Earth scientists have measured fractal dimensions of surfaces by different techniques, including the divider, box, triangle, slit-island, power spectral, variogram and distribution methods. We review these seven measurement techniques, finding that fractal dimensions may vary systematically with measurement method. We discuss possible reasons for these differences, and point to common problems shared by all of the methods, including the remainder problem, curve-fitting, orientation of the measurement plane, size and direction of the sample. Fractal measurements have been applied to many problems in the earth sciences, at a wide range of spatial scales. These include map data of topography; fault traces and fracture networks; fracture surfaces of natural rocks, both in the field and at laboratory scales; metal surfaces; porous aggregate geometry; flow and transport through heterogeneous systems; and various microscopic surface phenomena associated with adsorption, aggregation, erosion and chemical dissolution. We review these applications and discuss the usefulness and limitations of fractal analysis to these types of problems in the earth sciences.

·This report was prepared as an account of work sponso~ed by the United States Government. Neither the Un.'t~d States nor the United States Atomic Energy Co':"m•ss•on, nor any of their employees, nor any of theu contractors, sub~ontractors, or their employees makes _anr. warranty, express or implied, or Bssumes an ' tlegal hab1hty or responsibility for the accuracy cor: pleteness or usefulness of any information app~ratus ~ product or. pr?cess disclosed, or: represents 'that its us~ would not mfnnge privately owned rights.

Fluid and heat flow at temperatures approaching or exceeding that at the critical point (374°C for pure water, higher for saline fluids) may ·be· encountered in deep zones of geothermal systems and above cooling intrusives. In the vicinity of the critical point the density and internal energy of fluids show very strong variations for small temperature and pressure changes. This suggests that convective heat transfer from thermal buoyancy flow would be strongly enhanced at near-critical conditions. This has been confirmed in laboratory experiments.We have developed special numerical techniques for modeling porous flow at nearcritical conditions, which can handle the extreme non-linearities in water properties near the critical point Our numerical simulations show strong enhancements of convective heat transfer at near-critical conditions; however, the heat transfer rates obtained in the simulations are considerably smaller than data reported from laboratory experiments by Dunn and Hardee. We discuss possible reasons for this discrepancy and develop suggestions for additional laboratory experiments.

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