**5. Robust design under uncertainty**

The task of finding optimal system design while accounting for uncertainty leads to robust designs. In specific, risk can be controlled to stay below a given risk acceptance level, if robust design is performed under corresponding constraints. Obviously, the system performance and failure probability depends on uncertain parameters, but also depends on design variables. Evidently, the decision making will depend on the interplay between the response to design parameters, system uncertainty and, finally, the probability of failure or some statistical expectation of benefit. We propose to combine the model response to all design variables and uncertain parameters into a single approach based on an integrative (joint) response surface (obtained via PCE), which allows to reflect the non-linear dependence of the original model on all these parameters. This integrates the design task into the reduced stochastic model, allowing us to find robust designs with controlled risks at low computational costs. For example, designing system behaviors for maximum load at a guaranteed specific safety level can be achieved by quick Monte-Carlo methods on the resulting polynomials.

#### **5.1 Integrative response surface**

Fig. 7. Integrative response surface

22 Will-be-set-by-IN-TECH

The data-driven stochastic approach was validated on the basis of Monte Carlo simulation using a common 3D benchmark problem. The proposed approach yields a significant computational speed-up compared with Monte Carlo, and provides faster convergence than conventional polynomial chaos expansions. Even for small degrees of expansion, the data-driven expansion can be very accurate, which can save a lot of computational power for

Data-driven polynomial chaos expansion is based directly on raw data or other arbitrary sources of information without auxiliary assumptions. This increases the efficiency of chaos expansion and minimizes subjectivity, providing valuable support for risk-informed decision making as well as for robust design and control, allowing a better assessment and reduction

The task of finding optimal system design while accounting for uncertainty leads to robust designs. In specific, risk can be controlled to stay below a given risk acceptance level, if robust design is performed under corresponding constraints. Obviously, the system performance and failure probability depends on uncertain parameters, but also depends on design variables. Evidently, the decision making will depend on the interplay between the response to design parameters, system uncertainty and, finally, the probability of failure or some statistical expectation of benefit. We propose to combine the model response to all design variables and uncertain parameters into a single approach based on an integrative (joint) response surface (obtained via PCE), which allows to reflect the non-linear dependence of the original model on all these parameters. This integrates the design task into the reduced stochastic model, allowing us to find robust designs with controlled risks at low computational costs. For example, designing system behaviors for maximum load at a guaranteed specific safety level can be achieved by quick Monte-Carlo methods on the

probabilistic risk analysis.

resulting polynomials.

**5.1 Integrative response surface**

Fig. 7. Integrative response surface

**5. Robust design under uncertainty**

of risk.

Stochastic response surface methods (Isukapalli et al., 1998) deal with the characterization of uncertainties in systems describing the dependence of model output on the uncertain input parameters. Usually in many applied tasks, the model parameters can be classified in two classes: design or control variables that can be chosen by the operator of a system, and uncertain parameters that describe our (incomplete) knowledge of the system properties. On the other hand, the system's performance and failure probability will also depend on design parameters. Evidently, the decision-making for design parameters will depend on the interplay between the response to design parameters, system uncertainty and, finally, the probability of failure. In terms of system response, sensitivity and expansion, this distinction is artificial. Therefore, we drop the distinction between uncertain and design parameters. Instead, we project the model response to all design and uncertain parameters onto a single integrative model response surface. It is a multidimensional surface and contains the integral information about the system behavior under all possible conditions at all points in space and time. Thus, the notion of stochastic response surfaces introduced by Isukapalli et al. (1998) is expanded to integrative response surfaces forming an effective basis for robust design under uncertainty.

We investigate the influence of uncertain and design parameters ω on the model output Ω (integrative response surface approach). This means that the considered model has a multivariable input ω for the expansion Eq. (1):

$$
\omega\_{1\prime} \dots \omega\_{N\_{\rm lI}\prime} \omega\_{N\_{\rm lI}+1\prime} \dots \omega\_{N\_{\rm lI}+N\_{\rm D}} \tag{16}
$$

where *NU* is the number of uncertain parameters and *ND* is the number of design parameters. The total number of input parameters is *N* = *NU* + *ND*.

