Least squares estimation of hydraulic conductivity from field data

Least Squares Estimation of Hydraulic Conductivity from

Field Data

K. R. Bailey

,

B. G. Fitzpatrick y

,

and M. A. Jeris z

Center for Research in Scientic Computation

North Carolina State University

Raleigh, NC 27695-8205

Abstract

In this paper, we present some numerical re-sults of the determination of ow parameters in a groundwater model. The data used in this pa-rameter estimation is from the MADE experi-ments conducted on Columbus Air Force Base. Our results are based on least squares cost func-tional with a nite dierence scheme used to solve the ow equation.

1 Introduction

The modeling of groundwater ow is a cru-cial step in any quantitative analysis of subsur-face contamination. From site characterization to remediation, ow models provide important decision-making tools. The main diculty in ap-plying such ow models is the determination of ow parameters such as hydraulic conductivity, which may vary by several orders of magnitude over a particular eld site. In this paper, we ex-amine a least squares parameter estimation tech-nique which allows models to be calibrated to eld data.

The approach we take here begins with a sim-ple model of ow in a saturated porous medium. The coecient and boundary values must be es-timated as functions, from discrete observations made in the interior of the region of interest. A

Research supported in part by AFOSR grant F49620-93-1-0355 and by a Department of Education GAANN Fellowship, grant P200A40730

yResearch supported in part by AFOSR grant F49620-93-1-0153.

zResearch supported in part by AFOSR grant F49620-93-1-0153.

simple nite dierence scheme, containing ux balance conditions over conductivity discontinu-ities, is used to approximate the ow equation. Piecewise constant functions are used for the conductivities, and piecewise linear functions are used for the boundary values. For conductivity estimation, we use total variation constraints in the least squares t-to-data criterion.

The basic model of steady state ow of ground-water in saturated soil is the elliptic equation

r(krh) = 0 x2; (1)

where k is the hydraulic conductivity, h is the hy-draulic head, and is a subsurface region (in IR2

or IR3) of interest. The equation (1) is based on

Darcy's empirical law, which states that ground-water velocity is proportional to the hydraulic gradient

v =?krh; (2)

which, when combined with incompressibility and conservation of mass arguments, leads to (1). The references Freeze and Cherry (1975) and Bear (1972) provide a detailed discussion of the Darcy model.

For site characterization and remediation anal-yses, the groundwater velocity must be com-puted. In order to determine the velocity, one must solve (1), for which we need to know the function k and the boundary values of h. The inverse problem of interest here is the determina-tion of the funcdetermina-tion k and g = hj@

from

point-wise or distributed observations. In general one expects k to be discontinuous (due, e.g., to lay-ering in the subsurface), and it is not known a priori where these discontinuities lie. Thus, in the inverse problem we must search for k's from a very general class of functions.

Typical eld experiments yield data points

f^hign 1

i=1 and f^kjgn

2

j=1, which correspond to

observations of h at the points xi and of

(1=j!jj) R

!jk(x)dx in which !jdenotes a \small"

set in andjAjdenotes the Lebesgue measure of

A (the averaging nature of this observation oper-ator is based on the borehole owmeter measure-ment device: see Rehfeldt,et. al., 1992a, 1992b). To determine parameters based on these data, we dene the least squares cost functional

J(k;g) = n1 X

i=1

jh(xi;k;g)?^hij

2 (3)

+ n2 X j=1 1

j!jj Z

!j

k(x)dx?^kj 2

which is to be minimized over some appropriate collectionKGof functions k and g: The number

is a weighting parameter which must be used due to dierences (by orders of magnitude) in the values of h and k: In the next section, we examine theoretical aspects of this estimation problem.

