What Should We Do with Radioactive Waste?

Transcription

1 What Should We Do with Radioactive Waste? Andrew Cliffe School of Mathematical Sciences University of Nottingham Simulation of Flow in Porous Media and Applications in Waste Management and CO 2 Sequestration Radon Special Semester 2011 on Multiscale Simulation & Analysis in Energy and the Environment 4th October 2011 Andrew Cliffe What Should We Do with Radioactive Waste? 1/30

2 Collaborators and Acknowledgements Ian Dryden, Minho Park, Nicola Stone University of Nottingham, School of Mathematical Sciences Ingolf Busch, Oliver Ernst, Björn Sprunk TU Freiberg, Institute of Numerical Analysis and Optimziation Gerald van den Bogaart, Silke Konsulke TU Freiberg, Institute of Stochastics Rob Scheichl, Elisabeth Ullmann University of Bath, Department of Mathematical Sciences Andrew Cliffe What Should We Do with Radioactive Waste? 2/30

3 Outline Radioactive waste disposal Treatment of uncertainty Case study - WIPP Conclusions Andrew Cliffe What Should We Do with Radioactive Waste? 3/30

4 Radioactive waste disposal university-log Radioactive waste is waste that contains significant amounts of radioactive material. Andrew Cliffe What Should We Do with Radioactive Waste? 4/30

5 Radioactive waste disposal Radioactive waste is waste that contains significant amounts of radioactive material. Most radioactive waste comes from the civil nuclear power industry. Andrew Cliffe What Should We Do with Radioactive Waste? 4/30

6 Radioactive waste disposal Radioactive waste is waste that contains significant amounts of radioactive material. Most radioactive waste comes from the civil nuclear power industry. Other sources include: military, medical establishments, non-nuclear industries (such the oil industry) and educational and research establishments. Andrew Cliffe What Should We Do with Radioactive Waste? 4/30

7 Radioactive waste disposal Radioactive waste is waste that contains significant amounts of radioactive material. Most radioactive waste comes from the civil nuclear power industry. Other sources include: military, medical establishments, non-nuclear industries (such the oil industry) and educational and research establishments. Whatever your attitude to nuclear power, the existing waste has to be disposed of safely! Andrew Cliffe What Should We Do with Radioactive Waste? 4/30

8 Radioactive waste disposal university-log Total waste from UK civil nuclear power (assuming no new build): Andrew Cliffe What Should We Do with Radioactive Waste? 5/30

9 Radioactive waste disposal Total waste from UK civil nuclear power (assuming no new build): HLW - highly radioactive, heat generating ( 1,300m 3 ). Andrew Cliffe What Should We Do with Radioactive Waste? 5/30

10 Radioactive waste disposal Total waste from UK civil nuclear power (assuming no new build): HLW - highly radioactive, heat generating ( 1,300m 3 ). ILW - intermediate radioactivity, not heat generating ( 220,000m 3 ). Andrew Cliffe What Should We Do with Radioactive Waste? 5/30

11 Radioactive waste disposal Total waste from UK civil nuclear power (assuming no new build): HLW - highly radioactive, heat generating ( 1,300m 3 ). ILW - intermediate radioactivity, not heat generating ( 220,000m 3 ). LLW - slightly radioactive ( 2,100,000m 3 ). Andrew Cliffe What Should We Do with Radioactive Waste? 5/30

12 Radioactive waste disposal Total waste from UK civil nuclear power (assuming no new build): HLW - highly radioactive, heat generating ( 1,300m 3 ). ILW - intermediate radioactivity, not heat generating ( 220,000m 3 ). LLW - slightly radioactive ( 2,100,000m 3 ). All the waste would fill just over half of the new Wembley Stadium (4,000,000m 3 ). Andrew Cliffe What Should We Do with Radioactive Waste? 5/30

13 Radioactive waste disposal Total waste from UK civil nuclear power (assuming no new build): HLW - highly radioactive, heat generating ( 1,300m 3 ). ILW - intermediate radioactivity, not heat generating ( 220,000m 3 ). LLW - slightly radioactive ( 2,100,000m 3 ). All the waste would fill just over half of the new Wembley Stadium (4,000,000m 3 ). The HLW and ILW would stand approximately 17m deep on the pitch, with the HLW 10 cms deep. Andrew Cliffe What Should We Do with Radioactive Waste? 5/30

14 Radioactive waste disposal - issues university-log Some general principles governing management and disposal of radioactive waste: Andrew Cliffe What Should We Do with Radioactive Waste? 6/30