Of course design parameters do not have probabilistic distributions, but suitable weighting functions for such parameters can be described by user-defined feasibility functions that select the feasible range or preferences of the designing engineer concerning the values of design parameters. Feasibility functions provide a freedom for scenario analysis and can be used as an entry point for monetary punishment terms.

#### **5.2 Illustration: robust design for carbon dioxide storage**

#### **5.2.1 Problem formulation**

We again consider the problem of CO2 leakage (Class et al., 2009) presented in Section 4.3. In the current Section we will consider the design task of finding an optimal injection regime. As in the Section 4.3 we consider three uncertain parameters: reservoir absolute permeability, reservoir porosity and permeability of the leaky well. For the current case study we also included two design parameters for describing the injection strategy: the CO2 injection rate (fluctuating around 8.87 kg/s) and the size of the screening interval (up to 30m), see details in the paper by Oladyshkin, Class, Helmig & Nowak (2011b). The choice of the design parameters in this study is only exemplary and serves to demonstrate how engineering decision-making can be supported by the approach presented here. Both the injection rate and the screening interval directly affect the ratio of forces in the reservoir during the injection. The choice of feasibility functions is arbitrary, and modelers have full freedom to introduce feasibility functions and to weight them according to their personal experience or preferences. For example, we chose a beta distribution for the size of the screening interval with lower

Control Via Robust Design 25

Probabilistic Risk Assessment and Risk Control via Robust Design

<sup>341</sup> Polynomial Response Surfaces for

Fig. 9. Choice of design parameters based on caprock pressure after 1000 days: critical

injection regime can be chosen according to a maximum allowable failure probability. Figure 9 illustrates the choice of design parameters based on the caprock pressure after 1000 days. In this test case, a critical caprock pressure equal to 330 bar was chosen at a significance level of 5%, i.e. the maximal acceptable probability of failure is set to 0.05 (solid black line on surface). Figure 9 demonstrates acceptable strategies of injection where the caprock pressure does not exceed the limit of 330 bar, which corresponds to an injection-induced pressure build-up of

This Section explores a massive stochastic model reduction via polynomial chaos expansion for robust design. This approach offers fast evaluation for statistical quantities and their dependence on design or control parameters. In particular, we map the response of a model to changes in all the design parameters and uncertain parameters onto one single integrative response surface. Based on this integrative concept, the design task explicitly includes

We demonstrated that neglecting parametric uncertainty in design can be a strong simplification for modeling. Due to the non-linearity of processes, including uncertainty can lead to a systematic and significant shift of the predicted values, affecting both risk estimates and the design of injection scenarios. Thus, design under uncertainty, that openly admits uncertainty and seeks for roust solutions is much reliable in comparison to conventional

The authors acknowledge the German Research Foundation (DFG) for its financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the

uncertainty, which leads to robust designs with minimum failure probability.

pressure 330 bar at a significance level of 5 %

about 40 bar.

design.

**6. Acknowledgement**

University of Stuttgart.

**5.3 Conclusions to section 5**

and upper bounds of z=0[m] and z=30[m], respectively, reflecting the physical bounds of the reservoir.

#### **5.2.2 Design under uncertainty vs. conventional design**

The greatest challenge to modeling consists of finding a healthy and reasonable compromise between an accurate system representation and computational efforts. Unfortunately, neglecting uncertainty in design tasks can be a strong simplification for modeling, and the consequences can be stronger than when neglecting several physical phenomena.

