3.1 INTRODUCTION
In review ing inverse m ethods applied to solute tra n s p o rt m odels or tran sm issiv ity estim ation in aquifers, we will tak e advantage of th e fact th a t inverse m ethods for both system s tak e on th e sam e g en eral form, nam ely th e identification of d istributed p aram eters in advection-diffusion equations. We can thus exploit sim ilarities in the featu res and properties of inverse methodologies applied to solute tran sp o rt models on one hand, and aquifer models on the other. In view of this, we will restric t our atten tio n to the problem of aq u ifer id en tificatio n m entioning, w here a p p ro p ria te , connections w ith the solute tran sp o rt model.
Aquifer identification h as received a lot of atten tio n in recent years and th e n u m b er of p u b licatio n s d ealin g specifically w ith it is grow ing. F u rth e rm o re , th e p ro b lem h a s d ra w n th e a tte n tio n of a p p lie d m ath em atician s resu ltin g in some very useful th eo retical and num erical work being reported in the p a st few years.
Two survey papers by Yeh (1986) and Kool et al (1987) p resen t general resu lts w ith the form er providing an exhaustive review of th e lite ra tu re over th e p a s t tw enty years. Consequently, in th is ch ap ter, we sh all only expand in detail on aspects of inverse methodologies th a t are not covered in Yeh an d Kool et al. In addition, some useful re s u lts re p o rte d in th e m athem atical lite ra tu re regarding questions of uniqueness and stab ility of solutions are presented.
In th e n e x t se c tio n , we i ll u s t r a te th e r e le v a n c e of th e advection-diffusion e q u atio n to sev eral im p o rta n t problem s found in hydrology and hydrogeology, and confirm our above a rg u m e n t reg ard in g the sim ilarities th a t m ay exist betw een aquifer and solute tra n s p o rt model identification. We th en define and compare the two general types of possible approaches to aquifer identification i.e., on the one h an d , lin e a r m ethods asso c iated w ith th e so-called direct ap p ro ach es an d , on th e o th er, non-linear techniques often referred to as indirect m ethods. In th e la st
section, we review recently published m athem atical work dealing w ith the problem of d istributed param eter identification in diffusion-type equations. F u rth erm o re, a w eak form ulation of the aquifer identification problem is introduced.
In order to illu strate the relevance of the advection-diffusion equation to several im p o rta n t problem s found in hydrology a n d hydrogeology, it is useful to show how th e eq u atio n is derived from b asic co n serv atio n p rin cip les.
The application of m ass conservation principles to a solute in a fluid continuum leads to an equation of the form (Bear, 1972)
c is the solute concentration;
j is a vector rep re se n tin g th e in sta n ta n e o u s m ass flux of th e solute;
r is a source or sink of solute; and
V is the gradient operator.
In general, the flux j is assum ed to be the sum of two vectors : a flux
j j = cu due to advection w ith u being the advective velocity, an d a flux
j 2 = —KVc generated to first order by a concentration g radient Vc. H ere K
is a diffusivity m atrix whose coefficients m ay vary in tim e an d space and may also depend on c. U nder th is decomposition and th e assum ption th a t the fluid is incom pressible, i.e. V»u = 0, equation (3.2.1) becom es th e classical advection-diffusion equation
3.2 THE ADVECTION-DIFFUSION EQUATION
3.2.1 Derivation of the advection-diffusion equation
& .
ä +
= " r(3.2.1)
w here
V.KVc — r (3.2.2)
coefficients in a differential equation are called p a ra m eters. Accordingly, the advective velocity u in (3.2.2) may be regarded as a parameter even though, strictly speaking, it is not a physical quantity that measures an intrinsic property of the media in which the transport is taking place, such as the diffusivity embedded in the matrix K.
In order to illustrate the generality of the advection-diffusion equation (3.2.2) with regard to transport phenomena in subsurface and surface water systems, four important particular cases of (3.2.2) often found in practice are listed below. We start with the transport of water in an aquifer which is, for our purposes, the most important.