15 Radioactive waste disposal - issues Some general principles governing management and disposal of radioactive waste: Some radionuclides have long half-lives and remain dangerous for thousands of years Andrew Cliffe What Should We Do with Radioactive Waste? 6/30

16 Radioactive waste disposal - issues Some general principles governing management and disposal of radioactive waste: Some radionuclides have long half-lives and remain dangerous for thousands of years Each country is responsible for dealing with its own waste (no export of waste) Andrew Cliffe What Should We Do with Radioactive Waste? 6/30

17 Radioactive waste disposal - issues Some general principles governing management and disposal of radioactive waste: Some radionuclides have long half-lives and remain dangerous for thousands of years Each country is responsible for dealing with its own waste (no export of waste) Minimal assumptions about future development of human society (e.g. can not assume there will be a cure for cancer or other diseases caused by exposure to radioactivity) Andrew Cliffe What Should We Do with Radioactive Waste? 6/30

18 Radioactive waste disposal - issues Some general principles governing management and disposal of radioactive waste: Some radionuclides have long half-lives and remain dangerous for thousands of years Each country is responsible for dealing with its own waste (no export of waste) Minimal assumptions about future development of human society (e.g. can not assume there will be a cure for cancer or other diseases caused by exposure to radioactivity) Do not leave a legacy of hazardous and dangerous waste for future generations to manage Andrew Cliffe What Should We Do with Radioactive Waste? 6/30

19 Radioactive waste disposal - options university-log Surface storage Andrew Cliffe What Should We Do with Radioactive Waste? 7/30

20 Radioactive waste disposal - options Surface storage Disposal at sea Andrew Cliffe What Should We Do with Radioactive Waste? 7/30

21 Radioactive waste disposal - options Surface storage Disposal at sea Space disposal Andrew Cliffe What Should We Do with Radioactive Waste? 7/30

22 Radioactive waste disposal - options Surface storage Disposal at sea Space disposal Transmutation Andrew Cliffe What Should We Do with Radioactive Waste? 7/30

23 Radioactive waste disposal - options university-log Surface storage Disposal at sea Space disposal Transmutation Deep geological disposal Andrew Cliffe What Should We Do with Radioactive Waste? 7/30

24 Radioactive waste disposal - options university-log Surface storage Disposal at sea Space disposal Transmutation Deep geological disposal Repository deep underground in a stable geological formation Andrew Cliffe What Should We Do with Radioactive Waste? 7/30

25 Radioactive waste disposal - options university-log Surface storage Disposal at sea Space disposal Transmutation Deep geological disposal Repository deep underground in a stable geological formation Multiple barriers - mechanical, chemical, physical Andrew Cliffe What Should We Do with Radioactive Waste? 7/30

26 Radioactive waste disposal - options university-log Surface storage Disposal at sea Space disposal Transmutation Deep geological disposal Repository deep underground in a stable geological formation Multiple barriers - mechanical, chemical, physical Accepted as the long-term solution in the UK (2006) Andrew Cliffe What Should We Do with Radioactive Waste? 7/30

27 Radioactive waste disposal - options university-log Surface storage Disposal at sea Space disposal Transmutation Deep geological disposal Repository deep underground in a stable geological formation Multiple barriers - mechanical, chemical, physical Accepted as the long-term solution in the UK (2006) Preferred solution of most countries Andrew Cliffe What Should We Do with Radioactive Waste? 7/30

28 Radioactive waste disposal university-log Assessing the safety of a potential repository is a major scientific undertaking. Andrew Cliffe What Should We Do with Radioactive Waste? 8/30

29 Radioactive waste disposal Assessing the safety of a potential repository is a major scientific undertaking. Groundwater pathway - route for radionuclides to get to surface environment. Andrew Cliffe What Should We Do with Radioactive Waste? 8/30

30 Radioactive waste disposal Assessing the safety of a potential repository is a major scientific undertaking. Groundwater pathway - route for radionuclides to get to surface environment. Long timescales (many thousands of years) make modelling essential. Andrew Cliffe What Should We Do with Radioactive Waste? 8/30

31 Radioactive waste disposal Assessing the safety of a potential repository is a major scientific undertaking. Groundwater pathway - route for radionuclides to get to surface environment. Long timescales (many thousands of years) make modelling essential. Uncertainty is a major issue. Andrew Cliffe What Should We Do with Radioactive Waste? 8/30

32 Radioactive waste disposal Assessing the safety of a potential repository is a major scientific undertaking. Groundwater pathway - route for radionuclides to get to surface environment. Long timescales (many thousands of years) make modelling essential. Uncertainty is a major issue. Objective is to make a good decision. Andrew Cliffe What Should We Do with Radioactive Waste? 8/30

