Approximate solutions are obtained for the stress and displacement fields due to a pressurized spherical cavity in an elastic half‐space. The solutions take the form of series expansions in powers of ε = a/d, where a is the cavity radius and d is the depth. The leading‐order term in the expression for the surface uplift, which arises at O(ε3), recovers the well‐known result of Mogi for the response to a point dilatation. The first higher‐order correction accounts for a cavity of finite size and thus offers the possibility of fitting leveling data for not only the depth but also the radius and pressure increment. However, this correction is of O(ε6) and, consequently, is weak. The result provides a formal explanation for the success of the point dilatation model in representing uplift data even when it is known independently that ε is not small. The higher‐order correction causes the surface uplift to fall off more rapidly in the radial direction, implying that a fit of the point source solution tends to underestimate the depth d. In contrast to the surface displacement, the stress field near the cavity is affected profoundly by the proximity of the free surface. Three higher‐order corrections to the stress field are obtained, which result in a uniformly valid approximation to O(ε5). The hoop stress at the cavity exhibits a tensile maximum at the circle of tangency with a cone with its apex at the free surface. This result appears to be consistent with the locus of fractures radiating outward from the magma body inferred by seismic methods in Long Valley, California.

A linear theory for fluid‐saturated, porous, thermoelastic media is developed. The theory allows for compressibility and thermal expansion of both the fluid and solid constituents. A general solution scheme is presented, in which a diffusion equation with a temperature‐dependent source term governs a combination of the mean total stress and the fluid pore pressure. In certain special cases, this reduces to a diffusion equation for the pressure alone. In addition, when convective heat transfer and thermoelastic coupling can be neglected, the temperature field can be determined independently, and the source term in the pressure equation is known. Drained and undrained limits are identified, in which fluid flow plays no role in the deformation. In the drained case, the medium behaves as a simple thermoelastic body with the properties of the porous skeleton with no fluid present. In the undrained limit, the fluid is trapped in the pores, and the material responds as a thermoelastic body with effective compressibility and thermal expansivity determined in part by the fluid properties. The theory is further specialized to one‐dimensional deformation, and several illustrative problems are solved. In particular, the heating of a half space is explored for constant temperature and constant flux boundary conditions on the thermal field, and for drained (zero pressure) and impermeable (zero flux) conditions on the fluid pressure field. The behavior of these solutions depends critically upon the ratio of the fluid and thermal diffusivities, with very large and very small values of this parameter corresponding to drained and undrained responses, respectively.

No abstract

SUMMARYAn approximate, analytical solution is found for the gravity-induced stresses in the neighbourhood of an axisymmetric topographic feature on an elastic half-space. The solution is in the form of a perturbation expansion in powers of the characteristic slope, E . The leading order problem, at 0(1), is for a distributed normal load on a plane half-space. The O(E) correction is due to a distributed shear traction. The vertical variation of the near-surface stress perturbation and the rotation of the principal stress directions are O(E) eNects, and thus include contributions of like order from both the normal and shear loads. The method requires general solutions for axisymmetric, normal and shear tractions of arbitrary radial distribution, and these are found in terms of Hankel transforms.

No abstract

Surface displacements and tilts due to buried deformation sources are influenced by topography. The leading‐order corrections due to topography of arbitrary profile, but small slope, are determined for dip‐slip faults (edge dislocations) and magma bodies (lines of inflation) in a two‐dimensional elastic medium. The vertical displacement correction is simply the product of the topography and the horizontal normal strain due to the source in a flat half‐space. The correction for a normal fault beneath an idealized basin‐and‐range topography is a slight increase in the uplift on the range side. The effect of topography for a line of inflation beneath a symmetric volcano is to reduce the central uplift. Failure to account for topographic influences can bias estimates of source depth and geometry.

No abstract

Exact solutions are obtained for fluid flow induced by the heating of a borehole. The rock is modeled as a fluid‐saturated, porous, thermoelastic medium. The temperature and pore pressure fields are governed by a pair of diffusion equations, which are coupled through a source term in the pressure equation proportional to the temperature rate. The pressure profile exhibits a maximum that grows in magnitude and propagates away from the borehole. For a constant heat flux applied as an instantaneous step, the fluid flux to the borehole takes a finite initial value, and decays monotonically. When the heat flux exhibits a finite rise time, the fluid flux is initially zero, rises to a maximum, and then decays. At late time, the inverse of the fluid flux is linear in ln t; this observation can be exploited to estimate the permeability and fluid diffusivity of low‐permeability rock. Sample calculations are shown for Westerly granite.