Leak identification in a saturated unsteady flow via a Cauchy problem

Abstract

Highlights• A model for leak identification in pipes via the Cauchy problem for the heat equation is researched.• The model is reformulated to fit the application of a recently proposed regularising method.• Analyses of the regularizing method is presented.• The regularizing method is implemented using an open source Finite element code.• Conclusions and further research in this area are given. AbstractThis work is an initial study of a numerical method for identifying multiple leak zones in saturated unsteady flow. Using the conventional saturated groundwater flow equation, the leak identification problem is modelled as a Cauchy problem for the heat equation and the aim is to find the regions on the boundary of the solution domain where the solution vanishes, since leak zones correspond to null pressure values. This problem is ill-posed and to reconstruct the solution in a stable way, we therefore modify and employ an iterative regularizing method proposed in [13,14]. In this method, mixed well-posed problems obtained by changing the boundary conditions are solved for the heat operator as well as for its adjoint, to get a sequence of approximations to the original Cauchy problem. The mixed problems are solved using a Finite element method (FEM), and the numerical results indicate that the leak zones can be identified with the proposed method.

Missing boundary data reconstruction for the heat equation

“…The considered inverse problem encountered in many practical applications such as the identification of multiple leak zones in saturated unsteady flow, see previous work 13 and obstacle identification problems. In the framework of the parabolic equation, many approaches have been proposed in the literature for solving this ill‐posed inverse problem with varying degrees of success, we can cite the boundary integral equation approach, see Chapko and Johansson 14 and Onyango et al, 15 iterative regularization methods, we quote previous works 13,16 and Johansson, 17 quasi‐reversibility process, see Dardé, 18 the minimization of a cost functional, see Rischette et al, 19 and the decomposition approach see Lesnic and Elliott 20 …”