A w eak formulation o f the aquifer identification problem

Proofs of u n iq u en e ss such as t h a t of Chicone a n d G erlach view tra n s m is s iv ity e stim a tio n as a hyperbolic problem w ith th e solution constructed along characteristic lines (see equation (3.3.2)). In th is sense, Chicone an d G erlach are in terested in a point by point reconstruction of tran sm issiv ity along curves.

An altern ativ e approach is to use a weak form ulation of the aquifer flow equation to derive global properties of transm issivity. In th is section, we shall introduce such a concept for the aquifer flow equation (3.3.1) since it provides a n a tu ra l fram ew ork to stu d y th e problem of tra n sm issiv ity estim ation from a global point of view. Note th a t such a form ulation is used by F alk (1983) to derive an error analysis for the num erical identification of tran sm issiv ity . F u rth erm o re, the w eak form ulation u n d erlies th e direct m ethod presented in C hapter 6.

In order to introduce th is form ulation, let H denote a H ilb ert space endowed w ith the in n er product and let {i/a I i = 1,... } be a complete set

of orthonorm al elem ents w ith reg a rd to A fu n d am e n ta l r e s u lt of functional analysis is the following theorem (C ourant and H ilbert,1953)

T heorem 3.1

for any f e H , f = 0 in H <=> ( f KFi) = 0 for a lii = 1,2,...

In order to apply Theorem 3.1 to (3.3.1), consider a subdom ain cz Q and

H = L 2( W), i.e. the H ilbert space of real functions defined on IV w ith th e in n er product

(fg) = f f(z) g(z) dz

w

A direct consequence of Theorem 3.1 is th a t

V.TVy - q = 0in (3.4.1)

(V.TVcp - q ,T t) = 0 i = 1,2,... (3.4.2) The relationship (3.4.2) constitutes a weak form ulation of th e aquifer flow equation (3.3.1). Note th a t any solution to (3.3.1) satisfies (3.4.2) b u t the converse is not tru e.

H aving p resen te d th e w eak form ulation (3.4.2) to th e aq u ifer flow equation (3.3.1), we shall next provide a sim ple proof of u n iq u en ess of tran sm issiv ity . The practical consequences of th is re s u lt is t h a t u n d er some reg u larity assum ptions, transm issivity can be shown to be unique on subdom ains *W in the aquifer dom ain Q whose boundaries a re contour lines for the piezometric head or lines along which the flow eith er leaves or en ters th e subdom ain. This ensures th a t tran sm issiv ity can be uniquely reco v ered in n eig h b o u rh o o d of pu m p s or a ro u n d m o u n d s in th e piezom etric h ead surface.

Theorem 3.2 Consider the aquifer flow equation (3.3.1) defined on a bounded region Q. I f T is a solution of (3.3.1) for a given (p and q , then T is unique over a subregion ‘W <r Q with smooth boundary d if

i) in the subdomain (W, the transmissivityT is C1, i.e. T and its first derivatives are continuous, and (p is C2 , i.e. (p and its first and second derivatives are continuous;

ii) V(p does not vanish on a set of positive measure in (W;

iii) the normal derivative d(p/dn does not change sign along the boundary d IV; and

iv) the domain does not contain point recharges of opposite sign. This theorem can be viewed as a p a rticu la r case of the re s u lt rep o rted in Chicone an d G erlach (1987). However, we w ish to p re s e n t h e re a proof based on a sim pler argum ent.

Proof W ith d(pldn not changing sign on d"W, we can assum e w ithout loss of generality th a t d(p/dn > 0 on dM. If T 1 and T2 are two solutions of the equation V»TV(p = q , we have

V£Ti - T ^ V fp - 0 on <W _(3.4.3)

Define now the subsets

(W+ = [z e Vi] T t - T 2 > 0 } ; and

(W_ = [z e <w\T1- T 2 < 0}

W ithout loss of generality, we m ay assum e th a t (W_ h as positive m easure and (p is negative on *W_. The la tte r is alw ays possible since (p is only defined up to a constant and the domain <W does not contain point recharge of opposite sign. M ultiplying (3.4.3) by (p a n d in te g ra tin g over th e subdom ain CW_ we have

equation by p a rts using Green's form ula (De Lillo, 1982). This yields the rela tio n sh ip

Since fW_ is of positive m easure and T { - T 2 is C1, the boundary dcW_ can be sep arated into two smooth p arts as illu strated in Figure 3.3. Therefore, we have

C onsequently w ith (p and - T 2 both negative on d(W_ an d d(p/dn

positive on we have

0

W ith T being C1 and (p being C 2 in W_, we can in te g ra te th e above

(3.4.4)

dw_ = {d(W_ n dW) + - dvi)

> 0

dwjndw