Journal of Hydrology, 82 (1985) 265--284 Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands

265

[4]

A N A L Y S I S OF S T O C H A S T I C G R O U N D W A T E R FLOW PROBLEMS. P A R T II : S T O C H A S T I C P A R T I A L D I F F E R E N T I A L E Q U A T I O N S IN G R O U N D W A T E R FLOW. A F U N C T I O N A L - A N A L Y T I C A P P R O A C H

SERGIO E. SERRANO, T.E. UNNY and W.C. LENNOX

Civil Engineering Department, University of Waterloo, Waterloo, Ont. N2L 3G1 (Canada)

(Received May 6, 1985; accepted for publication July 5, 1985)

ABSTRACT

Serrano, S.E., Unny, T.E. and Lennox, W.C., 1985, Analysis of stochastic groundwater flow problems. Part II: Stochastic partial differential equations in groundwater flow. A functional-analytic approach. J. Hydrol., 82: 265--284.

Following the scheme and concepts presented in Part I, Part II uses functional- analytic theory to analyze the problem of stochastic partial differential equations of the type appearing in groundwater flow. Equations are treated as abstract stochastic evo- lution equations for elliptic partial differential operators in an appropriate functional Sobolev space. Explicit forms of solutions are obtained by using a strongly continuous semigroup. The deterministic and the stochastic problem can then be treated under the same theoretical framework. Use of the theory is indicated in an application to the solution of the stochastic analogue of the regional groundwater flow problem studied in Part I. Two cases are solved: The randomly forced and the randomly initiated equation. The solution is obtained by applying the properties of semigroups and expressing the Wiener process as an infinite basis in a Hilbert space composed of independent uni- dimensional Wiener processes with incremental variance parameters. The first two moments of the solution as well as sample functions for different cases are derived.

1. INTRODUCTION

The a m o u n t o f l i terature dedica ted to s tochast ic analysis o f ground- water f low is rapidly increasing: Early theoret ical analyses like the re- presentat ive e lementa ry vo lume (Bear, 1972) and to r tuos i ty recognize the r a n d o m na ture o f porous media flow. Some researchers like Beran (1968) a t t e m p t e d to s tudy the statistical characteris t ics of permeabi l i ty . Freeze (1975) s tudied the inheren t uncer ta in t ies of non -un i fo rm aquifers. Recen t ly , Sagar (1978b) developed a Galerkin finite e lement p rocedure for analysing f low th rough r a n d o m media. K o t t e g o d a and K a t u u k (1983) used a r a n d o m walk mode l to represent the ef fec t o f spatial variat ion o f hydraul ic con- duct iv i ty . The s tochast ic na ture of inputs has been s tudied by Gelhar (1974) , Maddock (1976) and Sagar (1978a) . Stochast ic analysis o f s teady ground- water f low has been done by Freeze {1975), Gelhar (1977) , Dagan (1979, 1981, 1982) and Det t inger and Wilson (1979) . S tochas t ic analysis o f

0022-1694185/$03.30 © 1985 Elsevier Science Publishers B.V_

266

transient groundwater flow has been done by Gelhar (1974), Sagar (1978a, b, 1979), Dagan (1979), Dettinger and Wilson (1979), and Townley and Wilson (1983). The effect of fluctuations in boundary conditions, sources or sinks have been studied by Gelhar (1974), Sagar (1978a, 1979) and Dettinger and Wilson (1979). The problem of non-stationary statistics has been studied by Delhomme (1979), Gutjahr and Gelhar (1981) and Dagan (1982). Finally, the effects of errors in measurement was treated by Wilson et al. (1978).

However, the theory and applications of stochastic partial differential equations to groundwater flow analysis has not received the deserved atten- tion. As stated in Part I of this series of articles, the main difficulties in the presentation of such a theory lie in the limitations of classical mathematics in treating equations with unbounded operators, such as the ones found in PDE, subject to random disturbances, for which a mathematical representa- tion is difficult to obtain in the classical sense.

The objectives of the present paper, Part II, is to show how the concepts of functional analysis, abstract evolution equations and semigroups of partial differential operators in appropriate Sobolev spaces, briefly presented in Part I, can be combined to present an integrated theory of stochastic PDE with a number of potential applications.

