Environmental Fluid Mechanics - Chapter 11 ppsx

11 Groundwater Flow and Quality Modeling 11.1 INTRODUCTION In Sec. 4.4 it was shown how laminar flow through porous media can be represented by a model of flow through small capillaries. Then the average flow rate per unit area, namely the specific discharge, is proportional to the hydraulic gradient. Therefore, though the flow is basically laminar, it can be modeled and simulated by methods applied to inviscid flows, where the specific 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. An aquifer 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 of aquifers, as illustrated in Fig. 11.1: (1) the confined aquifer,(2)thephreatic 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 aquifers shown in Fig. 11.1, a confined aquifer is an aquifer whose top and bottom consist of impermeable layers. A phreatic aquifer has an impermeable bottom and a free surface, and a leaky aquifer is an aquifer whose boundaries are leaky, i.e., there is flow across its boundaries. Figure 11.1c shows a leaky phreatic 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 the horizontal extent is of the order of kilometers. Therefore it can be assumed that in many cases the groundwater flow is approximately in the horizontal direction. Such an assumption leads to the Dupuit approximation, introduced Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. 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 the following 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 Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. assumption is that streamlines in groundwater flows are almost horizontal. Such an assumption with regard to a free surface (phreatic) aquifer greatly simplifies the boundary condition at the free surface. The free surface of the phreatic 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 free surface is not known prior to the performance of the calculations. Both of these difficulties are resolved by the employment of the Dupuit approximation. Figure 11.2 shows the basic differences between the presentation of the groundwater flow according to potential flow theory and the modification of that 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 is ignored, but the horizontal component of the specific discharge varies along the longitudinal x coordinate. It is assumed that due to the small curvature of the streamlines, the elevation of the free surface represents the piezometric head, which is constant along vertical lines, instead of along lines perpendicular to the free surface of the groundwater. Therefore the specific discharge vector is approximated as jqj³q x D ∂ ∂x ³K ∂h ∂x 11.2.1 where K is the hydraulic conductivity of the porous medium, q is the specific discharge,  is the potential function, and h is the elevation of the free surface, with regard to an arbitrary datum. In the particular case of Fig. 11.2, the bottom 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 for Fig. 11.2 may provide a complete linearization of the equation of flow and the surface boundary condition. The assumption of vertical lines of constant piezometric head implies that the specific discharge is uniformly distributed in a vertical cross section of the aquifer. Therefore the total discharge per unit width flowing through any vertical cross section of the aquifer, shown in Fig. 11.2b, is given by Q DKh ∂h ∂x D K 2 ∂ ∂x h 2 11.2.2 If there are no sources of water in the domain of Fig. 11.2, and the domain of this figure is subject to steady-state conditions, then due to the conservation of mass, the value of Q is constant for all vertical cross sections shown in Fig. 11.2. Under such conditions, the value of h varies only with Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. Figure 11.2 Differences between the potential flow theory and the Dupuit approxi- mation: (a) the potential flow presentation; and (b) the Dupuit approximation. the x coordinate, and the solution of Eq. (11.2.2) is given by Qx D K 2 h 2 C C11.2.3 where C is a constant of integration. Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. By applying two measured values of h at two arbitrary points, we can identify the value of the two constant coefficients of Eq. (11.2.3), namely Q and C. Consider that the measured value of h at x D 0ish 0 ,andatx D L the value of h is h L . Introducing these values into Eq. (11.2.3), we obtain Q D K h 2 0  h 2 L 2L 11.2.4 As previously noted, the Dupuit approximation basically neglects the component of the specific discharge in the vertical direction. Therefore, in the most general case, that approximation allows two horizontal components of the specific discharge, namely, q x DK ∂h ∂x q y DK ∂h ∂y 11.2.5 If the domain is subject to steady-state conditions, then by applying the proce- dure of Eq. (11.2.2), we obtain Q x DKh ∂h ∂x Q y DKh ∂h ∂y 11.2.6 By considering the conservation of mass under steady-state conditions, Eq. (11.2.6) yields ∂ ∂x  Kh ∂h ∂x  C ∂ ∂y  Kh ∂h ∂y  D 0 11.2.7 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 a streamline. However, this case also