33 Radioactive waste disposal Assessing the safety of a potential repository is a major scientific undertaking. Groundwater pathway - route for radionuclides to get to surface environment. Long timescales (many thousands of years) make modelling essential. Uncertainty is a major issue. Objective is to make a good decision. NB: Not necessarily interested in the worse possible case. Andrew Cliffe What Should We Do with Radioactive Waste? 8/30

34 Treatment of uncertainty university-log As we know, there are known knowns. There are things we know we know. We also know there are known unknowns. That is to say we know there are some things we do not know. But there are also unknown unknowns, the ones we don t know we don t know. Donald Rumsfeld Andrew Cliffe What Should We Do with Radioactive Waste? 9/30

35 Treatment of uncertainty university-log Five-fold strategy for dealing with uncertainty (Bailey 2005): Andrew Cliffe What Should We Do with Radioactive Waste? 10/30

36 Treatment of uncertainty university-log Five-fold strategy for dealing with uncertainty (Bailey 2005): 1. No impact on safety Andrew Cliffe What Should We Do with Radioactive Waste? 10/30

37 Treatment of uncertainty university-log Five-fold strategy for dealing with uncertainty (Bailey 2005): 1. No impact on safety eg. sorption coefficient for short lived radionuclides Andrew Cliffe What Should We Do with Radioactive Waste? 10/30

38 Treatment of uncertainty university-log Five-fold strategy for dealing with uncertainty (Bailey 2005): 1. No impact on safety eg. sorption coefficient for short lived radionuclides 2. Bounding uncertainty Andrew Cliffe What Should We Do with Radioactive Waste? 10/30

39 Treatment of uncertainty university-log Five-fold strategy for dealing with uncertainty (Bailey 2005): 1. No impact on safety eg. sorption coefficient for short lived radionuclides 2. Bounding uncertainty eg. solubility limit Andrew Cliffe What Should We Do with Radioactive Waste? 10/30

40 Treatment of uncertainty university-log Five-fold strategy for dealing with uncertainty (Bailey 2005): 1. No impact on safety eg. sorption coefficient for short lived radionuclides 2. Bounding uncertainty eg. solubility limit 3. Excluding uncertainty Andrew Cliffe What Should We Do with Radioactive Waste? 10/30

41 Treatment of uncertainty university-log Five-fold strategy for dealing with uncertainty (Bailey 2005): 1. No impact on safety eg. sorption coefficient for short lived radionuclides 2. Bounding uncertainty eg. solubility limit 3. Excluding uncertainty eg. meteorite impact Andrew Cliffe What Should We Do with Radioactive Waste? 10/30

42 Treatment of uncertainty university-log Five-fold strategy for dealing with uncertainty (Bailey 2005): 1. No impact on safety eg. sorption coefficient for short lived radionuclides 2. Bounding uncertainty eg. solubility limit 3. Excluding uncertainty eg. meteorite impact 4. Explicit quantification Andrew Cliffe What Should We Do with Radioactive Waste? 10/30

43 Treatment of uncertainty university-log Five-fold strategy for dealing with uncertainty (Bailey 2005): 1. No impact on safety eg. sorption coefficient for short lived radionuclides 2. Bounding uncertainty eg. solubility limit 3. Excluding uncertainty eg. meteorite impact 4. Explicit quantification eg. travel times in the groundwater pathway Andrew Cliffe What Should We Do with Radioactive Waste? 10/30

44 Treatment of uncertainty university-log Five-fold strategy for dealing with uncertainty (Bailey 2005): 1. No impact on safety eg. sorption coefficient for short lived radionuclides 2. Bounding uncertainty eg. solubility limit 3. Excluding uncertainty eg. meteorite impact 4. Explicit quantification eg. travel times in the groundwater pathway 5. Stylised approach Andrew Cliffe What Should We Do with Radioactive Waste? 10/30

45 Treatment of uncertainty university-log Five-fold strategy for dealing with uncertainty (Bailey 2005): 1. No impact on safety eg. sorption coefficient for short lived radionuclides 2. Bounding uncertainty eg. solubility limit 3. Excluding uncertainty eg. meteorite impact 4. Explicit quantification eg. travel times in the groundwater pathway 5. Stylised approach eg. biosphere (ICRP and IAEA projects) Andrew Cliffe What Should We Do with Radioactive Waste? 10/30

