hydraulic conductivity and water fluxes at the west boundary (Vosges mountains) and gives good results. Calibrated hydraulic conductivity and fluxes were ...
Calibration and Reliability in Groundwater Modelling (Proceedings of the ModelCARE 96 Conference held at Golden, Colorado, September 1996). IAHS Publ. no. 237,1996.
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Groundwater flow parameter identification using a downscaling parameterization: a case study
E. CHARDIGNY, R. MOSE, P. ACKERER Institut de Mécanique des Fluides, Université Louis Pasteur, URA CNRS 854, 2 Rue Boussingault, F-67000 Strasbourg, France
P. SffiGEL Institut fur Hydraulik, Gewàsserkunde und Wasserwirtschaft, Technische Universitât Wien, Karlsplatz 13/223, A-1040 Wien, Austria
F. MEHREB BURGEAP, 15 Rue du Parc, F-67205 Oberhausbergen, France
Abstract Model calibration for groundwater flow is very often required because the knowledge on the spatial distribution of parameters is very incomplete. Typically, most parameter identification procedures are based on minimizing an objective function which is equal to the sum of the quadratic difference between observed and computed piezometric heads. The adjoint state method has been used to calculate the gradients of the objective function. Parameterization is based on a linear interpolation of the parameter on a "parameter grid" which can be refined (downscaling). The model calibration consists in estimating the parameter at the nodes of this mesh. The domain studied in this case is part of the Rhine aquifer which is situated south of the town of Strasbourg, France. This aquifer is made of sand and gravel deposited in the Rhine erosion valley. The calibration procedure consists of the estimation of the hydraulic conductivity and water fluxes at the west boundary (Vosges mountains) and gives good results. Calibrated hydraulic conductivity and fluxes were consistent with the existing "soft" information. Reliability of the hydraulic conductivity has been estimated.
INTRODUCTION The knowledge of the required parameter distribution in an aquifer is one of the most difficult aspect in modelling groundwater flow. The natural way to get these required parameters would be to use a deterministic description of the heterogeneity. This requires a "sufficient" number of measurements and, if necessary, an upscaling procedure to find equivalent parameters which reproduce the average behaviour of the system discretized at the scale of elements or cells. Because of the high cost of measurements, the amount of data available is not enough to allow this type of description of the aquifer heterogeneity. Therefore, model calibration is very often necessary. The nature of calibration is not to model the processes in details but to model the bulk behaviour of the physical system (patterns of the parameters, in/out flows, etc.) which gives one possible description of the aquifer. Much work has been done in the field of automated
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flow model calibration. Detailed reviews are presented in Yeh (1986), Carrera (1988), Ginn & Cushman (1990) and Sun (1994). Our paper will focus on parameter identification for groundwater flow. We will present briefly the forward problem (simulation of head distribution), describe the objective function and its associated minimization algorithm and discuss in detail a new parameterization technique. This procedure is applied to a part of the Upper Rhine River aquifer south of Strasbourg (France).