2 Theoretical

considerations

In this section we recall some important results from analysis and approximation in elliptic par-tial dierenpar-tial equations such as (1). We denote by Hk _{= H}k_{() and H}k

0 = H

k

0() the usual

Sobolev spaces of weakly dierentiable functions. Throughout, we assume that is a bounded open subset of IRm: The notation fj@

is used in

gen-eral to denote the trace of an H1 function. We

also use the notation Hs_{(@) to denote the}

frac-tional order (i.e., s > 0 real) trace spaces of Lions (see, e.g., Wloka, 1987).

To solve the dierential equation (1) subject to the boundary condition hj@

= g; we seek a

function h 2 H

1 of the form h = u + G; where

G 2 H

1 satises G j@

= g and where u 2 H 1 0 satises r

kru?krG

= 0 (4) in : It is well known (see, e.g., Wloka, 1987) that there is a bounded linear extension operator Z:H1=2(@)

!H

1 such that the trace of Zf is

f; for each f2H

1=2(@): The determination of h

is then based on solving (4) for u: Mulitplying in

(4) by a test function and integrating by parts, we have

k(u;) = Fg;k(); (5)

in which the bilinear form k:H1 0

H

1 0

!IR is

given by

k(; ) =

Z

k(x)r(x)r (x)dx; (6)

and the bounded linear functional Fg;k:H1 0

!IR

is given by Fg;k() =R k(x)

rG(x)r(x)dx; in

which G = Z(g)2H

1: We summarize the

appli-cation of the Lax-Milgram theorem and the weak maximum principle in the following theorem (see Treves, 1975, or Wloka, 1987, for details of the proof).

Theorem 1

If the domain is a bounded open set with Lipschitz continuous boundary @, if

g2H

1=2(@) and k 2L

1(), and if there exist

positive numbers ; such that k(x)

a.e., then the equation (5) has a unique solution

u2H 1

0; and the equation (1) has a unique

solu-tionh2H

1 satisfyingh j@

= g:

We remark that in the proof of this theorem, which is essentially contained in the references Treves, (1975), and Wloka (1987), the coercivity and boundedness of are crucial: satises

kk 2

H1 0

k(;); jk(; )jkkH1

0 k kH1

0

in whichkk 2

H1 0

=R

jr(x)j

2dx is the norm on

H1 0():

Another crucial estimate involves the sup norm of the solution. Below we state a form of Har-nack's inequality, which may be found in full gen-erality in Gilbarg and Trudinger (1983). First, we need some notation. If 0 and 1are open sets,

we write 0

1if the closure 0is a compact

subset of 1:

Theorem 2

(Harnack's inequality) If is a bounded open set and if!;then there exist

constantsC; > 0such that

kukC (!)

CkukL2 ()

for each pair u2H 1;k

2 L

1() such that 0 <

k(x);such thatr(kru) = 0;whereC

As noted in Gilbarg and Trudinger (1983), the particular choice of the constants C and de-pends only on = and dist(@;!), and not on the particular coecient function k2L

1 or

so-lution u 2 H

1: It is sup norm regularity that

enables us to handle pointwise observation op-erators in the inverse problem, as we shall see below.

For the parameter estimation problem, we let

K = f2L

1() : 0 <

(x);

a:e:x2;TV () < g

where the total variation TV () is dened as TV () = supn

Z

(x)? rF

(x)dx: F2C

1(;IRm); sup

x2

jF(x)j1 o

; where IRm with m3: In Gutman (1990)

it is proved thatKis a compact subset of L 1();

when is a bounded open set in IRm_{: For the}

boundary conditions, we x a positive constant ; and we set

G=f2H 1(@):

kk 2

H1 (@)

g;

which is compact in H1

2(@) if the boundary @

is Lipschitz continuous (see, e.g., Wloka, 1987). The total variation constraints allow for discon-tinuous conductivity functions: this exibility is lacking in most other types of compactness con-straints.