can be completely linearized by the Dupuit approximation. In cases of flow through a confined aquifer, the boundary conditions of the domain are linear, but their shape may lead to some difficulties in solving Laplace’s equation. In such cases, the Dupuit approximation simplifies the calculations, as it leads to an assumption of unidirectional flow. If the domain of Fig. 11.2 is subject to unsteady conditions, then the groundwater free surface is subject to variations, as shown in Fig. 11.3. Then, consideration of Eq. (11.2.2) and the mass conservation for the elementary volume of unit width shown in Fig. 11.3 yields ∂ ∂x  Kh ∂h ∂x  D  ∂h ∂t 11.2.8 Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. Figure 11.3 Variation of groundwater surface in a phreatic aquifer. In the most general case of applying the Dupuit approximation, instead of Eq. (11.2.8), we obtain rÐKhrh D  ∂h ∂t 11.2.9 where  is the portion of the porosity which takes part in the water flow, namely the effective porosity of the aquifer; and the gradient vector refers only to the horizontal directions. 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 to the 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 thick- ness of the region of flowing groundwater is kept almost constant. However, variations of flow in the aquifer are accompanied by some compression of the water phase, as well as restructuring of the solid skeleton and porosity of Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. the porous medium. Such changes are characterized by the strativity or coef- ficient of storage, S. Therefore in the case of a confined aquifer, Eq. (11.2.9) is modified to yield rÐTrh D S ∂h ∂t 11.2.10 where T is the transmissivity of the aquifer. Its value is given by T D KB 11.2.11 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 T D Kh av 11.2.12 where h av is an average value of the aquifer thickness. By introducing Eq. (11.2.12) into Eq. (11.2.9), we obtain rÐTrh D  ∂h ∂t 11.2.13 Equations (11.2.10) and (11.2.13) are basically identical. Differences are between values of S and ,whereS D O10 2  and  D O10 1 ;andT of Eq. (11.2.13) results from the approximation given in Eq. (11.2.12). Characterization of aquifers is usually obtained by the analysis of various types of field tests by reference to Eqs. (11.2.10) and (11.2.13). A common procedure is the use of pumping tests, or sludge tests. Laboratory tests of permeability of core samples can provide information with regard to the type of the soil but cannot be useful for the prediction of the response of the large- scale aquifer to various types of flow conditions. With regard to confined and phreatic aquifers, the analysis of field tests yields the storativity and transmissivity of the aquifer. Contemporary methods are sometimes used to characterize phreatic aquifers by reference to more sophisticated parametric analysis. Sometimes such approaches are needed, mainly in cases of large variations of the phreatic aquifer thickness. With regard to leaky aquifers, an additional parameter, the leakage factor, is required for the complete para- metric presentation of the aquifer characteristics. Other types of aquifers, like fractured aquifers, require definitions of some other characteristic quantities. However, the topic of well hydraulics is based on practical uses of the Dupuit approximation, as exemplified by Eqs. (11.2.10) and (11.2.13), to characterize the capability of aquifers to supply required quantities of water to water supply systems. The Dupuit approximation is very useful in obtaining simplified approx- imate solutions of groundwater flow problems. It simplifies problems of flow between impermeable layers, like confined aquifers, by simplifying the format Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. of space-dependent coefficients. In cases of free surface flows, it linearizes the nonlinear surface boundary condition. In cases of immiscible fluid flows, it linearizes the nonlinear boundary of the interface between the immiscible fluids. In coastal aquifers, it is common to assume that the sea saltwater and freshwater of the aquifer are immiscible fluids. Then the Dupuit approxima- tion can be useful for the calculation of the movement and location of the interface between salt and fresh waters. It will be shown in the following sections of this chapter that the Dupuit approximation can often be useful for the solution of environmental problems associated with groundwater contamination and reclamation. Such topics are often defined as “contaminant hydrology.” Such a definition is suggested to separate topics of pure hydraulics, which refer only to flow through porous media, from topics related to the quality of groundwater. 11.3 CONTAMINANT TRANSPORT 11.3.1 General Introduction The basic equations of contaminant transport in any fluid system were introduced in Chap. 10. The same general approach, using elementary or finite 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 portion incorporates the fluid or fluids. The elementary volume of reference in a porous medium system also must be much larger than the characteristic pore size. Such a representative elementary volume (REV) is much larger than is usually required, according to continuum mechanics of single-phase materials. 