Fig. 8. Influence of design parameters on prediction of CO2 leakage rate after 1000 days: top surface (robust design) - expected CO2 leakage rate (average over uncertain parameters); bottom surface (conventional design) - CO2 leakage rate evaluated deterministically with expected values of parameters

Figure 8 demonstrates how the injection rate and screening interval influence the leakage rate of CO2. An important advantage of the integrative response surface approach is that parameter uncertainty is easily included in such predictions. The top surface in Figure 8 is the CO2 leakage rate expected after 100 days as a function of the design parameters, averaged over the uncertain parameters (robust design). The bottom surface in Figure 8 is the CO2 leakage rate using the expected values of the uncertain parameters, i.e. as in deterministic simulations (conventional design). It is easy to see that the impact can be extremely important for non-linear systems (here, a factor of about two), especially in long-term simulations. Instead of looking at mean values, robust design can also look at failure probability (see below). Thus, designs hat admit uncertainty are much more robust than design under deterministic assumptions.

#### **5.2.3 Robust design of failure probability**

The integrative response surface approach provides a constructive solution to the problem of robust design under uncertainty and provides valuable support for risk-informed decision making.

In a similar fashion as in the previous Section 5.2.2, the dependence of the leakage probability or any other statistical characteristics on design parameters can be evaluated, so that the 24 Will-be-set-by-IN-TECH

and upper bounds of z=0[m] and z=30[m], respectively, reflecting the physical bounds of the

The greatest challenge to modeling consists of finding a healthy and reasonable compromise between an accurate system representation and computational efforts. Unfortunately, neglecting uncertainty in design tasks can be a strong simplification for modeling, and the

Fig. 8. Influence of design parameters on prediction of CO2 leakage rate after 1000 days: top surface (robust design) - expected CO2 leakage rate (average over uncertain parameters); bottom surface (conventional design) - CO2 leakage rate evaluated deterministically with

Figure 8 demonstrates how the injection rate and screening interval influence the leakage rate of CO2. An important advantage of the integrative response surface approach is that parameter uncertainty is easily included in such predictions. The top surface in Figure 8 is the CO2 leakage rate expected after 100 days as a function of the design parameters, averaged over the uncertain parameters (robust design). The bottom surface in Figure 8 is the CO2 leakage rate using the expected values of the uncertain parameters, i.e. as in deterministic simulations (conventional design). It is easy to see that the impact can be extremely important for non-linear systems (here, a factor of about two), especially in long-term simulations. Instead of looking at mean values, robust design can also look at failure probability (see below). Thus, designs hat admit uncertainty are much more robust than design under deterministic

The integrative response surface approach provides a constructive solution to the problem of robust design under uncertainty and provides valuable support for risk-informed decision

In a similar fashion as in the previous Section 5.2.2, the dependence of the leakage probability or any other statistical characteristics on design parameters can be evaluated, so that the

consequences can be stronger than when neglecting several physical phenomena.

**5.2.2 Design under uncertainty vs. conventional design**

reservoir.

expected values of parameters

**5.2.3 Robust design of failure probability**

assumptions.

making.

Fig. 9. Choice of design parameters based on caprock pressure after 1000 days: critical pressure 330 bar at a significance level of 5 %

injection regime can be chosen according to a maximum allowable failure probability. Figure 9 illustrates the choice of design parameters based on the caprock pressure after 1000 days. In this test case, a critical caprock pressure equal to 330 bar was chosen at a significance level of 5%, i.e. the maximal acceptable probability of failure is set to 0.05 (solid black line on surface). Figure 9 demonstrates acceptable strategies of injection where the caprock pressure does not exceed the limit of 330 bar, which corresponds to an injection-induced pressure build-up of about 40 bar.

#### **5.3 Conclusions to section 5**

This Section explores a massive stochastic model reduction via polynomial chaos expansion for robust design. This approach offers fast evaluation for statistical quantities and their dependence on design or control parameters. In particular, we map the response of a model to changes in all the design parameters and uncertain parameters onto one single integrative response surface. Based on this integrative concept, the design task explicitly includes uncertainty, which leads to robust designs with minimum failure probability.