The mathematical model and its numerical formulation The mathematical formulation of the behaviour of a slightly compressible fluid in an undeformable porous media is based on Darcy's law and the equation of continuity of mass. Combining and integrating Darcy's law and continuity of mass over depth under Dupuit assumption yield the general formulation for a 2D unconfined aquifer: 5 — + V(TVh) + \(H-h)+Q dt
=0
(1)
subject to the following initial and boundary conditions: h(x,y,0) = hQ(x,y)
x,yEQ
h(x,y,t) = hAx,y,t)
x,yev,
(2)
dh an
K(h
where S is the specific yield (-), ZsTthe hydraulic conductivity tensor (L T"1), hh(x,y) the elevation of the bottom of the aquifer (L), X is the hydraulic conductance of the sediment under a river or of an aquitard under or over the aquifer (T 1 ), H is the head at the stream bottom or at the distal side of the aquitard (L),x,y are the space variables, t the time, albn the normal derivative, fi the flow region, Yx the boundary of the aquifer subject to a Dirichlet type boundary, T2 the boundary of the aquifer subject to a Neuman type boundary. The internal source/sink terms Q (L T"1) describe pumping wells (negative values) or injection wells (positive values) and groundwater recharge. Equation (1) is solved by a standard finite element method (e.g. Kinzelbach, 1986). Performing time integration using finite differences and an implicit scheme, the approximate solution of h is expressed in the form of: (R)+(P)h(t + At) h(t + At) = —(R)h(t)+(F) At AT
(3)
where the matrices P and R and the vector F are defined by:
P,
Ï
K(h-hh)
Rtj = J SoijUj dxdy
dco; do): da)j do): dx ax dy dy
dxdy (4)
Groundwaterflowparameter identification using a downscaling parameterization
*"• =
qu>tâxây •
139
qnoijds
The resolution of equation (3) requires a discrete representation of the aquifer parameters. This means that it is not feasible to compute exact head values for a continuous parameter distribution even if measurement errors are negligible. Because of the averaging of the small scale spatial and temporal variability of, say, hydraulic conductivity or recharge over the volume of an element or a time step, the predicted head distribution will always be smoother than the actual one. Objective function and minimization Because of the recognition that head and parameter measurements are subject to errors, the objective function is treated in a statistical way. The more popular approaches are weighted least squares (Neuman & Yakowitz, 1979) and maximum likelihood estimation (Carrera, 1984). Although built under more general assumptions, the maximum likelihood approach leads to a similar objective function to the weighted least squares method if the observation and modelling errors are assume to be Gaussian distributed and independent. We used the classical objective function/(f) defined as follows: J(p) = [hm-h{p)fC~hl[hm-h{p)}
(5)
where hm are the measured heads, h(p) are the heads computed with parameters/» and Ch is the covariance matrix of the head measurements. Estimation of the covariance matrices is a serious problem for practical cases. Head measurement errors are usually assumed to be statistically independent and identically distributed. With this assumption, the co variance matrix is a constant times the identity matrix. This assumption may be exact for steady-state conditions but needs discussion for transient flow systems. The minimization of J(p) is done by iteration. In each iteration step the parameter set is modified so that the objective function decreases: pi+1 =pi+pidl
(6)
where p' is the parameter set at iteration i, p' is a step size along the displacement direction d'. The most popular methods for calculating p1 and d' in groundwater inverse modelling are the quasi-Newton method which requires the computation of the derivatives of J(p) with respect to all model parameters and the Gauss-Newton or modified Gauss-Newton methods which require the computation of the sensitivity coefficient. The quasi-Newton method has been preferred because the number of calibrated parameters is much larger than the number of measurements. Since the method does not require the computation of the Jacobian matrix, it requires less computer time for calculating the gradient of the objective function. Because less information is provided for the optimization algorithm, more iterations are needed than with the Gauss-Newton method. The algorithm we have used allows simple bounds and is well adapted for solving large nonlinear optimization problems. It is based on the gradient projection method and uses a limited memory BFGS method to approximate the Hessian of the objective function (Byrd et al, 1994).
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The gradient of the objective function is calculated by the adjoint state method (Chavent, 1971). The function L(h,p) is defined by: L(h,p)
J(Pk)
(R) + (P)h(t + At) h(t + At) (7)
1 (R)h(t)-(F),y At which is the addition of the objective function and a scalar product equal to 0 when h is solution of equation (3). y is the a Lagrange multiplier. The differentiation of L(h,p) leads to:
8L(h,p) = F
^(h,p)bhJW,p) dh
dp
(8)
y
The Lagrange multipliers are the solution of the equation:
S
i
=0
(9)
dh and therefore: dJ(p) dp
[^W^\dp - {At\dp\
+
^ l \\]
h(t + At) (10)
1 IdR] [fJFl h(f)