A precise s tatement of the system we are treating is given by (see also the list of symbols):

Ou ~---[ + A ( x , t, co)u = g(x, t, co), (x, t, co) e G × [0, T] × ~ (1)

Q(x, t, co)u = J(co), (x, t, co) e ~ G × [0, t] × ~ (2)

u(x, O, co) = Uo(X ,co) , x e G × ~ (3)

where u e H~ (G) × ~ is the system output ; g e L2 (~, B, P) is a second-order random function; G C R n is an open domain with boundary 3G; t is the time coordinate; 0 ~ T ( ~ , Q is a boundary operator; x is the spatial domain; A is a formal partial differential operator given by:

A u = ~ (-- 1)l~lD k(pklDlu) (4) 141, Ill ~< m

and is assumed to be elliptic; H~0 is the ruth order Sobolev space of second- order random functions (see Oden, 1977; Showalter, 1977; Bensoussan and Iria-Laboria, 1977; Sawaragi et al., 1978; Hutson and Pym, 1980; Griffel, 1981; Serrano, 1985); ~ is the basic probabil i ty sample space of elements co; L : ( ~ , B, P) is the complete probabil i ty space of second-order random functions with the probabil i ty measure P and B the Borel field or class of co sets (see also Curtain and Pritchard, 1978); Pkz are real-valued coefficients and D is differentiation in Hi]bert space.

According to the way in which randomness enters the equation, we can

267

distinguish five basic problems in increasing order of complexity: ( 1 ) t h e random initial value problem, when u 0 is random; (2) the random boundary value problem, when J is random; (3) the random forcing problem, when g is random; (4) the random operator problem, when A or Q is random; and (5) the random geometry problem. We can also have any combination of the above problems. Solutions of abstract PDE considering one or several of the five problems have been obtained by Kempe de Feriet (1956), Adomian (1964), Samuels (1966), Beran (1968), Freidlin (1969), Curtain and Falb (1971), Chow (1972), Gopalsamy and Bharucha-Reid (1975), Becus (1977, 1978), Bensoussan and Iria-Laboria (1977), Chow (1978), Sawaragi et al. (1978) and Becus (1980) among others. Important contributions to the probabil i ty theory and stochastic processes in Hilbert space, and random equations in abstract spaces have been made by Bharucha-Reid (1964), Curtain and Falb (1971) and Curtain (1978).

The system of eqns. (1)--(3) is stated in an abstract form because u may represent a variety of physical systems, such as unsteady heat or electric potential as well as groundwater potential in an aquifer or contaminant transport, depending on the particular form of the partial differential operator (eqn. 4). System (1)--(3) is also representing a general sto- chastic performance since the forcing function g is a stochastic process accounting for the uncertainties in the source term, the partial differential operator A may be random due to uncertainties in the parameters or errors due to approximations and/or linearizations in the development of the model, and the boundary and initial conditions J and u may be random variables or stochastic processes due to uncertainties and errors in the representation or field measurement of these functions.

An abstract representation has the advantages of notational economy and generality. The analysis and solution of a complex system like the one des- cribed by eqns. (1)--(3) is best accomplished by the use of functional- analytic theory. Only functional analysis provides the advantages of the abstract representation, generality and rigorous mathematical t reatment. It is particularly useful for answering fundamental questions that arise when scientists a t tempt for the first t ime to solve a system equation, like the problem of existence of a unique solution and the determination of the properties of the solution. Once these concerns have been satisfied and a particular Sobolev space has been identified as composed of functions satisfying the system equation and boundary conditions, the topological and geometrical properties of this space may be used to produce a more important result, namely an expression for the general solution of the system. Finally, functional analysis provides an integrated theoretical frame- work from which both the deterministic and the stochastic problems may be studied. Important work has been done in this field by pure mathematicians and yet much of this theory remains unknown to engineers and hydrologists.

268

2. STOCHASTIC E V O L U T I O N EQUATIONS

In this section we briefly present some functional analytic theory describ- ing stochastic PDE. We define our abstract spaces of random functions, which are equivalent and follow some of the properties of the deterministic Sobolev spaces described in Part I. The objectives are to establish the condi- tions for existence and uniqueness of solutions of stochastic PDE. Then the form of the general solution of a stochastic evolution equation will be presented.

L2 (~) = L2 (~ , B, P) is the space of second-order real variables on ~ . It is a Hilbert space with an inner product given by:

( f ,g )a = ~ fgdP = E{(f, g)} (5) ~2

where E denotes the expectation operator. The norm is given by:

Ilfll~ = E { f 2 }1/2, for all f e L 2(~) (6)

Let M be the class of second-order random functions on G, that is M = {f: G -+ L2 (~)}. C~o [G; L2 (~2)], for 0 ~< m ~< ~ , is the space of functions f e M and with continuous derivatives in G up to the mth order, where the deriva- tives are defined in L 2 (~2) in the mean square sense.

