On the mechanical interaction between a fluid-filled fracture and the earth's surface

Abstract

The mechanical interaction between a fluid-filled fracture (e.g., hydraulic fracture joint, or igneous dike) and the earth's surface is analyzed using a two-dimensional elastic solution for a slit of arbitrary inclination buried beneath a horizontal free surface and subjected to an arbitrary pressure distribution. The solution is obtained by iteratively superimposing two fundamental sets of analytical solutions. For uniform internal pressure the slit behaves essentially as if it were in an infinite region if the depth-to-center is three times greater than the half-length. For shallower slits interaction with the free surface is pronounced: stresses and displacements near the slit differ by more than 10% from values for the deeply buried slit. The following changes are noted as the depth-to-center decreases: 1. (1) the mode I stress intensity factor increases for both ends of the slit, but more rapidly at the upper end; 2. (2) the mode II stress-intensity factor is significantly different from zero (except for vertical slits) suggesting propagation out of the original plane of the slit; 3. (3) displacements of the slit wall are asymmetric such that the slit gaps open more widely near the upper end. Similar changes are noted if fluid density creates a linear pressure gradient that is smaller than the lithostatic gradient. Under such conditions natural fractures should propagate preferentially upward toward the earth's surface requiring less pressure as they grow in length. If deformation near the surface is of interest, the model should account explicitly for the free surface. Stresses and displacements at the free surface are not approximated very well by values calculated along a line in an infinite region, even when the slit is far from the line. As depth-to-center of a shallow pressurized slit decreases, the following changes are noted: 1. (1) displacements of the free surface increase to the same order of magnitude as the displacements of the slit walls, 2. (2) tensile stresses of magnitude greater than the pressure in the slit are concentrated along the free surface. The relative surface displacements over a shallow vertical slit are downward over the slit and upward to both sides of this area. The tensile stress acting parallel to the free surface over a shallow vertical slit is concentrated in two maxima adjacent to a point of very low stress immediately over the slit. The solution is used to estimate the length-to-depth ratio at which igneous sills have gained sufficient leverage on overlying strata to bend these strata upward and form a laccolith. The pronounced mode II stress intensity associated with shallow horizontal slits explains the tendency for some sills to climb to higher stratigraphie horizons as they grow in length. The bimodal tensile stress concentration over shallow vertical slits correlates qualitatively with the distribution of cracks and normal faults which flank fissure eruptions on volcanoes. The solution may be used to analyze surface displacements and tilts over massive hydraulic fractures in oil fields and to understand the behavior of hydraulic fractures in granite quarries.