Theorem 3

Let(gn) be a sequence of functions

in G such that gn ! g in G. Let(kn) be a

se-quence of functions in Ksuch that kn!k 2K

in the L1-norm. For each n, dene u

n2H 1 0()

to be the Lax-Milgram solution of (6) using kn

and Gn, where Gn= Z(gn)2 H

1() is the

ex-tension of gn. Then, un ! u in H

1() where

u is the Lax-Milgram solution using k and G, and G = Z(g) 2 H

1() is the extension of g.

Moreover, we have that for each open set!0with

!0

, hn(;gn;kn) ! h(;g;k) uniformly

on!0; wherehn= un+ Gn;and h = u + G:

The proof of this result, contained in Fitz-patrick and Jeris (1994), relies in a crucial way on Harnack's inequality as stated above.

In general, there are two main obstacles to im-plementation of this approach: the function h

must be computed numerically and the setKG

must be approximated by a nite dimensional set. The paper of Fitzpatrick and Jeris (1994), contains analysis of this least squares problem in terms of existence of minimizers and convergence of numerical approximations to the least squares cost. The results contained therein pertain to standard nite element methods for solving the ow equation, together with rather smooth ap-proximations to the conductivity parameter. In the present work we use a simple nite dier-ence scheme with piecewise constant conductiv-ity functions. The scheme is easy to implement and captures the the behavior of the conductivity rather well. We do not, however, have a complete convergence analysis, and this matter is a topic of our current research. In the following section we give the details of our numerical approach for these approximations.

3 Numerical

Implementation

Our numerical scheme begins with the simplify-ing assumption that the ow is two dimensional. While it would seem a natural assumption for a shallow aquifer, in which the horizontal dimen-sions are 30 to 50 times the vertical dimension, we shall see in the computations below that this as-sumption may not be valid. However, in order to develop some insight into the ow at a particular site, using two dimensional slices is an instruc-tive rst step. Thus, we assume that the region of interest is a rectangle: = (a;b)(c;d): We

subdivide this rectangle into m1 m

2

subrectan-gles of equal lengths and widths, (b?a)=m 1 and

(d?c)=m

2; respectively. We assume that the

conductivity k is constant on these subdomains. The next step is to approximate the dieren-tial equation with a dierence equation. We di-vide the region into a grid of size `1

` 2; with

`i = simi+ 1 for i = 1;2 where the si's are

integers. This assures that the nite dierence grid is aligned with the conductivity grid. Solv-ing the dierential equation from the boundary values in the forward problem produces a system of (`1

?2)(` 2

?2) equations and unknowns. The

interior of a conductivity subregion, we have hi+1;j

?2hi;j+ hi ?1;j

x +hi;j+1

?2hi;j+ hi;j ?1

y = 0;

in which x and y denote the spatial discretiza-tion length and width for the nite dierence grid, and hi;j is the computed value of the

hy-draulic head at the (i;j) grid point. Note that the conductivity is constant in the subdomain containing all the grid points, so it may be di-vided out of the equation. If the point (i;j) lies on a conductivity subdomain boundary which is parallel to the x axis, we use the ux balance equation

ka(hi;j+1 ?hi;j)

y = kb(hi;j

?hi;j ?1)

y ;

in which ka denotes the value of conductivity

above the boundary and kbdenotes the value

be-low. Likewise, if the point (i;j) is on a boundary parallel to the y axis, we use

kl(hi;j?hi ?1;j)

x = kr(hi+1;j ?hi;j)

x ;

where kl and kr denote the conductivity values

from the subdomains to the left and right, respec-tively, of the boundary. Points on the \corners" of the subdomains appear in none of the other equations, so it is unnecessary to compute these values (we take the average of the four nearest neighbor points for the purposes of generating graphs below).

To solve the forward problem for a particular choice of k and g values, we use the GMRES al-gorithm (see Saad and Schulz, 1992) to solve this system of (`1

?2)(` 2

?2) linear equations. This

forward solver is then coupled with the FOR-TRAN optimization routineIFFCO(see Gilmore,

1993, and Gilmore and Kelley, 1992), which is a quasi-Newton iteration coupled with implicit l-tering (through careful dierencing) for avoiding local minima (Our original eorts usingLMDIF1,

a FORTRAN implementation of the Levenberg-Marquardt algorithm for nonlinear least squares problems, available through netlib, encountered some diculties due to local minima).

