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.

A rough‐walled fracture is modeled by fractal geometry. In the fractal fracture model, the rock surfaces are characterized by a fractal dimension D between 2 and 3, with lower D for smoother surfaces and higher D for rougher surfaces. The mismatch due to shear displacement between two mirror‐image fractal surfaces determines the fracture aperture distribution. An analytic equation is derived for the variogram γƒ(r) describing the spatial correlation of the aperture of a fractal fracture. The aperture of a smooth fracture with low D is highly correlated over distances much larger than the shear displacement. The aperture of a rough fracture with high D becomes uncorrelated within a range shorter than the shear displacement. Near the origin, r = 0, of the variogram, the variogram is proportional to r6–2D. For the special case of Brownian fractal with D = 2.5, the variogram is proportional to r, which is the same r dependence exhibited by the spherical model widely used in geostatistical analyses.

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Laboratory‐scale studies of discrete fractures lead to better understanding of the relationship between flow properties and pore geometry. Field‐scale studies of fracture networks test the validity of classical continuum approaches for transport processes in heterogeneous media. During 1987 to 1990, earth scientists have made significant advances in quantifying flow channeling and solute breakthrough in rough‐walled fractures, analyzing transitions between fracture and matrix flows under multiphase conditions, and characterizing fracture networks with hydrological and geophysical studies. In this review, the effect of heterogeneity and the issue of scaling are discussed. Many common features and similiar approaches can be identified in microcracks, discrete fractures, soil macropores, fractured rock masses, and heterogeneous reservoirs at different scales.