FUNCIONES EJECUTIVAS

Most of the analytical solutions to stream -aquifer system s have been given for confined aquifers or free-surface aquifers for w hich th e flow is essentially horizontal (Sahuquillo,1986). In such cases, the flow equation for a two-dimensional homogeneous aquifer w ithout recharge and perfectly connected to a stream is (Bear, 1972)

Ay = (4.5.1)

K Gt

w ith one of the boundary conditions being (p (x,0,t) = h(x,t) a t th e aquifer-stream boundary, and w here

A is the two-dimensional Laplacian operator;

<p(x,y,t) is the piezometric head;

k is the constant aquifer diffusivity;

h(x,t) is the stream stage height; and

(x,y) a coordinate system w ith the axis y perpendicular to th e stream direction x.

Once a solution for (p h as been obtained from equation (4.5.1), the inflow p(x,t) can be derived from Darcy's law (Bear, 1972)

p = (4.5.2)

dn

w here

(i is a constant to account for p artial penetration of the stream into th e aquifer;

T is the aquifer transm issivity; and

n rep resen ts the direction norm al to the river boundary.

Solution of (4.5.1) for the two-dimensional sem i-infinite aquifer y > 0

can be found in P olu b arin o v a-K o ch in a (1962). For o n e-d im en sio n al aquifers, solutions corresponding to general types of in itial an d boundary conditions are given in Carslaw and Jaeg er (1959) and B ear (1972).

These solutions show th a t if th e piezom etric head <p is e sse n tia lly driven by th e stream stage height, i.e. is a function of one of the boundary conditions only, th e n (p and h can be re la te d by a convolution-type integral. Since in practical situ atio n s the piezom etric h ead (p(x,y,t) a n d stage height h(x,t) are only m easured a t a few locations along th e stream , we will remove th e dependence of (p and h on x. Therefore, we m ay w rite (Brown,1961)

t

= jV vJ U(y, t - v)dv (4.5.3)

where U is the response of the piezometric head (p to a u n it im pulse in stage height h.

H all and Moench (1972) provide several analytical expressions for U

corresponding to different types of homogeneous one-dim ensional aquifers. For example, if th e aquifer is semi-infinite w ith the head (p converging to

zero w hen y goes to infinity, the impulse response U is given by

U(y,t) ---e x p (-

Kn^'V

' 2

(4.5.4)

Figure 4.8 illu stra te s the case where (4.5.4) m ay be applied. Note th a t th e situ atio n depicted in th is figure is th e one which will be assum ed for th e application of the solute tran sp o rt model presented in C hapter 5.

If we consider now a finite aquifer of w idth l w ith th e ad d itio n al boundary condition d(p/dy = 0 a t y = l, the impulse response is given by

U(y,t) U K

l2

1

(2n - 1) exp(-v2 Kt) sin(vy) (4.5.5)

where v = (2n - 1)k/ 21.

Since we are in terested in the inflow of w ater p from th e aquifer into the stream , we can use equation (4.5.2) in conjunction w ith (4.5.3) to provide the following model for p

p(t) (4.5.6)

The lim it is kept outside the integral since p erm u tatio n of in teg ratio n and lim it m ay not be p erm itted. For exam ple, th is h ap p en s for U given in (4.5.4) and is due to a singularity of U a t y = t = 0. This shows th a t even for very sim ple cases, the construction of a convolution betw een th e inflow

p and th e stage height h based on know n an aly tical im pulse response functions U m ay not be straightforw ard.

In o rd er to rem ove th is difficulty, one could a lte rn a tiv e ly use th e convolution in teg ral th a t links the head (p to the derivative h '(t) via th e step response function P as suggested by Venetis (1970). The relationship equivalent to (4.5.3) then becomes

t

(p(y,t) =

j

h'(v)P(y, t - v ) d v (4.5.7)

(4.5.2) provide relatively simple analytical expressions for the aquifer inflow p in terms of h '. For example, for the infinite and finite aquifers mentioned above, the respective solutions are

and

(4.5.8)

p(t)

j

h ’(v) e x p [ - l 2K ( t - v ) ] d v

n =1 0

(4.5.9)

The kernel in (4.5.8) characterises it as an Abel integral equation and thus it is only weakly singular in that the singularity is integrable. However, the difficulty now is that the input data h have to be

differentiated and this is, as we have already seen, an ill-posed operation.

outflow

inflow p

fäm im m m m m m m m sssm

1

D atum

Figure 4,8 Cross-section of a stream-aquifer system for a rectangular river bed where h is the stream stage height and (p the piezometric head measured from a z and y-axis system which ensures that (p goes to zero as y goes to infinity and p is taken at y = 0. The aquifer inflow into the stream is p while hr is the stream stage height measured from an arbitrary datum. The water level 1 indicates a situation where water flows from the aquifer into the stream while the reverse is indicated in 2 . Note that the vertical dimension of the figure is strongly exaggerated.