We demonstrated that neglecting parametric uncertainty in design can be a strong simplification for modeling. Due to the non-linearity of processes, including uncertainty can lead to a systematic and significant shift of the predicted values, affecting both risk estimates and the design of injection scenarios. Thus, design under uncertainty, that openly admits uncertainty and seeks for roust solutions is much reliable in comparison to conventional design.

#### **6. Acknowledgement**

The authors acknowledge the German Research Foundation (DFG) for its financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart.

Control Via Robust Design 27

<sup>343</sup> Polynomial Response Surfaces for

Grigoriu, M. (2002). *Stochastic Calculus: Applications in Science and Engineering*, Birkhauser,

Homma, T. & Saltelli, A. (1996). Importance measures in global sensitivity analysis of nonlinear models, *Reliability Engineering and System Safety* 52(1): 1–17. Isukapalli, S., S., Roy, A. & Georgopoulos, P., G. (1998). Stochastic response surface methods

Jang, Y., S., Sitar, N. & Kiureghian, A., D. (1994). Reliability analysis of contaminant transport in saturated porous media., *Water Resources Research* 30(8): 2435–2448. Keese, A. & Matthies, H. G. (2003). Sparse quadrature as an alternative to mc for stochastic

Kopp, A., Class, H. & Helmig, H. (2009). Investigations on co2 storage capacity in saline

Li, H., Sarma, P. & Zhang, D. (2011). A comparative study of the probabilistic collocation

Li, H. & Zhang, D. (2007). Probabilistic collocation method for flow in porous media: Comparisons with other stochastic methods, *Water Resources Research* 43: 44–48. Matthies, H., G. & Keese., A. (2005). Galerkin methods for linear and nonlinear

Maxwell, R. M. & Kastenberg, W. E. (1999). Stochastic environmental risk analysis: An

Nowak, W. (2009). Best unbiased ensemble linearization and the quasi-linear Kalman

Oladyhskin, S., Class, H., Helmig, R., Nowak, W., de Barros, F. P. J. & Ashraf, M. (2011).

Oladyshkin, S., Class, H., Helmig, R. & Nowak, W. (2011b). An integrative approach to robust

Oladyshkin, S., de Barros, F. P. J. & Nowak, W. (2011). Global sensitivity analysis: a flexible

Oladyshkin, S. & Nowak, W. (2011). Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion, *Reliability Engineering and System Safety* . Plischke, E. (2010). An effective algorithm for computing global sensitivity indices (easi),

uncertainty, SAMSI Geosciences Applications Opening Workshop 2011. Oladyshkin, S., Class, H., Helmig, R. & Nowak, W. (2011a). A concept for data-driven

*Stochastic Environmental Research and Risk Assessment* (13(1-2)): 27–47. Morris, M., D. (1991). Morris factorial sampling plans for preliminary computational

finite element techniques, *Proc. Appl. Math. Mech.* 3: 493U–494. ˝

(srsms) for uncertainty propagation: Application to environmental and biological

aquifers - part 1: Dimensional analysis of flow processes and reservoir characteristics,

and experimental design methods for petroleum reservoir uncertainty quantification,

elliptic stochastic partial differential equations, *Comp. Meth. Appl. Mech. Engrg.*

integrated methodology for predicting cancer risk from contaminated groundwater.,

Data-driven polynomial response surfaces as efficient tool for applied tasks under

uncertainty quantification and its application to carbon dioxide storage in geological

design and probabilistic risk assessment for co2 storage in geological formations,

and efficient framework with an example from stochastic hydrogeology, *Advances in*

Boston.

systems, *Risk Analysis* 18(3): 351–363.

Probabilistic Risk Assessment and Risk Control via Robust Design

*Int. J. of Greenhouse Gas Control* 3: 263–276.

experiments, *Technometrics* (33): 161–174.

*Computational Geosciences* 15(3): 565–577.

ensemble generator, *Water Resour. Res.* 45(W04431).

formations, *Advances in Water Resources* 34: 1508–1518.

