As the main objective of this dissertation, an inversion algorithm, SIRT-Cimmino, is developed to solve the shortcomings of SIRT. SIRT-Cimmino adopts the Cimmino iteration strategy with adjusted incremental corrections and an iteration-dependent relaxation parameter. The algorithm is verified by three numerical experiments and applied in a field test site in Göttingen, Germany. The SIRT-Cimmino shows its advantage to traditional inversion algorithm (i.e., SIRT), and a great potential for a more accurate and reliable characterization of diffusivity spatial distribution. The development of the algorithm, the analysis of the iteration step sensitivity, and the application of the tomographical idea shed lights on the understanding bridging mathematical science and practical problems.
Conclusion
The feasibility of SIRT-Cimmino and the residual based result selection rule are verified by three numerical models with different diffusivity distributions, i.e., inclined high-D layer, Y-shaped high-D zone, and aquifer analogue outcrop. In total, 44 pumping tests are simulated, and 520 travel times are obtained. The inversion results are compared to the true distribution, and the following conclusions are drawn.
• SIRT-Cimmino rebuilds the main feature (i.e., the high-D continuum) with better continuity and more precise values than SIRT.
• SIRT-Cimmino resolves the difficulty in determining the optimal number of iteration steps (NIS). The NIS, determined from the result selection rule in SIRT-Cimmino, provides a good agreement between structural similarity and numerical accuracy. Meanwhile in SIRT, the NIS with a satisfactory structure is not consistent with the NIS with an acceptable
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accuracy. This advantage has a great impact on the field study due to the lack of reference distribution.
• The investigation of SIRT-Cimmino under high resolution indicates that (a) the reconstructions are generally stable and reliable; (b) SIRT-Cimmino faces difficulties on the reconstruction of horizontal features, and deviations on the shape might occur.
Differs from the numerical studies, the field study is the first time of using travel time based hydraulic tomography in a fractured porous aquifer. 13 cross-well multi-level short term pumping tests are performed, and 13 travel times are obtained. The field study demonstrates that the distribution obtained by SIRT-Cimmino has a higher agreement with the result of the type curve analysis and previous thermal tracer test. This demonstration is concluded from two following observations.
• SIRT-Cimmino exhibits two high-D continua, which were found in the previous thermal tracer test. SIRT only rebuilds one continuum, i.e., the continuum at the lower part of the third screens.
• The values by using SIRT-Cimmino range in the proper interval, which is estimated by the type curve analysis of the cross-well multi-level pumping tests. SIRT overestimates diffusivities in some cells.
In overall, the models and field studies reveal that SIRT-Cimmino can offer great benefits over the traditional SIRT.
Outlook
The thoroughly examined the SIRT-Cimmino algorithm allows us to further understand and improve the travel time based hydraulic tomography. In particular, three concerns from mathematical, software and field perspectives can be addressed as follows.
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First, the travel time based hydraulic tomography is a nonlinear and ill-posed inverse problem. Three questions remain unanswered: (a) does a solution exist?
(b) is the solution unique? (c) how do the initial and boundary conditions affect the solution?
(a) We assume that the measurement is errorless, the mesh is fine enough, the ray-tracing technique simulates the trajectory precisely, thus the existence of the solution only depends on the line integral bridging the diffusivity and travel time.
Note that the derivation of the line integral is based on the smooth varying hydraulic parameter field. The feasibility of travel time based hydraulic tomography in a highly heterogenous test site is not promised. (b) SIRT-Cimmino and SIRT provide different approaches with small residuals. It implies the non-uniqueness of the solutions. Moreover, the relationship between non-non-uniqueness and grid resolution becomes complex due to nonlinearity. The oscillating behaviour in the residual need to be explained theoretically. (c) In general, the inversion is beneficially influenced by the appropriate initialization and constraint setting, which can be established through hydraulic test, geological survey, and previous research.
Second, ray-bending technique and 3D inversion could improve the inversion performance. The ray-bending technique provides curve segment and fines the trajectory simulation. The use of multiple raying techniques not only increase the accuracy, but alsoaccelerates the iteration. This work is limited to the 2D inversion for an aquifer profile, the estimated trajectory is therefore the projection of trajectory in the 3D aquifer on the 2D profile. This dimensional gap might lead to a mischaracterization.
Lastly, the field application of an algorithm contains a lot of challenges, which stem from the difference between theoretical and field works. Noise, well construction, interference of other wells, and geometry of the study profile are
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critical points. They restrict the utilization of some methods, which can improve the result quality in the numerical experiment. For instance, the early time diagnostic. In this dissertation, kinds of countermeasures are implemented to overcome the difficulties or reduce the influence, but some of them might not suit other test sites due to the different field conditions.
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