Now C~ [G; L2 (~2)1 = {f e C m (G); L2 (~2), support of f is compact in ~2}. In particular C~o [G; L2 (~2)] is the space of L2 (~2)-valued " smoo th" functions in G (see Part I). Let O(G) = {f: ~ R, f e C~o (G)}. Then O'[G; L 2 (~2)] is the space of L2(~2)-valued distributions on G [that is of continuous linear functions f ' : O(G) ~ L2 (~2); see Part I] . For f e M, D k f is defined as in Part I to be the kth order derivative of f in G in the sense of L2 (~2)-valued distri- butions. Hence, the kth weak derivative of f is:

(Dkf, dp) = (--1)k f fDk CpdX, dpeO(G) (7) G

Now introduce H = L 2 (G; L2 (~2)] = {f e M: [[flla e L 2 (G)}, where L 2 (G) denotes the usual space of square-integrable functions on G. H is a Hilbert space for the scalar product:

G G

The corresponding norm on H is defined by:

llflll = (L f)1/2 1 , f e l l (9)

For m~>0 we define / ~ = / ~ [G; L2(~ ) ] = { f e M : D k f e l l , Ikl<~m}. This is the ruth order Sobolev space of second-order random functions equivalent to the H m spaces of deterministic functions described in Part I. The inner product and norm corresponding to eqns. (5) and (6) are res- pectively:

2 6 9

(f, g)m

a n d :

IIfllm

= ~ fE{(Dkf'D~g)}dX I k l ~ m G

(lO)

Now /-/moo [G; L2(~2)] is a closed subspace of/-fin and H~' is the closure of C'~[G; L2 (~2)] in/- /~.

Let V be a real separable Hilbert space such that V C H, V is dense in H. The norm on V is denoted by II" IIv, H being identified with its dual, and V* denoting the dual of V with respect to the scalar product in H, we have V C H C V*.

For 0 ~ T ~ ~ we define: T

L2(0 , T; V) = {f: [0, T] -+ V: f Ilfll~dt < ~ } 0

If f c L 2 (0, T; V) then 3f/3t is the time derivative of f in the sense of V- valued distributions. Now define:

0 f W(O, T) -- {f e L2(O, T; Y): -~-

with the norm: T 2

I l f l l~ = I lf l l~ + s u p , e v I lvllv 0

then W(0, T) is a Hilbert space.

e L2 (0, T; V* )}

(12)

The abstract problem for a random evolution equation can be phrased as follows: Given g e L 2 (0, T; V* ) and u0 e H, find u e L 2 (0, T; V) such that eqns. (1)--(3) hold.

Let us consider the following theorem (Becus, 1980). There exists a unique stochastic process u eW(O,T) as a solution of eqns. (1)-- (3). Fur thermore the mapping {g, u0} -+ u is continuous from L 2 (0, T; V*) x H to L 2(0, T; V). A very detailed proof of the theorem is provided by Sawaragi et al. (1978). Becus (1980) proved the above theorem by spatial discretization providing a theoretical justification for many approximate methods of solution of random evolution equations like the Rayleigh-Ritz-Galerkin method (Sun, 1979) the finite difference method or the method of moments (Vorobyev, 1965; Lax, 1979; Lax, 1980). Becus (1980) also applied the above theorem to prove existence and unique- ness of the Dirichlet and the Neumann problem of the abstract random heat eqns. (1)--(3) for the five cases ment ioned in Section 1.

Let us now consider a special case: The randomly forced stochastic PDE, which is a problem that consti tutes an important application in engineering systems. Let us examine the stochastic evolution equation:

= (f, f)~: = ~ rj'E{ID kfP}dX (11) ik l~ m O

2 7 0

~u - - + A u = g + n (13) ~t

ul~c = 0 (14)

u(x , O) = Uo(X) + ~ (15)

The state of the system at t ime t is the function u(x, t). g and u are given deterministic inputs, whereas ~(x, t) and }(x) are perturbations of random type. In applications it is often assumed that K(x, t) is a white noise process in time (Jazwinski, 1970) and depends smoothly on the space variable x (Bensoussan and Iria-Laboria, 1977):

E{~} -- 0 (16)