*Water Resources* DOI: 10.1016/j.advwatres.2011.11.001.

*Reliability Engineering and System Safety* 95(4): 354–360.

Rubin, Y. (2003). *Applied Stochastic Hydrogeology*, Oxford University Press, Oxford.

*SPE Journal* pp. SPE–140738–PA–P.

194: 1295–1331.

#### **7. References**


26 Will-be-set-by-IN-TECH

Abramowitz, M. & Stegun, I., A. (1965). *Handbook of Mathematical Functions with Formulas,*

Anderson, M. & Burt, T. P. (1985). Modelling strategies, *in Anderson, M.G. and Burt, T.P. (eds.),*

Askey, R. & Wilson, J. (1985). *Some basic hypergeometric polynomials that generalize Jacobi polynomials*, Memoirs of the American Mathematical Society, AMS, Providence. Class, H., Ebigbo, A., Helmig, R., Dahle, H., Nordbotten, J. N., Celia, M. A., Audigane,

Crestaux, T., Le Maitre, O. & Martinez, J.-M. (2009). Polynomial chaos expansion for sensitivity

Dagan, G. (1988). Time-dependent macrodispersion for solute transport in anisotropic

de Barros, F. P. J., Ezzedine, S. & Rubin, Y. (2011). Impact of hydrogeological data

de Barros, F. P. J., Oladyshkin, S. & Nowak, W. (2011). An integrative data-adaptive approach

Efron, B. (1987). *The Jackknife, the Bootstrap, and Other Resampling Plans*, Society for Industrial

Fajraoui, N., Ramasomanana, F., Younes, A., Mara, T. A.and Ackerer, P. & Guadagnini,

Flemisch, B., J., F., Helmig, R., Niessner, J. & Wohlmuth, B. (2007). Dumux: a multi-scale

Foglia, L., Mehl, S., W., Hill, M., C., Perona, P. & Burlando, P. (2007). Testing alternative ground

Foo, J. & Karniadakis, G. (2010). Multi-element probabilistic collocation method in high

Ghanem, R. & Doostan, A. (2006). On the construction and analysis of stochastic models:

Ghanem, R. G. & Spanos, P. D. (1991). *Stochastic Finite Elements: A Spectral Approach*,

Ghanem, R. & Spanos, P. (1993). A stochastic galerkin expansion for nonlinear random

Ghanem, R. & Spanos, P. D. (1990). Polynomial chaos in stochastic finite elements, *Journal of*

dimensions, *Journal of Computational Physics* 229(5): 1536–1557.

˝

vibration analysis, *Probabilistic Engineering Mechanics* 8: 255–264.

on measures of uncertainty, site characterization and environmental performance

for global sensitivity analysis: application to subsurface flow and transport, *Geophys.*

A. (2011). Use of global sensitivity analysis and polynomial chaos expansion for interpretation of non-reactive transport experiments in laboratory-scale porous

multi-physics toolbox for flow and transport processes in porous media, *In A. Ibrahimbegovic and F. Dias, editors* ECCO3MAS Thematic Conference on Multi-scale Computational Methods for Solids and Fluids, Cachan, France, November 28–30,

water models using cross validation and other methods, *Ground Wate* 45(5): 627–641.

Characterization and propagation of the errors associated with limited data, *Journal*

˝

analysis, *Reliability Engineering and System Safety* 94(7): 1161–1172.

heterogeneous aquifers, *Water Resour. Res.* 24(9): 1491–1500. Dagan, G. (1989). *Flow and Transport in Porous Formations*, Springer Verlag, Berlin.

P., Darcis, M., Ennis-King, J., Fan, Y., Flemisch, B., Gasda, S., Jin, M., Krug, S., Labregere, D., Naderi, A., Pawar, R. J., Sbai, A., Sunil, G. T., Trenty, L. & Wei, L. (2009). Abenchmark-study on problems related to co2 storage in geologic formations,

*Graphs, and Mathematical Tables*, New York: Dover.