1) Flow of water in an aquifer:
An aquifer is a geologic formation that contains sufficient saturated permeable material to yield significant quantities of water to wells and springs. If the formation is bounded above and below by impervious layers and the porous media is fully saturated with water, the aquifer is said to be confined. If the porous media contains unsaturated zones, the aquifer is called a free-surface aquifer. Furthermore, if the local properties of the aquifer are invariant under rotations, the aquifer is termed isotropic.
For confined or free-surface isotropic aquifers Q for which the
Dupuit-Forsheimer assumption holds true (in other words the flow is essentially horizontal (Sahuquillo, 1986)), the governing equation is the following diffusion equation (Bear, 1972)
S * - V.TVtp = -q on Q (3.2.3)
dt
supplemented with appropriate initial and boundary conditions and where
S is the storativity;
T is the transmissivity;
q is a water source or sink term often referred to in this thesis as the aquifer recharge; and
(p is the piezometric head defined by the relationship
(p = y + 2 + const (3.2.4)
th e elev atio n (De W iest, 1967). The a rb itra ry c o n sta n t in (3.2.4) is d e te rm in e d once a m e a su re m e n t d a tu m is specified. N ote t h a t th e piezom etric head h as the dim ension of a length.
The storativity represents the capacity of the aquifer to hold w a ter an d is defined locally as the volume of w ater released or tak en up p er u n it a re a w hen th e piezom etric h e a d changes by one u n it. T he tra n s m is s iv ity m easu res the resistan ce to w a ter flow. Both p a ra m e te rs depend on th e characteristics of the porous m edia th a t constitutes the aquifer.
Once (3.2.3) is solved for (p , the w ater velocity u can be com puted using Darcy's law (Bear, 1972)
where k is the hydraulic conductivity related to the tran sm issiv ity T via the relationship T = kb w ith b being the aquifer thickness.
2) T ran sp o rt of solute in an aquifer:
In th is case the advection-diffusion (3.2.2) applies exactly as given w ith
K being th e hydrodynam ic dispersion tensor; and u is D arcy's seepage velocity vector obtained from the solution of (3.2.3) to g eth er w ith D arcy's law (3.2.5) (Um ari et al, 1979).
3) Flow of w ater in a stream :
For a stream w here p ro p erties are averaged over th e w etted riv er cross-section, the flow of w a ter is often given in term s of th e V e n a n t equations which express w a ter m ass and m om entum conservation (see B altzer an d Lai, 1968; Strelkoff, 1970; G u n a ratm an an d P erk in s, 1970; Becker and Yeh, 1972; Stepien, 1984). The full V enant equations are often d ifficult to solve n u m erically since th ey a re non lin e a r. T h erefo re, approxim ations to th em are often used, one being th e 'diffusion wave model' th a t can be applied w hen th e river is large w ith slowly v ary in g flows. In th is case the w a ter flow equation becomes (G u n a ra tm a n an d Perkins, 1970)
u = -kV(p (3.2.5)
which is of the general form (3.2.2). Here
h
is the river water surface height (also called river stage height) measured from some fixed datum; vis a constant cross-sectional velocity about which the equations are linearized; er is a diffusion coefficient that represents resistive or friction effects, and q is a source or sink of water along the stream.
4) Transport of solute in a stream:
Equation (3.2.2) applies again as given with K being the diffusivity tensor (Holly, 1975). In most practical situations, stream properties are averaged over the wetted cross-section. In such a case, the advection-diffusion equation (3.2.2) becomes
where c is the average cross-sectional solute concentration; u is the average cross-sectional velocity; D(x) is a longitudinal diffusion coefficient; and r is a source or sink of the solute along the stream.
The four examples given above illustrate the importance of the general advection-diffusion equation (3.2.2) as a fundamental tool to model transport phenomena in streams and aquifers. For the purpose of managing such water systems, modellers usually need to solve (3.2.2) for c. In other words they have to solve a forward problem as schematically indicated in Figure 3.1.
forw ard problem
Figure 3.1 The forward problem