E{~(x, , tl)~(x2, t2)} = q( t , ,X l , X2)5(t, - - t2) (17)

where q is a given symmetric positive funct ion and 5 is the delta function. Other practical situations involve the case when the noise appears on the boundary (Neumann or Dirichlet conditions). It is also possible to encounter the case when the system is governed by a stationary elliptic equation, where the noise corresponds to unknown parameters modeled as random variables. Equivalent to the deterministic problem (Part I), the solution of eqns. (13)--(15) is given by:

t t

u(t) = Jt{uo + ~} + j Jt_,g(s)ds + J Jt-,~(s)ds (18) o 0

Existence and uniqueness of this solution has been proved by Curtain and Falb (1971). Jt is again a strongly cont inuous semigroup in the Hilbert space associated with the evolution operator A (see Part I for a description of the properties of semigroups). The semigroup approach provides a con- sistent and convenient link between the deterministic and the stochastic evolution equation. It will be seen in the next section that knowing the explicit form of the semigroup associated with a particular operator, the solution to eqns. (13)--(15) subject to any form of u0, g, ~ or ~ can be easily found from eqn. (18). Even if the bounday conditions (random or non-random) are changed, it is possible to use the same analytic semigroup.

Now u(t) is a strong solution of eqns. (13)--(15) if u(t) e Domain(A) w.p. 1, u(t) satisfies eqn. (18) almost everywhere on [0, t] × Ft and u(t) has con- t inuous sample paths, u(t) is unique if whenever u l (t) is another solution:

p{su4ptllu(t) - u~ (t)II :# O} = 0

In general eqn. (18) is not a strong solution and does not have cont inuous sample paths. However, it is a well-defined stochastic process and is con- t inuous in mean square. This is then the "mild" solution of eqns. (13)-- (15).

271

3. APPLICATION TO STOCHASTIC ANALYSIS OF GROUNDWATER FLOW

Let us now consider the stochastic version of the one-dimensional ground- water flow equation described in Part I:

~h T ~2 h I ~t (x' t' ~z) S ~x 2 (x' t" ~) = ~ + wl(t) (19)

subject to:

h(0, t , w ) = C 0 < t (20)

Oh ~ x ( L ' t ' ¢ ° ) = 0 0 < t (21)

and:

bh(x , 0,¢o) = ~ ( x ) + w 2 ( x ) 0 < x < L (22) 0x dx

The parameters and variables are defined as in Part I, and the physical situ- ation of the problem is represented by fig. 1 in Part I. The groundwater level fluctuates responding to the combined action of deep percolation, evaporation, pumping, groundwater depletion and the local conditions discussed in Section 3 in Part I, and can be adequately described as a random process. For the present analysis we will assume the gradient of the inital water-table condition ~h/~x (x, O, ¢o) is represented by a deterministic function dh 0/dx (x), plus a random term w2 accounting for the uncertainties in the fluctuation and errors in the determination of the initial hydraulic gradient dho/dx (x). Usually ho(x) is obtained by interpolation of a few piezometric measurements across the watershed, w2 (x) is a spatial white noise process with parameter q2 attempting to model the uncertainties in this hydrologic routine. As for the deep percolation input I and the para- meters S and T, uncertainties in this calculations and errors inevitably involved are modeled by the time-white noise w 1 with parameter q l •

In justifying the use of white Gaussian noise to model the stochastic parts of our differential equation, we might add that these stochastic parts arise as a result of a large number of random minor environmental effects. The combination of all this fluctuations follows the Gaussian law, according to the central limit theorem. It is also useful to assume that the stochastic parts come from wide band processes with almost flat frequency spectrum up to very high frequencies. The correlation stucture of these wide band processes are almost delta-correlated and they can be approximated by white-noise processes (Unny and Karmeshu, 1984). Hence it is justified and appropriate to model the stochastic parts of the PDE as white Gaussian processes with zero mean and delta-correlated correlation structure:

E{wl(t)} = 0 (23)

272

E { w l ( t l ) " wx(t2)} = q l f i ( t l - - t2)

and:

E{w2(x)} = 0

E{w2(x l ) "w2(x2)} = q 2 5 ( x l - - x 2 )

(24)

(25)

(26)

