Project

• Description
• Aims
• Personnel
• References

Project description

Stochastic modelling of the subsurface is a critical element in many aspects of underground engineering design. It is becoming increasingly important in environmental applications including the design and assessment of underground waste repositories and landfill sites and associated risk analyses.

Modelling and prediction are based on data. Expense and access mean that direct measurements of primary variables (e.g. porosities, permeabilities, fracture densities, elastic modulii) are very sparse (e.g. several vertical boreholes on a large horizontal grid, often very irregular). At all unsampled locations the values of primary variables are unknown and must be predicted. Any model of the subsurface thus depends on the prediction method and the amount, quality and location of the data. All models are, therefore, subject to uncertainty and, a priori, there can be no unique model.

In the applications described here the sparse primary data are supplemented by indirect measurements of secondary (geophysical logs and seismic), variables. Problems arise in calibrating the secondary variables to meaningful primary variables and integrating the different types of data for use in modelling and prediction. The most widely used approaches to these problems are deterministic and essentially assume that the solutions are unique. The primary objective of this project is to develop stochastic solutions to the calibration and integration of geophysical and seismic data and thereby incorporate more of the effects of non-uniqueness in any subsequent modelling. Data are generally available from three different sources:

• borehole cores
• geophysical logging of boreholes
• surface-recorded 2D seismic measurements or full 3D multi-source seismic surveys

each representing a different scale of measurement, a different level of precision and a different method of measurement employing fundamentally different physical principles.

For the sake of a common example, suppose that a model of the subsurface porosity is required. Direct measurements of porosity can be made on borehole cores. Geophysical logging of the boreholes will yield, inter alia, compressional velocity and bulk density, the product of which defines acoustic impedance (AI) more or less continuously along the boreholes. A predictive relationship can be established between AI and porosity. A 3D seismic survey will yield travel times to, and amplitudes of, reflection events. The seismic times and amplitudes can be inverted to give relative acoustic impedances which can then be calibrated to the borehole AI. Core sample properties are obtained by laboratory measurements and are usually measured on plugs, or sub-cores, of around 25mm diameter and rarely greater than 100mm diameter. These properties thus refer to relatively precise measurements on relatively small, specific, identifiable volumes. Data derived from the geophysical logging of boreholes are based on the interpretation of signals; for example, a wireline logging tool measures bulk density by monitoring the attenuation of gamma rays as they respond to the mean electron density of the surrounding formation between a gamma source and detector. Wireline velocity logs operate at a range of wavelengths up to several tens of centimetres and properties derived from these logs refer to volumes of up to several cubic metres of in situ rock. The precision of wireline measurements depends on many factors including logging speeds, borehole environment, tool design, operational parameters and calibration of equipment. Seismic-derived data are based on seismic velocities and acoustic impedance and contain wavelengths up to hundreds of metres; properties derived from seismic surveys thus have relatively large-scale resolution. Seismic data are wave-length dependent, band-width limited and noise-contaminated.

The problems are:

1. to filter out noise from seismic data
2. to calibrate the secondary, indirect data (e.g. AI from geophysical logs) to the values of relevant primary variables (e.g. porosity from core measurements)
3. to integrate (or synthesise) the various forms of secondary data taking account of the different scales of resolution and the different levels of precision

Most noise filtering techniques are based on Fourier analysis but these approaches ignore the spatial nature of the variables. Various approaches, both deterministic and stochastic, have been developed to integrate and calibrate the secondary data. Stochastic approaches must recognise the spatial nature of the variables and include models of multivariate spatial variability; these approaches are largely geostatistical. However, almost all of the geostatistical approaches developed to prior this project relate to sedimentary hydrocarbon reservoirs and not to crystalline rocks. This is not to say that these geostatistical approaches could not be adapted to such situations and we have evaluated the applicability of some of them in these rock types.

Geostatistical methods include co-kriging, kriging with an external drift, Markov-Bayes calibration and conditional simulation as a means of integrating seismic data as prior conditioning data. These approaches all have limitations. For example, kriging and co-kriging tend to yield smoothed models; kriging with an external drift ignores any spatial correlation between the variables; the Markov-Bayes method requires a prior calibration that is often difficult to establish with any reliability. Conditional simulation, however, shows significant promise.

Aims, beneficiaries and methodology

Overall aims

The overall aim of the project was to develop a general stochastic approach for the treatment of the various forms of data used in the assessment and modelling of the subsurface particularly for underground waste repository and landfill applications. The stochastic treatment included calibration and integration of primary, direct data and secondary, indirect data and the filtering of noise from the latter. In these applications the former are almost always derived from core, or analogous, physical measurements and the latter are almost always geophysical logging and seismic data.

Measurable objectives were:

Documented algorithms and associated software for:

• filtering seismic data
• calibration of primary and secondary variables
• the integration and calibration of the various types of secondary data
• generating stochastic images to assist risk perception and management

The provision of a “turn-key” interactive stochastic data treatment package for industrial use.

The provision of documented, quantitative methods of risk assessment relating to the safe disposal of hazardous wastes.

