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Cooperation with: S.B. Hazra, G. Wittum (Universität Heidelberg)
Supported by: DFG: ``Inverse Modellierung von Strömungs und Transportvorgängen im heterogenen Untergrund auf der Basis von Mehrgitterverfahren'' (Inverse modeling of flow and transport in a heterogeneous subsurface with multigrid methods)
Description:
Multiphase flow is of high importance, e.g., for waste removal in the subsurface. The basic mathematical models are described by certain nonlinear partial differential equations which involve also empirical relationships. For example, the capillary pressure saturation relationship is described by a BrooksCorey formulation with two constitutive parameters: the entry pressure p_{d} and another parameter . These parameters are a priori unknown and cannot be measured or derived from other relationships. For realistic computational simulations of planned in situ remediation processes, these parameters have to be determined via parameter estimation from other magnitudes which can be directly measured.
In our problem, they are to be determined from pointwise measurements of the capillary pressure or saturation of water at several time instances. We employ an outputleastsquares approach resulting in a nonlinear optimization problem with PDE constraints. This yields in particular two difficulties: the high dimension of the discretized optimization problem and its nonlinearity. Since it is wellknown that single shooting strategies result in numerical instabilities, we have developed a multiple shooting approach, where we have to take into account that we are dealing with an instationary PDE rather than with an ordinary differential equation involved in the classical multiple shooting formulation. Therefore we have developed a variant employing the weak formulation of the continuity conditions. In order to cope with the high dimensionality of the problem, reduced GaussNewton methods are used, where derivatives are computed according to the principle of internal numerical differentiation.
We have started out in the year 2000 with academic formulations of parameter estimation problems for the isothermal case. The multiple shooting approach has been implemented within the PDE software toolbox ug. Now we have enlarged the problem class so that we are able to treat nonisothermal threephase threecomponent models, which leads to even more pronounced nonlinearities. On the other hand, we now also incorporated real measurement data. Fig. 1 presents one such result where the parameters and a (scaled absolute permeability) are determined using the measurements of saturation from a onedimensional column experiment carried out at the VEGAS, University of Stuttgart. The figure shows the comparison of computed saturation, using the parameters determined by our method, with the experimental ones. The latter developments are described in detail in [2].
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