46 Case study - WIPP university-log WIPP - Waste Isolation Pilot Plant. Figure: Cross section through the rock at the WIPP site Andrew Cliffe What Should We Do with Radioactive Waste? 11/30

47 Case study - WIPP WIPP - Waste Isolation Pilot Plant. US DOE repository for radioactive waste situated in New Mexico. Figure: Cross section through the rock at the WIPP site Andrew Cliffe What Should We Do with Radioactive Waste? 11/30

48 Case study - WIPP WIPP - Waste Isolation Pilot Plant. US DOE repository for radioactive waste situated in New Mexico. Located at a depth of 655m within bedded evaporites, primarily halite. Figure: Cross section through the rock at the WIPP site Andrew Cliffe What Should We Do with Radioactive Waste? 11/30

49 Case study - WIPP WIPP - Waste Isolation Pilot Plant. US DOE repository for radioactive waste situated in New Mexico. Located at a depth of 655m within bedded evaporites, primarily halite. The most transmissive rock in the region is the Culebra dolomite. Figure: Cross section through the rock at the WIPP site Andrew Cliffe What Should We Do with Radioactive Waste? 11/30

50 Case study - WIPP WIPP - Waste Isolation Pilot Plant. US DOE repository for radioactive waste situated in New Mexico. Located at a depth of 655m within bedded evaporites, primarily halite. The most transmissive rock in the region is the Culebra dolomite. Culebra would be the principal pathway for transport of radionuclides away from the repository in the event of an accidental breach. Figure: Cross section through the rock at the WIPP site university-log Andrew Cliffe What Should We Do with Radioactive Waste? 11/30

51 Case study - WIPP measurement point Region containing data is 20km by 30km with the WIPP repository in the centre y (m) x (m) university-log Andrew Cliffe What Should We Do with Radioactive Waste? 12/30

52 Case study - WIPP y (m) measurement point Region containing data is 20km by 30km with the WIPP repository in the centre. Measurements of transmissivity, T, and freshwater head, h, are available at 39 locations x (m) university-log Andrew Cliffe What Should We Do with Radioactive Waste? 12/30

53 Case study - WIPP y (m) measurement point Region containing data is 20km by 30km with the WIPP repository in the centre. Measurements of transmissivity, T, and freshwater head, h, are available at 39 locations. Measurements are irregularly spaced, most concentrated around the repository in centre of region x (m) university-log Andrew Cliffe What Should We Do with Radioactive Waste? 12/30

54 Case study - WIPP university-log One scenario at WIPP is a release of radionuclides by means of a borehole drilled into the repository. Andrew Cliffe What Should We Do with Radioactive Waste? 13/30

55 Case study - WIPP One scenario at WIPP is a release of radionuclides by means of a borehole drilled into the repository. Radionuclides are released into the Culebra dolomite and then transported by groundwater. Andrew Cliffe What Should We Do with Radioactive Waste? 13/30

56 Case study - WIPP One scenario at WIPP is a release of radionuclides by means of a borehole drilled into the repository. Radionuclides are released into the Culebra dolomite and then transported by groundwater. Flow is two-dimensional to a good approximation. Andrew Cliffe What Should We Do with Radioactive Waste? 13/30

57 Case study - WIPP One scenario at WIPP is a release of radionuclides by means of a borehole drilled into the repository. Radionuclides are released into the Culebra dolomite and then transported by groundwater. Flow is two-dimensional to a good approximation. The time taken to travel from the repository to the boundary of the WIPP site is an important quantity. Andrew Cliffe What Should We Do with Radioactive Waste? 13/30

58 Case study - WIPP university-log Find h(x) such that: Andrew Cliffe What Should We Do with Radioactive Waste? 14/30

59 Case study - WIPP Find h(x) such that: T h = 0, x D, h = h 0, x D, T (x i ) = T i, x i D, i = 1,..., 39. Andrew Cliffe What Should We Do with Radioactive Waste? 14/30

60 Case study - WIPP Find h(x) such that: Then solve T h = 0, x D, h = h 0, x D, T (x i ) = T i, x i D, i = 1,..., 39. ζ = T (ζ) eφ h(ζ), ζ(0) = ζ 0, and compute the time, t f, at which ζ(t) D W, where e is the thickness of the layer, φ the porosity and D W is the boundary of the WIPP site. Andrew Cliffe What Should We Do with Radioactive Waste? 14/30