The ~1 = f0 t wl ds and f12 = fd w2ds will then be Wiener processes (or Brown- ian motion) having values in the Hilbert space H e L2 (~2, P; H). In eqns. (19)--(22) 6) has been included in the list of independent variables along with x and t to distinguish them from ordinary deterministic functions. The output of our system will be h(x, t, ¢~), a stochastic function defined in this case for 0 ~< x ~< L, t ~> 0, and over the probability space (~2, B, P). At each point ( . , . , co), h ( ' , ", w) = h (~ ) is a random variable, where co is a point in the sample space ~2. For a fixed point ( ' , t, co), h i ' , t, w) is a stochastic process (Papoulis, 1965) and in general h(x, t, co) is a family of stochastic processes. A probability p e P is defined for each oz. Events like {h(co) ~< c} consti tute a Borel field B. Alternatively, for a given co, h(t, x, ") is a sample function, co will be omit ted from now on for notational con- venience, but all the functions are stochastic unless specified otherwise.

Now by defining y = h -- V, where V is the steady state function given by (see Serrano, 1985):

V(x) = ~ x - - ~ + C (27)

it is possible to transform system (19)--(22) in h e H 1 (0, L) into an equiva- lent one y e HI(0 , L), which is the first-order Sobolev space of second-order random functions with compact support:

0y T 32y ~t S ~x 2 - wl ( t ) (28)

y(0, t) = 0 (29)

0 y ( L , t ) = 0 (30) 0x

y(x,O) = h 0 ( x ) - - V ( x ) + 3 2 ( x ) = Y0(X)+W2(X) (31)

Now we can treat eqn. (28) as a random evolution equation in the Sobolev space H10(0, L) of the form of eqn. (13) with the operator A given by A = T/S 32 y /3x 2 . It is an operator on the Hilbert space:

I D_fly T 82y H = L2[0, L ] withDomain(A) = y e l l : ~x' S 8x 2 e H ; y ( O , t )

= 0 ;~x (L, t) = 0

273

Based on our previous discussion of funct ional analysis, we can state that there exists a unique stochastic process y satisfying eqns. (28)--(31), the mapping {Y0 + w2 } -+ y is cont inuous f rom L2 (0, T; V* ) to L2 (0, T; V) and V = H~(0, L). Generally, y has cont inuous sample paths, it is a well-defined stochastic process cont inuous in mean square (see Soong, 1973) and with cont inuous sample paths.

A generates a semigroup given by (see Serrano, 1985): L

2 ~ sin(knx)exp ( k s T t ) f y ( u ) s i n ( X . u ) d u (32) J t (Y ) ~- n n = l

o

where the eigenvalues k. are:

= ( 2 n - 11 (33)

The solution of eqn. (28) is given by (18) as: t

y(x, t) = JtYo + j Jt-ud[Jl(u) (34)

0

It seems natural to take (see Curtain, 1978; Curtain and Pritchard, 1978; Chow, 1979):

f31 (t) = ~ b, (t)en (35) n = l

where en is a basis for H and {bn(t)} is a sequence of independent , identical copies of the standard Brownian mot ion in one dimension. However the process (35) cannot be realized in H for otherwise the variance E{[~I (t)] 2 } would be infinite. Instead, since:

1

n = l

one may regard eqn. (35) as a Hi*-va lued process for which:

1 g{ l l~ , ( t ) l l 2} = t ~..=,

is defined. Now, using eqn. (35), we obtain the explicit solution of our original

problem (19)--(22):

h(x, t) = -~ x -- + C + sin(knx)exp -- n----1

274

Z ( x , t ) =

and:

L

12 I ~- I [h°(x) +/32 (x)- V(x)]sin(k,,x)dx }

o

+ exp dbn(u)sin(knX ) (36) =1 S

0

From this expression we will a t t empt to s tudy the effect of two indepen- dent cases of randomness of the groundwater level in the aquifer: The effect of a random initial condi t ion and the effect of a random forcing function. We will s tudy the mean and correlation structure of both cases.

If V(x) is given by eqn. (27):

b,,exp ( - ~'~ T-----t ) (37) rl=l S

h(x, t) =

where b.r

2 bnr L

assuming Part I.