The upper and lower bound constraints are implemented directly in IFFCO. The integral and total variation constraints on g and k, how-ever, are more dicult to implement directly and

must be incorporated into the cost functional as penalty terms. The cost functional which we use is given by

J(k;g) = n1 X

i=1

jh(~xi;k;g)?^hij 2

+1 Z

@ jg

0 j

2

ds +2

X

(i;j) X

(l;m)2N(i;j) q

+jki;j?kl;mj 2;

+3

n2 X

j=1

jk(~yj)?^kjj 2;

in which 1;2; and 3 are weighting parameters

that must be chosen, and is a smoothing pa-rameter. The nal term in the cost functional approximates the total variation of the function k. The notationN(i;j) denotes the k subdomain

indices having sides adjacent to the (i;j) subdo-main. Since the approximation takes k to be a piecewise constant function, we use the notation k(i;j) and k(x) for values of k, the former being

the value on the (i;j) subdomain. The nal piece of notation is the point ~yj, denoting the center of

the region !jin which the borehole owmeter

op-erates. We now proceed to an examination of the MADE data.

4 Conductivity

Estimates for the MADE

site

The papers of Rehfeldt et. al. (1992a, 1992b) and Boggs et. al. (1993) describe in detail the MADE experiments as well as some rather ex-tensive data analysis. The site over which the measurements were taken is roughly 400600

feet in horizontal area and about 30 feet in depth. The site consists of a very heterogeneous mixture of soil types on the Columbus Air Force Base is Mississippi. Approximately 60 borehole owme-ter measurements from the MADE-1 experiment are used in our analysis, with about 10 head mea-surements taken from multilevel sampling well.

in the horizontal slab between 60 and 61 meters above sea level. The dramatic dierence in veloc-ity elds indicates the possible need for three di-mensional computations, which we are currently pursuing. These computations used 1010

sub-domains for the conductivity, with 5151 grid

points for the head computations. Three linear spline functions per side were used to estimate the boundary values at the edge of the rectangle. After many sensitivity studies, values of ;1;2;

and 3 were chosen at 10

?2; 8,2, and 8 for our

computations.

In Figures 1 and 2, the asterisks indicate the vertically averaged data from the MADE experi-ments, while in Figures 4 and 5, the asterisks in-dicate the actual data in the 60-61 meters above sea level vertical range. The velocity plots con-tain contour lines of the hydraulic head, subdo-main boundaries for the conductivity estimates, and circles to indicate the borehole owmeter locations (where conductivity data are taken). We remark that the heterogeneous nature of the medium is particularly evident in the velocity plots. In a homogeneous medium one would ex-pect \closeness" of level curves of head to indicate high velocity regions. However, the variation in the conductivity is also a factor, one which seems in this study to be the driving inuence.

5 Acknowledgements

The authors would like to thank Dr. T. B. Stauf-fer and Dr. S. Kang, AL/EQ, Tyndall AFB, for providing the MADE experiment data, and for their helpful comments concerning the exper-iment and the data collection and analysis.

6 References

H.T. Banks and K. Kunisch, 1989. \Estima-tion Techniques for Distributed Parameter Systems," Birkhauser, Boston.

J. Bear, 1972. \Dynamics of Fluids in Porous Media," Elsevier, New York.

J. M. Boggs, L. M. Beard, S. E. Long, M. P. McGee, W. G. MacIntyre, C. P. Antworth, and T. B. Stauer, 1993. Database for the Second Macrodispersion Experiment

(MADE-2), Electric Power Research Insti-tute Technical Report TR-102072, EPRI, Palo Alto.