61 Case study - WIPP Find h(x) such that: Then solve T h = 0, x D, h = h 0, x D, T (x i ) = T i, x i D, i = 1,..., 39. ζ = T (ζ) eφ h(ζ), ζ(0) = ζ 0, and compute the time, t f, at which ζ(t) D W, where e is the thickness of the layer, φ the porosity and D W is the boundary of the WIPP site. Let s f = log(t f ). university-log Andrew Cliffe What Should We Do with Radioactive Waste? 14/30

62 Case study - WIPP university-log Problem on previous slide is ill-posed since we don t know T for all x D. Andrew Cliffe What Should We Do with Radioactive Waste? 15/30

63 Case study - WIPP Problem on previous slide is ill-posed since we don t know T for all x D. Limited information on the transmissivity leads to uncertainty. Andrew Cliffe What Should We Do with Radioactive Waste? 15/30

64 Case study - WIPP Problem on previous slide is ill-posed since we don t know T for all x D. Limited information on the transmissivity leads to uncertainty. It is important to understand and quantify the effect of this uncertainty on the results - in particular on the travel time. Andrew Cliffe What Should We Do with Radioactive Waste? 15/30

65 Case study - WIPP Problem on previous slide is ill-posed since we don t know T for all x D. Limited information on the transmissivity leads to uncertainty. It is important to understand and quantify the effect of this uncertainty on the results - in particular on the travel time. One approach to uncertainty quantification is to use probabilistic techniques. Andrew Cliffe What Should We Do with Radioactive Waste? 15/30

66 Case study - WIPP Problem on previous slide is ill-posed since we don t know T for all x D. Limited information on the transmissivity leads to uncertainty. It is important to understand and quantify the effect of this uncertainty on the results - in particular on the travel time. One approach to uncertainty quantification is to use probabilistic techniques. Probabilistic solution - distribution function for s f, F Sf (s). Andrew Cliffe What Should We Do with Radioactive Waste? 15/30

67 Case study - WIPP Problem on previous slide is ill-posed since we don t know T for all x D. Limited information on the transmissivity leads to uncertainty. It is important to understand and quantify the effect of this uncertainty on the results - in particular on the travel time. One approach to uncertainty quantification is to use probabilistic techniques. Probabilistic solution - distribution function for s f, F Sf (s). Interpretation of probability is important - a subjective Bayesian approach is used here. Andrew Cliffe What Should We Do with Radioactive Waste? 15/30

68 Probabilistic Uncertainty Quantification university-log Goal is to compute F Sf (s). The following steps are required: Andrew Cliffe What Should We Do with Radioactive Waste? 16/30

69 Probabilistic Uncertainty Quantification Goal is to compute F Sf (s). The following steps are required: Step 0 - collect data. Andrew Cliffe What Should We Do with Radioactive Waste? 16/30

70 Probabilistic Uncertainty Quantification Goal is to compute F Sf (s). The following steps are required: Step 0 - collect data. Step 1 - build stochastic model. Andrew Cliffe What Should We Do with Radioactive Waste? 16/30

71 Probabilistic Uncertainty Quantification Goal is to compute F Sf (s). The following steps are required: Step 0 - collect data. Step 1 - build stochastic model. Step 2 - solve the stochastic model. Andrew Cliffe What Should We Do with Radioactive Waste? 16/30

72 Probabilistic Uncertainty Quantification Goal is to compute F Sf (s). The following steps are required: Step 0 - collect data. Step 1 - build stochastic model. Step 2 - solve the stochastic model. Step 3 - post process - calculate the quantities of interest - travel times. Andrew Cliffe What Should We Do with Radioactive Waste? 16/30

73 Probabilistic Uncertainty Quantification Goal is to compute F Sf (s). The following steps are required: Step 0 - collect data. Step 1 - build stochastic model. Step 2 - solve the stochastic model. Step 3 - post process - calculate the quantities of interest - travel times. Step 4 - interpret the results. Andrew Cliffe What Should We Do with Radioactive Waste? 16/30

74 Stochastic model for WIPP Represent log transmissivity field, Z = log(t ), as a Gaussian random field with mean and covariance: E[Z (x)] = β 0 + β 1 x 1 + β 2 x 2 Cov[Z (x), Z (x )] = ω 2 C(x, x ) = ω 2 exp ( x x /λ ). Andrew Cliffe What Should We Do with Radioactive Waste? 17/30

75 Stochastic model for WIPP Represent log transmissivity field, Z = log(t ), as a Gaussian random field with mean and covariance: E[Z (x)] = β 0 + β 1 x 1 + β 2 x 2 Cov[Z (x), Z (x )] = ω 2 C(x, x ) = ω 2 exp ( x x /λ ). The joint probability distribution, f Θ (θ), for the hyperparameters θ = (β 0, β 1, β 2, ω 2, λ) is estimated using a standard Bayesian analysis and MCMC to generate samples of θ drawn from f Θ (θ). Andrew Cliffe What Should We Do with Radioactive Waste? 17/30

