In order to te s t the m ethod presented in C hapter 6, we shall construct piezom etric h ead m easu rem en ts by solving, u n d er varying conditions, th e aquifer flow equation
V»TV(p (p Q 0 qp + qb on Ü on dQ (7.2.1) w here Ü is a 6 x 6 k m 2 domain;
T is the transm issivity in m 21 day;
(p is the piezometric head expressed in m;
qp is the sum of two point sinks located a t (3.6km, 1.2km ) , and
(4.65km ,4.65km ). W hen tu rn e d on, the two pum ps w ithdraw w ater a t a ra te of 450 m 31 day and 675 m 31 day, respectively; an d
qb is a smooth background flow given in m l day . For all cases qb
will be the plane
qb = -11.20x10-5 + 1.35x10-% - .90x10-% w ith (u,v) e [0,6] x[0,6] km 2.
E quation (7.2.1) is solved for (p by a finite difference schem e, th a t involves a discretization of the dom ain Q into 41 x 41 nodes. For th is purpose, a NAG su b ro u tin e w as used (NAG, 1987). Note t h a t we have chosen to discretize the forw ard problem w ith a very fine m esh in order to m inim ize d is c re tiz a tio n e rro rs t h a t m ig h t su b s e q u e n tly affect th e tran sm issiv ity identification procedure.
As a firs t set of experim ents, we sh all consider th e fo u r following combinations of recharge q and transm issivity T :
Com bination 1: smooth T and smooth 7- Com bination 2: smooth T and rough <7-
Combination 3: rough T an d smooth V-
H ere, th e te rm 'sm ooth' denotes functions th a t are analytic w hile 'rough' d e n o te s fu n c tio n s w ith s in g u la r itie s su ch as d is c o n tin u itie s in tran sm issiv ity an d Dirac distributions modelling pum ps.
In all su b seq u en t resu lts, the aquifer ß , or any square sub-region contained in ß , a n d the equation (7.2.1) are im plicitly norm alised to th e u n it square [—1,1] x [—1,1] for th e associated com putations. F urth erm o re, in constructing an in terp o lan t for (p from (6.4.14) and (6.4.15), we will use the fact th a t according to th e two inequalities in (6.5.15), we only need to compute those coefficients £pq w ith p < R + 1 + 4 and q < S + J + 4,
w here R a n d S are th e h ig h est values of th e su b scrip ts r and s appearing in the te s t functions Gr(x)Gs(y), and I and J are th e highest values of th e subscripts i and j in the tensor product C i(x)C fy) in the tran sm issiv ity expansion
N i
T(x,y) = X X n f f x ) C ( y ) (7.2.2)
n = 0 t +j= n i j > 0
E quation (7.2.2) is th e Chebyshev expansion (6.5.1) for tran sm issiv ity w ith the ordering of the subscripts as illu strated in Figure 6.4. Recall th a t if, for example N j = 2, there are 6 coefficients % ordered as follows
Voo, Vio, Voi, rl20, Vu, V02
It is straig h tfo rw ard to see th a t th e num ber of coefficients r/y for a given integer N j is equal to (Nj + l)(N j + 2)/2.
We have observed th at, generally, the best results for the reconstruction of T are obtained w hen the subscripts of the te s t functions Gr(x)Gs(y) in (6.5.16) and (6.5.17) satisfy the relationship
r + s < N j + 1 (7.2.3)
w ith N j as defined in (7.2.2). Indeed, th e num erical resu lts have shown th a t w ith a n increasing num ber of te s t functions whose orders r and s
are such th a t r + s > N j + 1, an increasing num ber of outliers ap p ear in the resid u als associated w ith (6.5.18). C onsequently, th e re is a point, approxim ately given by (7.2.3), beyond which th e use of ad d itio n al te st functions only w orsens th e accuracy of th e tra n s m is s iv ity coefficient estim ates. Clearly, th is is due to the presence of certain frequencies of large
A A
m a g n itu d e in th e re sid u a l q - V»TV(p w hich a re picked up by th e corresponding te s t functions. The reasons for th e presence of those high m agnitude frequencies are not yet clear and are still being investigated.
In view of th is, in all su b seq u en t n u m erical ex p erim en ts, th e te s t functions used to recover tran sm issiv ity coefficients are those determ ined by (7.2.3).
7.3 COMBINATION Is SMOOTH TRANSMISSIVITY AND