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 the present section, we present the basic modeling approach to the analysis and calculation 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 unit volume of the water phase, consistent with the definition of mass concentra- tion in Chap. 10. Also as in Chap. 10, a binary mass system of water and the constituent is assumed here. The total mass of the constituent is assumed to be very small in comparison to the quantity of water. Therefore the introduc- tion of the minute quantity of constituent into the water phase does not affect Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. the original volume of that water phase. The constituent may be present as a dissolved 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. Refer- ring to an elementary representative volume of the porous medium, the basic equation of mass conservation of the dissolved constituent in the groundwater can be obtained as ∂ ∂t C 1 CrÐEqC 2 DrÐ Q DrC 3 f C  4  5 PCC 6 RC R 7 11.3.1 It should be noted that this equation is valid for a porous medium saturated with water. In the unsaturated zone the porosity, , should be replaced by the water saturation. Each of the terms included in Eq. (11.3.1) requires some consideration, as presented in the following paragraphs, where the number of the paragraph corresponds to the number of the term in the equation. (1) This term represents the rate of change of constituent mass per unit volume in the elementary control volume of the porous medium. It is usually assumed that variations of the porosity, , may be neglected. (2) This term represents the difference between advective fluxes of contaminant leaving and entering the elementary control volume of the porous medium through its surfaces; q is the specific discharge of the flowing water phase. (3) This term represents the effects of molecular diffusion and hydrody- namic dispersion on fluxes of contaminant entering and leaving the elementary control volume through its surfaces. Fluxes of diffusion and dispersion are proportional to the gradient of the constituent concentration. D is a second- order hydrodynamic dispersion tensor, which is represented by a matrix of the nine coefficients of dispersion. The values of the dispersion coefficients depend on the type of the porous medium, its isotropy and homogeneity, and its Peclet number. The Peclet number is defined by Pe D VL D d 11.3.2 where V is the interstitial average flow velocity, which is also the specific discharge divided by the porosity; L is a characteristic length of the pores, and D d is the coefficient of molecular diffusion of the constituent in the water phase. Appropriate expressions for the hydrodynamic dispersion tensor are well presented in the scientific literature for cases of homogeneous and isotropic porous media. For an isotropic porous medium, i.e., one in which the permeability is identical in all directions, if Pe − 1, then the hydrodynamic dispersion tensor is an isotropic tensor, whose main diagonal components are Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. smaller than the molecular diffusion coefficient. If 0.4 < Pe < 5, then some anisotropy characterizes the hydrodynamic dispersion tensor, and it becomes a symmetric second-order tensor. The principal directions of this tensor are parallel and perpendicular to the velocity vector. If Pe > 5, then the effect of molecular diffusion is minor, and the dispersion tensor can be represented by D ij D a T jVjCa L  a T  V i V j jVj 11.3.3 where a T and a L are the transverse and longitudinal dispersivity, respectively. Studies report that the longitudinal dispersivity is between 5 to even 100 times larger than the transverse dispersivity. The common ratio between the longitudinal and transverse dispersivity is considered to be between 20 and 40. (4) This term represents phenomena of sorption–desorption. A positive value of f indicates larger quantities of the constituent adsorbed to the solid skeleton of the porous medium than those desorbed from the solid skeleton. It is common to analyze phenomena of sorption–desorption using linear isotherm models, such as developed by Langmuir or Freundlich. These models provide approximate linear relationships between the concentration of the constituent dissolved in the water phase and its mass quantity adsorbed to the solid skeleton of the porous medium. Such a presentation of the adsorption process leads to incorporation of the fourth term of Eq. (11.3.1) with the first term as ∂C ∂t C f D R ∂C ∂t 11.3.4 where R is called the retardation factor. (5) This term refers to the constituent added to the water phase, as a result of chemical reactions inside the elementary control volume. It incor- porates the decay of the constituent mass and possible microbial uptake. The value 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 porous medium. (6) This term represents the artificial removal of the constituent, which may consume water with the constituent. The consumed water leaves the system with the current concentration level of the constituent. (7) This term represents the artificial recharge of the constituent, which supplies water with constituent. The constituent concentration of the recharged water is C R . 