*Hydrological Forecasting* pp. 1–13.

*Computational Geosciences* 13: 451U–467.

metrics, *Advances in Water Resources* .

*of Computational Physics* 217: 63–U81.

Springer-Verlag, New York.

*Applied Mechanics* 57: 197–202.

Mathematics.

2007.

*Res. Abstr.*, Vol. 13, EGU General Assembly 2011.

media, *Water Resour. Res.* (doi:10.1029/2010WR009639).

**7. References**


28 Will-be-set-by-IN-TECH

344 Novel Approaches and Their Applications in Risk Assessment

Saltelli, A., Ratto, M. & Andres, T. (2008). *Global Sensitivity Analysis: The Primer*, John Wiley &

Siebert, W. M. (1986). Circuits, signals, and systems, *MIT Press: Cambridge, MA* pp. 410–411. Siirila, E., R., Navarre-Sitchler, A., K., Maxwell, R., M. & McCray, J., E. (2011). A quantitative

Sobol, I. M. (1990). On sensitivity estimation for nonlinear mathematical models, *Mathem.*

Soize, C. & Ghanem, R. (2004). Physical systems with random uncertainties:

Stieltjes, T., J. (1884). Quelques recherches sur la théorie des quadratures dites méchaniques.,

Sudret, B. (2008). Global sensitivity analysis using polynomial chaos expansions, *Reliability*

USEPA (December 1989). Risk assessment guidance for superfund volume i: Part a, human

USEPA (December 2001). Risk assessment guidance for superfund: Volume iii – part a, process for conducting probabilistic risk assessment, *Tech. Rep. Rep.EPA 540/R-02/002,* . Villadsen, J. & Michelsen, M. L. (1978). *Solution of differential equation models by polynomial*

Wackernagel, H. (1998). *Multivariate Geostatistics, An Introduction With Applications*, Second,

Wan, X. & Karniadakis, G., E. (2006). Multi-element generalized polynomial chaos for arbitrary probability measures, *SIAM Journal of Scientific Computing* 28(3): 901–928.

Winter, C. L., Guadagnini, A., Nychka, D. & Tartakovsky, D. M. (2006). Multivariate sensitivity

Witteveen, J. A. S. & Bijl, H. (2006). Modeling arbitrary uncertainties using gram-schmidt

Witteveen, J. A. S., Sarkar, S. & Bijl, H. (2007). Modeling physical uncertainties in dynamic stall

Zhang, D. & Lu, Z. (2004). An efficient, high-order perturbation approach for flow in random

˝ Xiu, D. & Karniadakis, G. E. (2003). Modeling uncertainty in flow simulations via generalized

analysis of saturated flow through simulated highly heterogeneous groundwater

polynomial chaos, *44th AIAA Aerospace Sciences Meeting and Exhibit* Reno,

induced fluidUstructure interaction of turbine blades using arbitrary polynomial

media via karhunen-loeve and polynomial expansions, *Journal of Computational*

methodology to assess the risks to human health from co2 leakage into groundwater.,

chaos representations with arbitrary probability measure, *SIAM J. Sci. Comput.*

Sons.

*Advances in Water Resources* .

*Mod.* 2(1): 112–118.

26(2): 395–U410. ˝

Springer, Berlin.

*Oeuvres I* pp. 377–396.

*approximation*, Prentice-Hall.

Nevada: AIAA–2006–896.

*Physics* 194: 773–794.

˝

*Engineering and System Safety* 93(7): 964–979.

health manual., *Tech. Rep. Rep.EPA/540/1-89/002* .

Wiener, N. (1938). The homogeneous chaos, *Am. J. Math* 60: 897–936.

chaos, *Computers and Structures* 85: 866U–878.

aquifers, *Journal of Computational Physics* 217(1): 166–175.

polynomial chaos, *Journal of Computational Physics* 187: 137–167.
