DETERMINISTIC SENSITIVITY ANALYSIS FOR A MODEL FOR FLOW AND
TRANSPORT IN POROUS MEDIA
E. Marchand1,2, F. Clément2, J. Roberts2
1ANDRA, France; 2INRIA Rocquencourt, France
estelle.marchand@inria.fr
This work is devoted to the deterministic sensitivity analysis of a mathematical model for the
simulation of flow and transport in porous media. The method used is based on the singular value
decomposition of the Jacobian matrix of the mathematical model of the problem. Since the method uses a first
order approximation, it provides local results around particular input and output parameters. For this reason we
also study the variability of the results of the computation of sensitivities due to the variations of the input
parameters of the model. Numerical results are compared with those obtained through a statistical analysis.
The questions of safety and uncertainties are central to feasibility studies for an underground nuclear
waste storage site. One of the important points to be considered is the problem of the evaluation of uncertainties
concerning safety indicators (“output parameters”) which are due to uncertainties concerning properties of the
subsoil, such as hydraulic conductivity, or of the contaminants (“input parameters”). Uncertainties about the
input parameters are due to imprecisions of measurement techniques or to spatial variability. Safety indicators
may be for instance the measure of the water flow through outlet channels or concentration measures.
Two different aspects of the quantification of the influence of the parameters on the safety indicators
are uncertainty analysis, which corresponds to the quantification of the uncertainty concerning the indicators
(for example in the form of distributions or quantiles) and sensitivity analysis, which corresponds to the
identification of the weight of the input parameters with respect to their influence on the indicators. For
sensitivity analysis, probabilistic approaches, such as Monte-Carlo methods, can be used. These methods give
good results and are relatively easy to implement, but they are expensive because they require a large number of
simulations (see for example [1,2]). The deterministic method investigated here is much less demanding in
computing time but it gives only local information: for nonlinear problems, the results obtained will be correct
only for small variations of the input parameters (relatively to the nonlinearity) around a particular set of input
parameters.
For deterministic methods, “first order” uncertainties are computed from the derivatives of the function
F relating the output parameters to the input parameters. Different methods can be used for differentiation:
divided differences, automatic differentiation, analytic differentiation for possibly implicit problems. For each
method, direct mode or reverse mode (reverse mode is equivalent to the adjoint state method), can be used.
These derivatives can be used combined with interval arithmetic to evaluate uncertainties in the form of
intervals containing the image under F of input intervals, see [3].
The derivatives of F are stored in the Jacobian matrix J. The hierarchization of the influences of the
inputs of the function upon their influence on the outputs of the function is provided by the singular value
decomposition (SVD) of J. The Singular Value Decomposition can also be used in probabilistic sensitivity
analysis ([4]).
Thus the probabilistic and deterministic approaches are complementary and both deserve to be
developed. In our study, we apply both approaches to the same problem.
We will present our discretized model for flow and transport in porous media. The flow equation is
based on the stationary Darcy law and is discretized with a mixed hybrid finite element method. The transport
equation consists of a mass balance equation for each contaminant and a law of exchange between liquid and
solid phases for each contaminant.
We will give some details about the computation of the derivatives used for the deterministic analysis.
For the flow equation we use a C++ code which has been differentiated with analytical formulas, using the
adjoint state method. For the transport equation we combine manual differentiation and automatic
differentiation. For automatic differentiation we use the library AdolC ([5]). The singular value decomposition
of the Jacobian matrices, computed with LAPACK, provide a (weighted) hierarchical list of directions in both
the space of input parameters and in the space of safety indicators. Then a truncated computation of the
uncertainties on the outputs is possible.
For the flow model, various deterministic studies have been computed for a 3 dimensional test case.
We observe a weak variability of the local sensitivities when the choice of the input parameters varies in the
spectrum of possible input parameters. This is due to the weak nonlinearity of the model, assured by the choice
of a logarithmic parameterization. In this case, probabilistic and deterministic studies provide similar results.
We also confront numerical results obtained using a deterministic analysis with those obtained using a
probabilistic analysis for the full model.
Acknowledgement:
This work is partially supported by ANDRA, the French National radioactive waste management agency, and
by the GDR MOMAS.
References
[1] Helton JC, Johnson JD, Sallabery CJ, Storlie CB: Survey on sampling-based methods for uncertainty and
sensitivity analysis. Technical report, Sandia National Laboratories SAND2006-2901 (2006)
[2] Cacuci DG, Ionescu-Bujor M: A comparative review of sensitivity analyis of large scale systems—ii:
Statisticla methods. Nuclear Science and Ingineering 147, 204-214 (2004)
[3] Rump SM: Estimation of the sensitivity of linear and nonlinear algebraic problems. Lin. Algebra Appl. 153,
1-34 (1991)
[4] Gerbrands JJ: On the relationships between SVD, KLT and PCA. Pattern Recognition 14, 375-381 (1981)
[5] http://www.math.tu-dresden.de/~adol-c/