Beneficiaries of the outputs from this research project are:

• the academic community - access to research results and (non-commercial) use of software products.
• the waste disposal industry - integrated software products for stochastic treatment of data and for risk assessment.
• the data logging industry - integrated software products for stochastic treatment, synthesis and calibration of data.
• regulators - an additional tool for use in risk analysis and risk management and a means of assessing the validity of data synthesis.
• the civil engineering/geotechnical engineering industry - the methods can be used in any applications of engineering design and assessment based on seismic data.
• the general public - the output from the simulation methods proposed here provides a powerful means of perceiving and managing risk (Dowd, 1997a, 1998).

Methodology

1. Noise filtering

We have used deconvolution by factorial kriging to filter noise from seismic data. Factorial kriging (Matheron, 1982) is a form of spatial factor analysis and is essentially a method for decomposing experimental variograms into a sum of component models each of which is allotted to a factor. The principle is to identify a nugget (random) variogram component that quantifies the noise. This approach has been used by a number of authors (e.g. Zhang and Galli, 1992) but it has generally been restricted to two-dimensional random function models. We have extended the method to three-dimensional random functions and to its co-kriging form.

2. Calibration of primary and secondary data

We have used co-kriging to establish relationships between the primary and secondary variables (e.g. porosity and acoustic impedance). Unlike standard techniques, such as regression, this approach allows spatial inter-relationships to be taken into account. We have used standard co-kriging and lag-displaced co-kriging to examine spatial relationships that may be offset by displacements along boreholes.

3. Integration and calibration of the secondary data

We have developed two methods for integration and calibration:

(i) Multiple indicator co-kriging
The seismic data (e.g. seismic velocity) will be are incorporated as values of a variable spatially correlated with the borehole data (e.g. acoustic impedance). Both variables will be are represented as indicators with models defined for representative ranges of thresholds. This approach will allows for explicit modelling of the bivariate spatial variability at all relevant scales. We will investigate have extended ing this approach to any number of relevant variables by using multivariate, multiple indicator co-kriging; collocated forms of co-kriging will be considered are used to maintain feasible computation and modelling times.

(ii) Conditional simulation
This method generates synthetic seismograms for the boreholes by simulating values of the borehole variable from the borehole data and then matching the synthetic seismograms to the seismic data. The method was first suggested by Bortoli et. al. (1992) and avoids many of the scale (resolution) effect problems inherent in inversion techniques.

We have extended the work of Bortoli et. al. (1992) and Dowd (1994, 1996) to include stochastic geological controls on the simulations using the methods described in Dowd (1994, 1996).

4. Risk analysis, perception and management

In this project we have built on the work of Dowd (1997a, 1998) to provide, through the conditional simulation techniques in 3(ii) above, tools for the analysis, perception and management of risk in applications relating to the consequences of underground waste disposal. We have used conditional simulation to provide a means of probabilistic risk analysis based on the integration of the various quantifiable sources of risk.

Quantification is, however, only way of assessing and reporting risk. In general, risk is perceived and this perception must be addressed when conducting risk analyses and reporting the results, especially in public forums. The modelling and visualization capabilities of conditional simulation provide a means of dealing with perception in risk analysis.

Project personnel

Professor P A Dowd, Principal Investigator.

Dr E Pardo-Iguzquiza, Post-Doctoral Research Assistant.

References

1. Bortoli, L.J., Alabert, F., Haas, A. and Journel, A. (1992) Constraining stochastic images to seismic data. Geostatistics Troia ‘92, volume 1; pp 325 - 337.Kluwer Academic Publishers, Quantitative Geology and Geostatistics series.
2. Dowd, P.A. (1994) Geological controls in the geostatistical simulation of hydrocarbon reservoirs. The Arabian Journal for Science and Engineering, volume 19, no. 2B, April 1994, pp 237 - 247.
3. Dowd, P.A. (1996) Structural controls in the geostatistical simulation of mineral deposits. Geostatistics Wollonging ’96 (Proceedings of the fifth International Geostatistics Congress). Kluwer Academic publishers, Dordrecht, Netherlands. volume 2, pp 647-657.
4. Dowd, P.A. (1997a) Risk in minerals projects: analysis, perception and risk. Trans. Instn. Min. Metall 106, pp A9-18.
5. Dowd, P.A. (1997b) The geostatistical characterization of three-dimensional spatial heterogeneity of rock properties at Sellafield. Trans. Instn. Min. Metall. 106 , pp A133-147.
6. Dowd, P.A. (1998) The assessment and analysis of financial, technical and environmental risk in mineral resource exploitation. Invited Lecture at the NATO Advanced Study Institute on Resource Exploitation and Environmental Security. Mátraháza, Hungary, 6-18 September 1998. Published in 2003 in Deposit and Geo-environmental Models for Resource Exploitation and Environmental Security (eds: A.G. Fabbri, G. Gaal, R.B. McCammon) NATO Science Series 2. Environmental Security, Volume 80. Kluwer Academic Publishers, Dordrecht, The Netherlands. ISBN HB 1-4020-0989-5, PB 1-4020-0990-9. pages 187-221.
7. Matheron, G. (1982) Pour une analyse krigeante des donées régionalisées. Internal report N-732, Centre de Géostatistique, ENSNP, Fontainebleau, France; 22p.
8. Zhang, Z. and Galli, A. (1992) Getting better quality seismic sections. Geostatistics Troia ‘92, volume 1; 285-297pp . Kluwer Academic Publishers, Quantitative Geology and Geostatistics series.