Hence:

L 2

| [ho(x) -- V(x)] sin(knx)dx b. = ~- o

then the case of random initial condi t ion in eqn. (36) can be wri t ten as:

Z(x , t) V ( x ) + Z ( x , t) + - - bnr b. is now the random funct ion: L

~ sin(XnX)~2 (x)dx o

(38)

(39)

(40)

that ho(x) is the same sinusoidal funct ion given by eqn. (45) in

L

2 I E{bnr} = ~ sin(~,,x)E{fi2(x)}dx = 0 (41) o

by using the interchangeabili ty proper ty between the integral and the expectat ion operator. Now with x, ~< x2 :

L L

4II s 1 E{b2nr} = ~-~ E sin(X~xl)f i2(xl)dxl • sin(X~x2)fi2(x2)dx2 (42) 0 0

4q2 { sin(XnL)cos(XnL) sin(knL) E{b2r} - L 2 X 2 Xn - - L c ° s 2 ( X n L ) )k n

+ Lcos(X~ L) } (43)

275

Hence , the mean o f the hydrau l ic head for p rob lem (36) and r a n d o m initial cond i t ion is equal to the de terminis t ic solut ion in Par t I. The cor responding cor re la t ion f u n c t i o n will be:

E{h(x 1, t)h(x2, t)} = V(x 1)V(x2) + V(xl )Z(x2, t )+ V(x~)Z(xl , t)

q-Z(Xl, t)Z(x2, t) + Z(xl" t)Z(x2, t).E{b~r } (44)

I t is also interes t ing to ob ta in some sample func t ions o f h fo r d i f fe ren t cases o f r a n d o m initial condi t ions . The sample water- table prof i le is given by eqn. (39) whenever bnr is genera ted by some discre t iza t ion p rocedure .

F o r the c o m p u t e r rou t ine , the Wiener process was deve loped as a l imiting fo rm of the r a n d o m walk, fo r which x ( ' ) = kA, where k is the n u m b e r of steps given in the x d i rec t ion and A = xk -- xk - 1 • The mean and variance of this process are given by (Papoulis , 1965) :

E{~ ' (x)} = 0 (45)

and:

x8 E{/3 '2(x)} = ~ (46)

where s is the ins tan t ver t ical d i sp lacement at each step x. As s and A t en d to zero for c ons t an t x , the var ianee of /3 ' (x ) will remain finite and d i f fe ren t f r o m zero if s t ends to zero as A 1/2 . Thus it is possible to assume (Papoulis , 1965) :

s2 = q2 A (47)

This re la t ionship was used to genera te sample func t ions of the Wiener process fo r given q2 and A.

I t was f o u n d t ha t the accuracy o f the m e t h o d was in inverse p r o p o r t i o n to the step size A. Fo r the purposes o f the present s tudy A was chosen to be 1 m and c onsequen t l y s was equal to the square r o o t of q2 • The p a r am e te r q2 was chosen so as to give a reasonable smoo thness to the shape of the initial wate r table , bu t o f course this depends on the par t icu lar aquifer cons idered and on the variance of the g ro u n d w a te r level f l uc tua t ion at cer ta in boreho les across the watershed .

S ome sample func t ions of the t rans ien t f ree-surface prof i les were gener- a ted in o rde r to observe some fea tures o f the mode l . Figures 1, 2 and 3 show some sample- t rans ient f ree-surface prof i les fo r the case of a r a n d o m initial cond i t ion c o m p o s e d o f a Wiener process super imposed to a sine wave, which is c ons t ruc t e d on a s traight line whose slope is similar to tha t of the g round surface. The values o f N in the sine func t i o n were again 1.09, 2.64 and 3.39, respect ively; these values preserve the b o u n d a r y cond i t i on o f h0 at x = L. T o observe a m o r e deta i led var ia t ion in the shape o f the prof i le , the m o d e l was run on a dal ly basis as in Fig. 5. In this case N is equal to 5.41. N o te tha t

276

15--

14

13 E

12t-

" " I I - -

N = 1.09

• o O • ~ 8 ~ • ~ ÷~o ~o

• • •

• • 0 @ 0 • o o

• o • o

o 0 + + + + + +

0 + +

L I i I L I 0 20 4 0 60

D I S T A N C E (rn)

e t = O

o t : I month + t = 5months

I l I 8O I00

Fig. 1. Sample free-surface profiles for random initial conditions.

15 - -

14

13 E

12

W

- r I I -

I0

N = 2 . 6 4

o~+~ + + @

~o ° o o O

, I , I i I 0 20 4 0 60

D I S T A N C E (rn)

o o o o ~°

o 0 + + + + + ~,°+ e+

• • e t = O o t = I month + t = 5months

i I , I 80 I00

Fig. 2. Sample free-surface profiles for random initial conditions.