76 Stochastic model for WIPP Density 0e+00 2e 05 4e 05 6e 05 8e lambda Figure: Probability density function for λ from MCMC calculation. university-log Andrew Cliffe What Should We Do with Radioactive Waste? 18/30

77 Stochastic model for WIPP university-log The measurements of transmissivity are also used to condition the random field model (so that each realisation agrees with the measured values at the measurement points). Andrew Cliffe What Should We Do with Radioactive Waste? 19/30

78 Stochastic model for WIPP The measurements of transmissivity are also used to condition the random field model (so that each realisation agrees with the measured values at the measurement points). This leads to a low-rank modification of the covariance operator (so that the variance is zero at the measurement points). Andrew Cliffe What Should We Do with Radioactive Waste? 19/30

79 Stochastic model for WIPP The measurements of transmissivity are also used to condition the random field model (so that each realisation agrees with the measured values at the measurement points). This leads to a low-rank modification of the covariance operator (so that the variance is zero at the measurement points). Note there is uncertainty in the hyperparameters as well as that inherent in the random field model for T. Andrew Cliffe What Should We Do with Radioactive Waste? 19/30

80 Stochastic model for WIPP The measurements of transmissivity are also used to condition the random field model (so that each realisation agrees with the measured values at the measurement points). This leads to a low-rank modification of the covariance operator (so that the variance is zero at the measurement points). Note there is uncertainty in the hyperparameters as well as that inherent in the random field model for T. This uncertainty must be taken into account in the analysis. Andrew Cliffe What Should We Do with Radioactive Waste? 19/30

81 Stochastic model for WIPP university-log Gaussian RF Z (x, ω) can be written in the form Z (x, ω) = z(x) + λi ψ i (x)ξ i (ω). i=1 where the ξ i are iid standard normal random variables. Andrew Cliffe What Should We Do with Radioactive Waste? 20/30

82 Stochastic model for WIPP university-log Gaussian RF Z (x, ω) can be written in the form Z (x, ω) = z(x) + λi ψ i (x)ξ i (ω). i=1 where the ξ i are iid standard normal random variables. We have E[Z ] = z(x), Cov[Z (x), Z (y)] = λ i λ j ψ i (x)ψ j (y). i=1 [Karhunen, 1947], [Loève, 1948] Andrew Cliffe What Should We Do with Radioactive Waste? 20/30

83 Stochastic model for WIPP Gaussian RF Z (x, ω) can be written in the form Z (x, ω) = z(x) + λi ψ i (x)ξ i (ω). i=1 where the ξ i are iid standard normal random variables. We have E[Z ] = z(x), Cov[Z (x), Z (y)] = λ i λ j ψ i (x)ψ j (y). i=1 [Karhunen, 1947], [Loève, 1948] Here Cov[Z (x), Z (y)] is the covariance function of the conditioned field. university-log Andrew Cliffe What Should We Do with Radioactive Waste? 20/30

84 Stochastic model for WIPP Solve [T (x, ξ) h(x, ξ)] = 0 h(x, ξ) = h 0 (x) x D, a.s. x D, a.s. where log T (x, ξ) = z(x) + λi ψ i (x)ξ i. i=1 Andrew Cliffe What Should We Do with Radioactive Waste? 21/30

85 Stochastic model for WIPP Solve [T (x, ξ) h(x, ξ)] = 0 h(x, ξ) = h 0 (x) x D, a.s. x D, a.s. where log T (x, ξ) = z(x) + λi ψ i (x)ξ i. i=1 Compute the log of travel time, s f, from h and T. Andrew Cliffe What Should We Do with Radioactive Waste? 21/30

86 Stochastic model for WIPP Solve [T (x, ξ) h(x, ξ)] = 0 h(x, ξ) = h 0 (x) x D, a.s. x D, a.s. where log T (x, ξ) = z(x) + λi ψ i (x)ξ i. i=1 Compute the log of travel time, s f, from h and T. Thus s f depends on ξ and θ, and the problem is to compute the stochastic law of s f from the joint law of ξ and θ. Andrew Cliffe What Should We Do with Radioactive Waste? 21/30

87 Solution of stochastic model university-log Taking account of uncertainty in hyperparameters: Andrew Cliffe What Should We Do with Radioactive Waste? 22/30