W hile th e analytical models p resen ted above are based on some strong assum ptions (homogeneity and one-dim ensional aquifer) th a t are not likely to be m et in practice, th ey do suggest th a t a reasonable m odel for th e re la tio n b etw een th e inflow p an d th e stre a m stage h e ig h t h is a convolution of the form

t

p(t) =

J

h(v)K(t-v) d v (45.10)

— o o

The unknow n k ern el K m ay th e n be id en tified in d ire ctly from th e a v a ila b le s tre a m -a q u ife r d a ta . B efore p ro ce ed in g to th e k e rn e l identification, note th a t m ea su re m e n ts of th e stre am stage h e ig h t are usually m ade from a p articu lar point of reference. For example, Figure 4.8 illu strates the fact th a t th e river h eig h t m ay be m easured as h r w ith the d atu m being a sta n d a rd d atu m such as th e A u stra lia n H eig h t D atum (AHD). Therefore, in such cases we have

h = hr - y (4.5.11)

where y is a constant th a t determ ines the vertical location of the z-y axis system. This transform ation ensures th a t the second boundary condition in (4.5.1) is sa tisfie d . F or exam ple, for a se m i-in fin ite a q u ife r w ith

limy->o<, <P = 0 as second boundary condition, y h as to be such th a t (p goes to zero as y goes to infinity (see Figure 4.8). In m any practical situation y m ay be unknown. Therefore, equation (4.5.10) h as to be w ritten as

t

p(t) =

J

[hr(v) - y ] K ( t - v) dv (45.12)

—oo

where the constant y is a p a ra m e te r th a t needs to be estim ated together w ith the kernel K(t).

4.5.2 Identification o f th e k e rn e l K(t)

As m entioned in C h a p te r 2, th e identification of K(t) from noisy input/ouput d ata using the convolution (4.5.10) is a m ildly ill-posed inverse problem which can be cast w ith in th e general fram ew ork of firs t kind Volterra equations (Linz, 1980).

T here are ways of tran sfo rm in g (4.5.10) into a second kin d V olterra equation by differentiation th a t yield a problem which from the outset m ay ap p ear b e tte r posed (Lee, 1979; Sloan, 1980; Linz, 1980). However, th e resu ltin g form ulation eith er requires a differentiation of the o u tp u t d ata p

or a differentiation of h thereby introducing d a ta errors in th e first case an d model errors in the second. Therefore, such tran sfo rm atio n s will not be considered here. In stead , we will ad d ress th e ill-posed n a tu re of the identification of K(t) by using th e fact th a t from physical considerations,

K(t) can be assum ed to be reasonably smooth and decaying w ith tim e. This m eans th a t we can expand K(t) in term s of a few functions of the form

exp(at) w here a is a complex n u m b er w ith 5Ke(a) < 0 to e n su re exponential decay w ith tim e. In other words, assum e th a t

N

K(t) = ^ rnexp(ant) (4.5.13)

n = 1

w ith < 0.

In order to obtain an expression am enable to p a ra m eter estim ation in (4.5.13), we adopt the classical re s u lts of lin e a r system s th eo ry by considering the Laplace transform of (4.5.10) and (4.5.13) so th a t we have respectively (Brown, 1961)

w here

p(s) = K(s) h(s)

K(s)

E quation (4.5.15) can be rew ritten as

K(s) * M (S) PN(s)

(4.5.14)

(4.5.15)

(4.5.16)

where (Pj/s) and (Pj/s) are polynomials of degree N and M, respectively, with M < N. Inversion of (4.5.14) and (4.5.16) back to the tim e dom ain now yields the model

¥Nm P(t) = *PJ<D)h(t) (4.517) w here CD is the differential operator d[...]Idt. Since field d a ta are usually only available on a discrete tim e basis, equation (4.5.17) can be replaced by its discrete tim e equivalent to yield

A ( z 1) p k = <B(z 0 hk or P k Sfz‘0 .

---

K

A(z~‘) (4.5.18)

where A and CB are the following polynomials

Afe-1) = 1 + a z -1 + cl^z~2 + ... + CL^jZ~^ (4.5.19)

CB(z-1) = b + b z -1 + b z~2 + ... + bM z M (4.5.20)

0 1 2 M

and z~l is the backw ard shift operator defined as z~1p k = . E q u a tio n (4.5.18) expresses the fact th a t p k is linked dynam ically to th e stage discharge h k via th e rational tran sfer function CB( z~l) / A ( z~l).