1 5 - -

14

13 E

,~ 12

W

I 0

0

N =3.39

oo o o •

o • o O O 4_ + + +

• • • ~ + + + +

• +" + • t 0

@ 8 + o o • • o t = I month

• • + | = 5months

i I I ~ i I L I A I 20 4 0 60 80 I00

D I S T A N C E (m)

Fig. 3. Sample free-surface profiles for random initial conditions.

277

1 5 1 ~ N = 4.57

1 4 L l =0.1 mm h -I +

+ +

~ 1 3 + + E + •

~ 1 2 + + ~: 8 o

l~J • • + 0 0 "r II + o °

• • 0 •

÷ 0 0 0

I0 •

9 L I A I I I 0 20 40 60

DISTANCE (m)

+ + + + + + + e •

+

~ o o o

o o

o • o

0 • o o

,I:0

o t = I monlh

+ t : 5months

L I , i

80 I00

Fig. 4. Sample free-surface profiles for random initial conditions and 0.1 mmh -1 of constant deep recharge.

15--

N = 5.41 14

13 E

12 h i

-z- il -o o • o

I0 ~ ++

i I l I i I I

0 20 40 60

DISTANCE (m)

+ O + + ° + o o

o o

• 0 o + +

, t = O o t = I doy + t = 5 doys

8O I00

Fig . 5. S a m p l e f r e e - s u r f a c e profiles for random initial conditions.

in a matter of hours the shape of the water table smoothes up and that the strong difference between the values of h at t = 0 and at t - 1 day for x approaching to the origin indicates that the model is not very accurate in areas surrounding the stream.

The irregular effect of the randomness in the initial condition vanishes quickly and as time passes the profile shape resembles the shape of the water-table profile generated by the corresponding deterministic model (see Part I).

Figure 4 shows the water-table profiles when a 0 . 1 m m h -1 of constant deep percolation is applied. Note again that as time increases the profile approaches to the corresponding one developed in Part I.

By altering the number of cycles and the value of q2, the effect of the ground depressions can accordingly be modeled.

An important effect which still remains to investigate is the proximity of

278

the water table to the ground surface. The present model does not consider this fact, and a field investigation would be needed to determine the func- tional relationship between ground-surface level and the corresponding groundwater level.

Now let us consider the case when the transient flow equation is randomly perturbed by a white Gaussian noise process in time only. The mean value of eqn. (36) coincides with the deterministic solution (Part I), since the expected value of the random term in it is equal to zero. Noting that dbn = wdt and that (see Frost and Kailath, 1969):

1 E{w~w~} = ~ 6(u-~)

the correlation function is:

E{h(x, t, )h(x, t2 )} = V(x)[V(x) + Z(x, t, ) + Z(x, t2 )1 + Z(x, t, )Z(x, t2 )

+ 2qlS ~ ~ Mn(tl)Mm(t2)sin(XnX)sin(XmX) L2T n=l m:

1 "X2X 2 - [ M n ( - 2 t l ) - l ] n m " [ 1 - - C°S(X~L)] [ 1 - - COS(Xm L)] (48)

where:

Similar to the previous case of random initial conditions, sample functions are easily obtained by using:

= V(x) + Z(x, t) + q,

• U.(k)sin(XnX) (50)

where Un (k) is a normally distributed random number with mean equal to zero and variance 02, = 1/Xn. Again the accuracy of this discretization scheme of the stochastic integral depends on the size of time steps taken.

Now, the model will have to reproduce the white noise variance parameter q 1, which is an input data obtained from an analysis of field measurements in piezometers of the oscillations of groundwater level. Hence the following relationship holds in agreement with the central limit theorem (Papoulis, 1965):

q , = On 2 = - - ( 5 1 ) n =1 =1 ~n

279

In the computer routine, the program calls the IMSL subroutine GGNML for generation of a set of standard normal deviates R(k). Then the discrete series of white noise are obtained as:

U , ( k ) = R ( k ) " (52)

As the counter n changed, the cumulative variance was checked against the value of q 1 and the series was t runcated when both figures coincided.

The resulting sample water-table profiles are a family of sinusoidal curves of the type presented in Part I with a t ime-random perturbat ion affecting the entire watershed. Generally this model presents an adequate description of the evolution of the water table with time under uncertain conditions, and with a few adjustments could be easily implemented in applications of hydrologic analysis of regional groundwater flow. The main adjustments would consist of a determination of the range of values of the parameter q~ or an investigation of alternative stochastic process appropriate for groundwater modeling, and an inclusion of a functional relationship between the ground-surface level and the groundwater level. This task would require a field investigation of the water-table behaviour of a particular watershed with an extensive piezometric network.