88 Solution of stochastic model Taking account of uncertainty in hyperparameters: Now F Sf (s) = s f (ξ,θ) s f Ξ,Θ (ξ, θ)dξdθ, Andrew Cliffe What Should We Do with Radioactive Waste? 22/30

89 Solution of stochastic model university-log Taking account of uncertainty in hyperparameters: Now F Sf (s) = s f (ξ,θ) s ( = f Ξ,Θ (ξ, θ)dξdθ, ) s f (ξ,θ) s f Ξ Θ (ξ θ)dξ f Θ (θ)dθ, Andrew Cliffe What Should We Do with Radioactive Waste? 22/30

90 Solution of stochastic model university-log Taking account of uncertainty in hyperparameters: Now F Sf (s) = s f (ξ,θ) s ( = = f Ξ,Θ (ξ, θ)dξdθ, ) s f (ξ,θ) s F Sf Θ(s θ)f Θ (θ)dθ. f Ξ Θ (ξ θ)dξ f Θ (θ)dθ, where F Sf Θ(s θ) = s f (ξ,θ) s f Ξ Θ (ξ θ)dξ. Andrew Cliffe What Should We Do with Radioactive Waste? 22/30

91 Solution of stochastic model university-log F Sf Θ(s θ) is obtained from the solution of the stochastic PDE for each value of θ. Andrew Cliffe What Should We Do with Radioactive Waste? 23/30

92 Solution of stochastic model university-log F Sf Θ(s θ) is obtained from the solution of the stochastic PDE for each value of θ. We do not have an explicit expression for f Θ (θ), just a (large) sample, {θ i } N θ i=1, of equally probable values of θ generated by the MCMC calculation. Andrew Cliffe What Should We Do with Radioactive Waste? 23/30

93 Solution of stochastic model university-log F Sf Θ(s θ) is obtained from the solution of the stochastic PDE for each value of θ. We do not have an explicit expression for f Θ (θ), just a (large) sample, {θ i } N θ i=1, of equally probable values of θ generated by the MCMC calculation. Use Monte-Carlo to compute F Sf (s) as N θ F Sf (s) = 1 F N Sf Θ(s θ i ) θ i=1 where {θ i } N θ i=1 is the MCMC sample. Andrew Cliffe What Should We Do with Radioactive Waste? 23/30

94 Solution of the stochastic PDE university-log Probably the most expensive part of the calculation Andrew Cliffe What Should We Do with Radioactive Waste? 24/30

95 Solution of the stochastic PDE Probably the most expensive part of the calculation Currently a very active area of research Andrew Cliffe What Should We Do with Radioactive Waste? 24/30

96 Solution of the stochastic PDE Probably the most expensive part of the calculation Currently a very active area of research Curse of dimension is the main issue (see later slide) Andrew Cliffe What Should We Do with Radioactive Waste? 24/30

97 Solution of the stochastic PDE university-log Probably the most expensive part of the calculation Currently a very active area of research Curse of dimension is the main issue (see later slide) Options include: Brute force Monte-Carlo Quasi Monte-Carlo methods Generalised polynomial chaos methods Multi-level Monte-Carlo methods Andrew Cliffe What Should We Do with Radioactive Waste? 24/30

98 Solution of the stochastic PDE university-log Probably the most expensive part of the calculation Currently a very active area of research Curse of dimension is the main issue (see later slide) Options include: Brute force Monte-Carlo Quasi Monte-Carlo methods Generalised polynomial chaos methods Multi-level Monte-Carlo methods Many challenges remain in this area, with several promising lines of research. Andrew Cliffe What Should We Do with Radioactive Waste? 24/30

99 Solution of the stochastic PDE university-log Truncated KL expansion N Z N (x, ω) = z(x) + λi ψ i (x)ξ i (ω). i=1 Andrew Cliffe What Should We Do with Radioactive Waste? 25/30

100 Solution of the stochastic PDE Truncated KL expansion Solve N Z N (x, ω) = z(x) + λi ψ i (x)ξ i (ω). i=1 [T (x, ξ) h(x, ξ)] = 0 h(x, ξ) = h 0 (x) x D, a.s. x D, a.s. where N log T (x, ξ) = z(x) + λi ψ i (x)ξ i. i=1 Andrew Cliffe What Should We Do with Radioactive Waste? 25/30

101 Solution of the stochastic PDE Truncated KL expansion Solve N Z N (x, ω) = z(x) + λi ψ i (x)ξ i (ω). i=1 [T (x, ξ) h(x, ξ)] = 0 h(x, ξ) = h 0 (x) x D, a.s. x D, a.s. where N log T (x, ξ) = z(x) + λi ψ i (x)ξ i. i=1 Compute distribution function for s f. university-log Andrew Cliffe What Should We Do with Radioactive Waste? 25/30

