) into Eq (11. 3.26), apply