摘要：We introduce an algorithm to solve an inverse problem for a non-linear system of
partial differential equations, which can be used to estimate oil water
displacement functions. The direct model is non-linear because the sought for
parameter is a function of the solution of the system of
equations. Traditionally, the estimation of functions requires the election
of a fitting parametric model and thus the optimum curve depends on that
election. We develop an algorithm that does not require a parametric model
and thus provides a more objective fit. The estimation procedure is carried out
linearizing the solution of the direct model with respect to the parameter
and then computing the least squares solution in functional spaces. We
present the partial differential equations that are used to compute the
Fréchet derivative. The resulting method has shown convergence in numerical
tests, and because of its general theoretical formulation has the potential
to be extended to solve more complex problems. The main contribution of this
work is the formulation and application of the algorithm described above to
estimate non-linear parameters in functional spaces. This algorithm obtains
the sought-after parameters without the imposition of a priori parametric
models. Though the use of such models is currently the common practice among
field engineers, different models yield different results and there is no
objective criterion to choose among them.