4. CONCLUDING REMARKS

Similar to the deterministic case, the analysis of stochastic partial differ- ential equations in groundwater flow is best accomplished by using functional- analytic theory. Questions concerning the existence of uniqueness of the solution to a particular problem can only be approached from this rigorous point of view. The authors believe these concepts will find increasing accep- tance among water resources researches.

The form of the solution to a stochastic partial differential equation is best obtained by treating the equation as an abstract stochastic evolution equation for an elliptic partial differential operator in an appropriate Sobo- lev space. As for the deterministic problem described in Part I, the use of semigroup theory presents encouraging results in finding solution expressions of abstract stochastic evolution equations. An important conclusion drawn from the exercises of Part I and Part II is that the deterministic and the stochastic cases of boundary-value problems can be studied, analyzed and solved under the same theoretical framework.

The application presented as a stochastic analogue of the regional ground- water flow problem solved in Part I shows how the theory can be used to obtain solutions to a randomly forced and a randomly initiated stochastic partial differential equation. The random case of the sinusoidal boundary condit ion proposed by Toth (1963} seems to be an adequate representation

280

of the r a n d o m free-surface b o u n d a r y cond i t ion subject to variable soil depressions. The explici t so lu t ion derived by expressing the Wiener process as a basis func t ion in a Hilbert space c o m p o s e d of infinite and independen t unid imensional Wiener processes with incrementa l variance pa ramete r seems to be a useful device.

The mode l presented could be imp lemen ted for hydro log ica l analysis af ter fu r ther field research is done concern ing the func t iona l relat ionship be tween the g round surface and the g roundwa te r level. Fu r the r research should be devoted to s tudy the appropr ia te s tochast ic processes to mode l the shape of the water table and the magn i tude o f its parameters for actual field c i rcumstances .

LIST OF SYMBOLS

A bn bnr bi(t) B C Co[G;L2(~)]

D D k f ei e(.) Ilfll

(f, g)a

(f, g)l g(x, t, ~ ) G bG h ho(x) H H m

H~ I Jt J(co) K L L1 L2(~ , P;H) M Mn tl

A partial differential operator Fourier coefficient Random Fourier coefficient Unidimensional Brownian motion processes A Borel field Constant head at left boundary Space of infinitely differentiable second-order random functions in G with compact support Differentiation in Hilbert space kth order derivative of f Orthonormal basis in H Expectation operator Norm of f Norm of f in the sense of Hilbert space L2 (~) Inner product in the Hilbert space L2 (~) of random functions, it is also written as E{(f, g)) Hilbert space scalar product for random functions f Random forcing function, generally Spatial domain in R s. Boundary of G Piezometric head Initial water table Separable Hilbert space mth order Sobolev space of second-order random functions equivalent to the deterministic H m space mth order Sobolev space with compact support Deep percolation A strongly continuous semigroup Random boundary condition Hydraulic conductivity Aquifer length Space of inte~able distributions Space of square-integrable random functions from ~ × ( ' ) into H Borel set, second-order random functions on G A sequence of exponential functions An index

281

o(o ) O'{G;L2(~)} P( ') ql Q n(k) S

S s u p

t T u(x, t, ~ ) uoCx, ~) Un

V V' 1'I~ 0, T) wl(t) w2(x) x

y ~(t) 6 A ki kn ~(x)

,P ff

Kit) On

Z O9

R ~n

C oo

Space of test functions f from G into R Space of L~ (~)-valued distributions on G A probabi l i ty measure White noise variance parameter A boundary operator Standard normal deviates Dummy variable, random walk vertical displacement Specific storage Supreme value (mimum of maximum value) Time coordinate Time domain, aquifer transmissivity State of system at t ime t Random initial condit ion Normally distributed random number with mean zero and variance l/~n A real separable Hilbert space, steady state function The dual of V A Hilbert space in [0, T] White noise process in time White noise in space Spatial coordinate A stochastic process Brownian (or Wiener) process in time Delta function Increment or interval Inverse of Brownian mot ion variance parameter Eigenvalues White noise process in space Dummy variable A test function, empty set 3 . 1 4 1 5 . . . White noise process in time Variance Summation Sample elements of probabil i ty space Probabil i ty space One-dimensional real space nth dimensional real space Partial derivative Subset of Infinity

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