11.3.3 Various Issues of Interest Solutions of Eq. (11.3.1) can be developed, provided that the appropriate forms for each of the terms (4) through (7) are known, values of dispersivities are Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. [...]... 11. 3.7a 11. 3.7b 11. 3.7c The Laplace transform of Eqs (11. 3.6), (11. 3.7b), and (11. 3.7c), respectively, yields ∂2 C ∂x 2 V ∂C D ∂x pC D 0 where C0 p CD0 CD C D C x, p at xD0 11. 3.9a at xD1 11. 3.9b 11. 3.8 where p is the Laplace transform variable; and the Laplace transform of C is defined by 1 C x, p D C exp pt dt 11. 3.10 0 The solution of the differential Eq (11. 3.8), subject to the boundary conditions (11. 3.9),... (11. 5.5), subject to the boundary conditions of Eqs (11. 5.7) and (11. 5.8), is C D Cs Cs exp x 1 2aL 1 C 4aL Kf 11. 5.9 In most cases, the effect of dispersion on the NAPL dissolution process is very minor Therefore Eq (11. 5.5) can usually be approximated by dC D Kf Cs dx 11. 5.10 C Direct integration of Eq (11. 5.10), and use of the boundary condition given by Eq (11. 5.7), give C D Cs 1 exp Kf x 11. 5 .11. .. for υ rather than for C, as υD0 at xD0 11. 3.30 In general, an initial condition may be specified as υD0 at tD0 11. 3.31 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-side term of Eq (11. 3.26), we introduce Eq (11. 3.27) into Eq (11. 3.26), apply Copyright 2001 by Marcel... the same pressure Therefore, according to Fig 11. 7, B s D hf C B 11. 4.2 f where f and s are the specific weights of freshwater and saltwater, respectively Rearrangement of Eq (11. 4.2) yields f BD s 11. 4.3 hf f Introducing Eq (11. 4.3) into Eq (11. 4.1), and performing a single integration with respect to x, we find s QDK s hf f dhf C dx N dx 11. 4.4 Equation (11. 4.4) is subject to two boundary conditions... given by ∂2 C ∂C ∂C CV D Dy 2 ∂t ∂x ∂y 11. 4.13 For steady-state conditions, we apply the following approximations (refer to Eqs 11. 3.14 and 11. 3.27): Dy D aT V C D C0 1 (11. 4.14a) Á n y ÁD υ (11. 4.14b) 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... the x coordinate, quasi-steady-state conditions may be assumed, with constant NAPL saturation Copyright 2001 by Marcel Dekker, Inc All Rights Reserved Therefore, Eq (11. 5.3) is simplified as d2 C dC D aL 2 C Kf Cs dx dx 11. 5.5 C where Kf D kf ab sw V 11. 5.6 The denominator of Eq (11. 5.6) represents the specific discharge of the groundwater Equation (11. 5.5) is an ordinary second-order differential equation... obtaining approximate solutions of Eq (11. 3.5) For example, consider the one-dimensional contaminant transport problem represented by Eq (11. 3.6), subject to the boundary conditions given by Eq (11. 3.7) By adopting a moving longitudinal coordinate, x1 D x Vt 11. 3.19 Equation (11. 3.6) becomes ∂C ∂2 C DD 2 ∂t ∂x1 11. 3.20 Now, referring to the C, x plane shown in Fig 11. 5, consider the buildup of two boundary... then direct integration yields Qx D K 2 s s f 2 hh C C 11. 4.7 where C is an integration constant The boundary condition of Eq (11. 4.5) indicates that C D 0 By applying the boundary condition of Eq (11. 4.6), QD K 2L s s f 2 hL 11. 4.8 Also, according to Eqs (11. 4.7) and (11. 4.8), hf D hL x L Copyright 2001 by Marcel Dekker, Inc All Rights Reserved 11. 4.9 The results of the preceding paragraphs show that... in Eq (11. 3.12) is the longitudinal dispersion, which, according to Eq (11. 3.3), is given by D D DL D aL V 11. 3.14 Introducing this expression into Eq (11. 3.12), we obtain C x, t D x Vt x C Vt x 1 C0 erfc p C exp erfc p 2 aL 2 aL Vt 2 aL Vt 11. 3.15 If x/aL is sufficiently large, the second term of this expression can be neglected and Eq (11. 3.15) reduces to C x, t D 1 x Vt C0 erfc p 2 2 aL Vt 11. 3.16... S1/2 D ∂C ∂t D tDt1/2 C0 V 2 Dt1/2 11. 3.17 where S1/2 is the slope of the breakthrough curve at t D t1/2 Therefore, by measuring S1/2 , the value of the longitudinal dispersion coefficient and dispersivity can be calculated, using Eqs (11. 3.14) and (11. 3.17), DL D C2 V2 0 4 t0.5 S2 aL D DL V Copyright 2001 by Marcel Dekker, Inc All Rights Reserved 11. 3.18 Figure 11. 4 11. 3.4 Breakthrough curves Application . 0att>0,x D1 11. 3.7c The Laplace transform of Eqs. (11. 3.6), (11. 3.7b), and (11. 3.7c), respectively, yields ∂ 2 C ∂x 2  V D ∂ C ∂x  p C D 0whereC D Cx, p 11. 3.8 C D C 0 p at x D 0 11. 3.9a C. conditions are not needed. Under steady-state conditions, and neglecting the first right-hand-side term of Eq. (11. 3.26), we introduce Eq. (11. 3.27) into Eq. (11. 3.26), apply Copyright 2001 by Marcel. namely, q x DK ∂h ∂x q y DK ∂h ∂y 11. 2.5 If the domain is subject to steady-state conditions, then by applying the proce- dure of Eq. (11. 2.2), we obtain Q x DKh ∂h ∂x Q y DKh ∂h ∂y 11. 2.6 By considering