exception of the approach proposed by Woodbury ej. aL (1987), these appear to have been built with demonstration purposes in mind and lack generality.
ModelCARE 90: Calibration and Reliability in Groundwater Modelling (Proceedings of the conference held in The Hague, September 1990). IAHS Publ. no. 195, 1990.
INVERSE MODELLING OF COUPLED FLOW AND SOLUTE T R A N S P O R T PROBLEMS
A. MEDINA, J. CARRERA k G. GALARZA E.T.S.E. Camins, Canals i Ports, Universitat Politècnica de Catalunya, C / Jordi Girona Salgado, 31, 08034, Barcelona, Spain
ABSTRACT Considering concentration data during the calibration of aquifer models improves the reliability of results because identifiability may be improved parameter uncertainty can be reduced, and concentration data are usually cheap to obtain so that large amounts of data can often be taken into account. The drawbacks arise from the numerical and conceptual difficulties associated to solute transport. While these may lead to intractable problems in some cases, they can be overcome in others. An algorithm for estimation of flow and solute transport, including matrix diffusion and adsorption processes, is presented. The formulation of the problem is based in Maximum Likelihood Estimation theory, in which prior information in model parameters can be included. Prior information may come from qualitative sources or may be hard-data based, including geostatistically based. Minimization of the objective function is achieved by Gauss-Newton type algorithms, which work very well after suitable modifications. One example is presented in order to illustrate the efficiency of the proposed algorithm. INTRODUCTION The predictive capabilities of a model depend on several factors. First, the model should include all the physico-chemical processes that are relevant to the simulated output. Second, its structure should be sound, in the sense that the geometric aspects of the model, its boundary conditions and variability patterns should be similar to those of the real system. Third, the values of model parameters should be representative of their physical counterparts. And, finally, the solution of the involved equations should be accurate. The last requirement is of a strictly numerical nature and, while accuracy is not always easy to ensure, we shall assume that the solution algorithms are sufficiently accurate. The choice of numerical values for model parameters is made during calibration, which consists of finding those values that grant a good reproduction of the past and are consistent with geological information and available tests data. Calibration is rarely straight-forward. Data come from various sources, with varying degrees of accuracy and levels of representativeness. Some parameters can be measured directly in the field, but such measurements are usually scarce and prone to error. Furthermore, since measurements are most often performed on a scale different from that required for modelling purposes, they tend to be b o t h numerically and conceptually different from model parameters. The most dramatic example of this is dispersivity, whose representative value increases with the scale of mesurement, so that dispersivities derived from tracer tests cannot be used directly in a model. As a result, model parameters are calibrated by ensuring t h a t simulated heads and concentrations are close to the corresponding 185
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measurements. Calibration can be tedious and time consuming because many combinations of parameters have to be evaluated, which also makes the process prone to be incomplete. This, coupled to the difficulties in taking into account the reliability of different pieces of information, makes it very hard to evaluate the quality of results. Therefore, it is not surprising that a very significant effort has been devoted to the development of automatic calibration methods. The idea of automatic calibration is very old (see Yeh (1986) or Carrera (1987) for detailed reviews). However, because of various conceptual and numerical difficulties, most efforts have been concentrated on the flow equation. Work on the inverse problem of the solute transport equation is rather scattered. The contributions most relevant to the work reported here are those of Strecker & Chu (1986), Chu et ah (1987), Wagner & Gorelick (1986), Medina e l ah (1987), Medina e t aL (1989), Carrera et aL (1987), Woodbury et ah (1987), Woodbury & Smith (1988) and Graham & McLaughlin (1989). However, with the possible exception of the approach proposed by Woodbury ej. aL (1987), these appear to have been built with demonstration purposes in mind and lack generality. The objective of this paper is to present an algorithm that can be used in a wide number of instances, thus including the most common physico-chemical processes and various alternatives for the type of regime,%nd yet is efficient. DIRECT PROBLEM EQUATIONS In subsequent derivations and discussions, we will concentrate on flow and transport of a single specie. Groundwater flow is governed by: V{KVh)
+ w=
5S—
on Q,
(1)
where K is hydraulic conductivity tensor; h is head; to is a distributed sink/source term; V is the "del" operator (divergence or gradient); Ss is storativity; t is time. Equation (1) is solved on the domain 0 subject to the boundary conditions: (KVh)n
= a(H-h)
+Q
on T
(2)
where n is the unit vector perpendicular to Y and pointing outwards; H is an external head; Q is prescribed flow; a is a coefficient controlling the type of boundary condition ( a = 0 for prescribed flow; a = oo for prescribed head; and a yé 0, oo leakage condition). Initial conditions have to be specified on Q, though they are often chosen as the steady-state of (1). Solute transport of a single radioactive non-reacting specie is governed by the advection-dispersion equation: BC V(D VC)-V(qC)-\cj>RC+w{Ce-CyaDm4>m-~
dC = 4>R-^
on 0
(3) where D is the dispersivity tensor; q is Darcy's velocity; is porosity; C e is the concentration of water entering the aquifer(C e = C for the discharging portions of the aquifer); A is the radioactive decay constant; R is the retardation coefficient; m and Dm are matrix porosity and diffusion coefficient,