In order to identify the orders N and M of the polynomials A and CB

and to estim ate the coefficients an , n = 1,...,N and bm , m = 0,...,M, we need discrete m easurem ents of p k and h k tak e n on equally spaced tim e intervals k. River stage height d a ta are usually available on a daily basis and are often reasonably accurate. On the other hand, aquifer inflows m ost often are unknow n. This is indeed the case for the study reach investigated h e re .

In the next chapter, we shall show th a t it is possible to identify and calibrate A and CB indirectly from stream flow and salinity d a ta available a t the u p stream an d dow nstream locations.

4.6 SUMMARY OF THE RESULTS

In th e previous sections, a model for solute tra n sp o rt an d subm odels for th e solute trav el tim e Tky the in teg ral of th e inverse flow J k and th e

aquifer inflow p k have been derived. I t is useful now to sum m arize our resu lts. F irst, the m ajor assum ptions are

i) th e sa lt concentration in the aquifer is large in comparison w ith th a t of the stream ;

ii) stream salinity advective fluxes dom inate diffusive fluxes;

iii) a t all locations along th e stream th ere is a single-valued relationship betw een the stream w etted cross-sectional area A and stream flow Q ; iv) the stream flow a t the u p stream location rem ains subcritical;

v) sources or sinks of w ater along the river are sm all enough so th a t they do not affect th e dow nstream propagation of stream flow; and

vi) aquifer properties do not vary along the stream and the aquifer-stream interaction can be represented by a convolution betw een aquifer inflow into the stream p and stream stage height h.

Second, if th e source of sa lt from th e aquifer is continuously d istrib u ted along the reach we have the solute tran sp o rt model

ck ~ Cl . r = $Pk J k + ^ (4.6.1) w here c k c°_k-Tk s Pk Jk

is the dow nstream salinity concentration a t tim e k\

is the u p stream salinity concentration a t tim e k -T k ; is the aquifer salinity assum ed to be constant;

is the flow of salty w ater from the aquifer into the stream ; is an error term ; and

is given by L

s

dx Q(x,t(x)) o (x,t(x)) e r k w ith

Q(x,t(x)) th e stream discharge m easured along th e ch aracteristic p a th r k\ and

L the length of the reach.

I f th e source of sa lt coming from the aquifer is a point source located a t

x = x 0 the above model becomes

- c

k-Tk (4.6.2)

w here Q ^Xq) is the stream discharge m easured a t tim e k and location Xq.

T he subm odels req u ire d for e ith e r of (4.6.1) a n d (4.6.2) a re now su m m a riz e d .

The w ater wave travel tim e £k is

Ck = ß , L Q k 2

w here L is the length of the reach; and ß2 are constants w ith values

ß 1 = .0758 and ß2 = -1 .6 7 for the stretch betw een E uston and Colignan; and Qk is the discharge a t the dow nstream location m easured a t tim e k.

The solute travel time rk, is

h = (<*, + «2 <*»

w here th e discharge Q° is m easu red u p stre a m an d is a tim e average given for any spatially fixed quantity X by

<*>* I X(t)dt

k-Tt

The integral of the inverse flow J k is

const(zk - Z k) < j t > k

w here A ° is th e u p stre am cross-sectional a re a. I f th e cross-sectional velocity and the w ater wave celerity do not vary significantly over the time interval [k - Xk,k], J k simplifies to

const<-A > y *

The stream -aquifer inflow p k is

Pk W z -') ---K Mz~') or more explicitly Pk + 7 t , + V i - 2 + - + aNPk-W b h. 0 k b hl k-i b h2 k -2 bM h k-M

w here h k = h rk - y w ith h rk th e stream -stage h eig h t and y a co n stan t th a t m ay need to be estim ated together w ith the p aram eters a v ...,aN and

b0>...,bM..

In th e next ch ap ter we p re s e n t n u m erical re s u lts asso ciated w ith testin g of the 'diffuse source' accession model (4.6.1) and the 'point source' accession model (4.6.2). U sing the expressions given above we have

the 'diffuse source' accession model

c_{k - c}

k - \

% z-1) X Z ~ J)

(4.6.3)

the 'point source' accession model

c_k - c

k - T k

W z'1) hk + ,

svz~')

(4.6.4)

Note th a t the constant in J k h as been subsum ed in the definition of th e polynomial t y z ” 1) .

APPLICATION OF THE SOLUTE TRANSPORT MODEL TO

In document Capítulo 2: Etiologías del Daño Cerebral (página 178-197)