Conclusion

We propose an identification strategy for discrete channel flow structures typical of fractured media. We define a parameterization based on the hierarchical organization of the flow channels. Channels are determined iteratively by decreasing order of importance. We set up the methodology on a set of 20 configurations ranging from a couple of straight channels to more intricate interconnected configurations having up to six independent channels. All other flow settings and optimization algorithms have been taken as simple as possible. Boundary conditions are derived from a uniform head gradient. Data are made up of steady-state hydraulic heads and distances from the wells to the nearest channel.

The objective is to identify both geometrical and hydraulic characteristics of the conducting structures. Channels are identified by decreasing order of importance by using successive optimizations of an objective function. The identification strategy takes advantage of the hierarchical flow organization to restrict the dimension of the solution space of each individual optimization. Because of the successive optimizations, main flow channels determine strongly the characteristics of secondary channels. Additionally, main flow channels can be slightly modified by secondary channels through the introduction of a regularization term on the main channel characteristics in the objective function. The regularization term is weighting the rate of variations of the formerly identified structures compared to the newly added structures. Refinement of main channels is performed first before introducing additional less essential channels. The identification is stopped when both the improvements of the channel structures and of the channel number become marginal. A matrix has been introduced in order to replace the channels that should be identified later. The matrix permeability sharply decreases when the identification proceeds. Modifications of the objective function have been introduced to allow marginal modifications of the main flow structures by secondary flow structures. The classical simulated annealing method has been chosen as the optimization algorithm because of the strong non-convex nature of the objective function. For each configuration, the identification strategy has been run 50 times yielding 50 solutions. A post-processing algorithm extracts the prevalent channels from the 50 solutions.

The simplest configurations made up of a couple of straight channels are identified with only the steady-state head data. Identification of similar configurations having tortuous rather than straight channels requires additionally the knowledge of the head deviations. For intricate interconnected configurations made up of up to three tortuous channels, the identification

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strategy needs additional geometrical data taking the form of the distances from the wells to the closest channel. The tested configurations consisting in more than four complex channels were not identified because of the absence of strong enough hierarchy in the channel structure.

The hierarchical channel identification strategy has only been tested on steady-state data. Its extension to transient-state data requires critical improvements of the optimization algorithms.

We argue that this may be possible because transient-state data are more informative than their steady-state counterpart. The other advantage of well test or flowmeter data is to be less dependent of boundary conditions.

Appendix A. Simulated annealing parameterization

Simulated annealing performs a random walk in the parameter space directed towards the minimization of the objective function [Kirkpatrik et al., 1983]. The acceptance of parameter sets leading to an increase of the objective function is necessary to get out of local minima. Its probability however decreases slowly in order to force the algorithm to converge to the global minimum. The simulated annealing algorithm is parameterized by the acceptance mechanisms of worse parameter sets, by the definition of the random walk in the parameter space and by its condition of termination. The probability of acceptance is managed by an energy criterion characterized by a temperature. The temperature scheduled has been chosen according to Ingber [1993a]:

nŽ = nOMO ∙ exp s−zO∙ $^9?u (13)

where T_init is the initial temperature, D the dimension of the parameter space, k the number of accepted states and c_i a user-defined value that can be adapted to improve the algorithm performances and which default value is set to:

zO = − log 10^8’∙ exp(− log 100 ∗ 1/ó) (14)

The acceptance test defines if a new parameter state is accepted or rejected. It writes:

t < exp s−@G− @G8?

nG u (15)

where u is a random value drawn from a uniform distribution on [0;1], Fi is the value of the objective function with the new parameters, Fi-1 is the value of the objective function at the last accepted state and T_i is the current temperature. Thus, if the new parameters induce a decrease of the objective function, they are accepted for sure. Otherwise, they are accepted with a probability that depends on how they penalize the solution. This probability decreases when the temperature becomes smaller.

The random walk within the parameter space is characterized by the parameter modifications from steps i to i+1. For the parameter p^j:

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f_O^W= f_O8?^W + !O∙ Yf•4ªW − f_•OM^W \ (16)

where f_•4ª^W and f_•OM^W are the minimal and maximal possible value for p^j and y_i is the following scaling factor:

!O = sign(t − 0.5) ∙ nO∙ vs1 + 1

nOu^wL∙x8?w− 1y (17)

where u is a random value drawn from a uniform distribution on [0;1]. With this formulation, the random walk shifts progressively from global to local thanks to the decrease of temperature. The optimization stops when the objective function becomes smaller than a fixed minimal value F_i≤Fmin or when the maximal number of allowed iteration is reached.

Acknowledgements

The French National Research Agency ANR is acknowledged for its financial founding through the MOHINI project (ANR-07-VULN-008) and for its contribution to the development of numerical methods through the MICAS project (ANR-07-CIS7-004).

Additional funding was provided by the French Association for Research and Technology ANRT (CIFRE-747/2006).

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Chapitre : Simulated annealing parameterization

B. Influence des conditions aux limites sur la sensibilité des données de charges