102 Solution of the stochastic PDE university-log How big should N be? Andrew Cliffe What Should We Do with Radioactive Waste? 26/30

103 Solution of the stochastic PDE How big should N be? Possibly too large for some current methods.. 1 ECDF of log travel time, MC samples log 10 t KL 5 KL 10 KL 15 Full Figure: Distribution function for the log travel time for various N. university-log Andrew Cliffe What Should We Do with Radioactive Waste? 26/30

104 Solution of stochastic model university-log In this work we used a Gaussian process emulator to approximate F Sf Θ(s θ) by F Sf Θ(s θ). Andrew Cliffe What Should We Do with Radioactive Waste? 27/30

105 Solution of stochastic model In this work we used a Gaussian process emulator to approximate F Sf Θ(s θ) by F Sf Θ(s θ). This involves solving the stochastic PDE for a set (Latin hypercube) of values of θ (solves done by brute force for Monte-Carlo). Andrew Cliffe What Should We Do with Radioactive Waste? 27/30

106 Solution of stochastic model In this work we used a Gaussian process emulator to approximate F Sf Θ(s θ) by F Sf Θ(s θ). This involves solving the stochastic PDE for a set (Latin hypercube) of values of θ (solves done by brute force for Monte-Carlo). (Describing Gaussian process emulators would be another talk - think of them as a way of doing interpolation with scattered data in high dimensions - similar to RBF methods.) Andrew Cliffe What Should We Do with Radioactive Waste? 27/30

107 Solution of stochastic model In this work we used a Gaussian process emulator to approximate F Sf Θ(s θ) by F Sf Θ(s θ). This involves solving the stochastic PDE for a set (Latin hypercube) of values of θ (solves done by brute force for Monte-Carlo). (Describing Gaussian process emulators would be another talk - think of them as a way of doing interpolation with scattered data in high dimensions - similar to RBF methods.) Then use the emulator to evaluate the sum F Sf (s) = 1 N θ N θ i=1 F Sf Θ(s θ i ) to obtain the required solution. university-log Andrew Cliffe What Should We Do with Radioactive Waste? 27/30

108 WIPP Results university-log The distribution function, F Sf (s) contains a description of the uncertainty due to the hyperparameters as well as the uncertainty due to the random field model of transmissivity. Andrew Cliffe What Should We Do with Radioactive Waste? 28/30

109 WIPP Results The distribution function, F Sf (s) contains a description of the uncertainty due to the hyperparameters as well as the uncertainty due to the random field model of transmissivity F(s) design points emulator mean 99% bound 1% bound CDF from MC s Figure: Distribution function for the log travel time. university-log Andrew Cliffe What Should We Do with Radioactive Waste? 28/30

110 Summary university-log Radioactive waste disposal is an important problem with many scientific challenges. Andrew Cliffe What Should We Do with Radioactive Waste? 29/30

111 Summary Radioactive waste disposal is an important problem with many scientific challenges. Characterising and quantifying uncertainty is necessary for the safety assessment of a potential radioactive waste repository. Andrew Cliffe What Should We Do with Radioactive Waste? 29/30

112 Summary Radioactive waste disposal is an important problem with many scientific challenges. Characterising and quantifying uncertainty is necessary for the safety assessment of a potential radioactive waste repository. Uncertainties in the hyperparameters of the stochastic model for transmissivity can be dealt with in a Bayesian framework. Andrew Cliffe What Should We Do with Radioactive Waste? 29/30

113 Summary Radioactive waste disposal is an important problem with many scientific challenges. Characterising and quantifying uncertainty is necessary for the safety assessment of a potential radioactive waste repository. Uncertainties in the hyperparameters of the stochastic model for transmissivity can be dealt with in a Bayesian framework. The central computational/mathematical problem is solving elliptic PDEs with random coefficients. Andrew Cliffe What Should We Do with Radioactive Waste? 29/30

114 Summary Radioactive waste disposal is an important problem with many scientific challenges. Characterising and quantifying uncertainty is necessary for the safety assessment of a potential radioactive waste repository. Uncertainties in the hyperparameters of the stochastic model for transmissivity can be dealt with in a Bayesian framework. The central computational/mathematical problem is solving elliptic PDEs with random coefficients. Exciting and important area of research with many promising approaches. Andrew Cliffe What Should We Do with Radioactive Waste? 29/30