B. G. Fitzpatrick and M. A. Jeris, 1994. \Pa-rameter Estimation in Groundwater Flow Models with Distributed and Pointwise Observations," Technical Report CRSC-TR94-17, Center for Research in Scientic Computation, North Carolina State Uni-versity, Raleigh.

R. A. Freeze and J. Cherry, 1979. Groundwater,

Prentice-Hall Englewood Clis.

S. Gutman, 1990. Identication of Discontinu-ous Parameters in Flow Equations,SIAM Journal on Control and Optimization

28

(5), pp. 1049-1060.

D. Gilbarg and N. S. Trudinger, 1983. Ellip-tic Partial Dierential Equations of Second Order,Springer-Verlag, Berlin.

P. Gilmore, 1993. \IFFCO: Implicit Filter-ing for Constrained Optimization, Users' Guide, Technical Report CRSC-TR93-7, Center for Research in Scientic Compu-tation, North Carolina State University, Raleigh.

P. Gilmore and C. T. Kelley, 1992. \An Im-plicit Filtering Algorithm for Optimiza-tion of FuncOptimiza-tions with Many Local Min-ima," Technical Report CRSC-TR92-22, Center for Research in Scientic Compu-tation, North Carolina State University, Raleigh; to appear in SIAM Journal on Optimization.

Y. Saad and M. H. Schultz, 1986. \ GM-RES: A Generalized Minimal Residual Al-gorithm for Solving Nonsymmetric Linear Systems",SIAM Journal on Scientic and Statistical Computing,

7

(3), pp. 856{869. F. Treves, 1975. Basic Linear Partial Dieren-tial Equations, Academic Press, Orlando. K. R. Rehfeldt, L. W. Gelhar, S. C. Young,

E. E. Adams, L. M. Beard, and J. M. Boggs, 1992a. \Fieid Study of Dispersion in a Heterogeneous Aquifer: 1. Oveview and Site Description," Water Resources Research,

28

(12): 3281-3291.

Analysis of Hydraulic Conductivity, Wa-ter Resources Research,

28

(12), pp. 3309-3324.

Wloka, J., 1987 Partial Dierential Equations,

Cambridge University, Cambridge.

-300 -200 -100 0 100 200 300

0 200 400 600 202

202.5 203 203.5 204 204.5 205 205.5 206

X (ft)

Y (ft)

head (ft)

Least squares estimate of head, Beta2=2, alpha=1e-2, Averaged K data

Figure 1: Hydraulic head estimate using ver-tically averaged data

-300 -200 -100 0 100 200 300

0 200

400 600

0 0.1 0.2 0.3 0.4 0.5

X (ft)

Y (ft)

K(X,Y) ft/min

Least squares estimate of Conductivity, Beta2=2, alpha=1e-2, Averaged K data

Figure 2: Hydraulic conductivity estimate using vertically averaged data

-200 -100 0 100 200

-100 0 100 200 300 400 500 600

X (ft)

Y (ft)

Velocity vector field, Beta2=2, alpha=1e-2, averaged K data

Figure 3: Velocity eld estimate using verti-cally averaged data

-300 -200 -100 0 100 200 300

0 200 400 600 202

202.5 203 203.5 204 204.5 205 205.5 206 206.5

X (ft)

Y (ft)

head (ft)

Least squares estimate of head, Beta2=2, alpha=1e-2, K data with 60<=z<=61

Figure 4: Hydraulic head estimate using ver-tical data, 60-61 meters

-300 -200 -100 0 100 200 300

0 200

400 600

0 0.2 0.4 0.6 0.8 1 1.2

X (ft)

Y (ft)

K(X,Y) ft/min

Least squares est. of K(x,y), Beta2=2, alpha=1e-2, K data with 60<=z<=61

-200 -100 0 100 200 -100

0 100 200 300 400 500 600

X (ft)

Y (ft)

Velocity vector field, Beta2=2, alpha=1e-2, K data with 60<=x<=61