Estimating fluid-induced stress change from observed deformation

Abstract

S U M M A R YObserved deformation is sensitive to a changing stress field within the Earth. However, there are several impediments to a direct inversion of geodetic measurements for changes in stress. Estimating six independent components of stress change from a smaller number of displacement or strain components is inherently non-unique. The reliance upon surface measurements leads to a loss of resolution, due to the attenuation of higher spatial frequencies in the displacement field with distance from a source. We adopt a technique suited to the estimation of stress changes due to the injection and/or withdrawal of fluids at depth. In this approach the surface displacement data provides an estimate of the volume change responsible for the deformation, rather than stress changes themselves. The inversion for volume change is constrained by the fluid fluxes into and out of the reservoir. The distribution of volume change is used to calculate the displacements in the region above the reservoir. Estimates of stress change follow from differentiating the displacement field in conjunction with a geomechanical model of the overburden. We apply the technique to Interferometric Synthetic Aperture Radar (InSAR) observations gathered over a petroleum reservoir in the San Joaquin Valley of California. An analysis of the InSAR range changes reveals that the stress field in the overburden varies rapidly both in space and in time. The inferred stress variations are found to be compatible with the documented failure of a well in the field.

Satellite-based monitoring of groundwater depletion in California’s Central Valley

“…A related issue is the non-uniqueness, whereby multiple models of aquifer volume changes lead to the same set of surface displacements. The problem is particularly acute when volume changes can occur at various depths, as there can be trade-offs between the changes at different depths within the aquifer 20 .…”

“…These GSAs vary in size greatly but are generally smaller than a full hydrological basin. However, the general approach described in this study can be conducted at a much smaller scale 20 . For example, one can examine areas of a few square kilometers in size, with grid block dimensions of tens to hundreds of meters as was done for oil and gas applications.…”

“…In particular, the material within a grid block may experience elastic and inelastic deformation due to compaction produced by groundwater withdrawal 3,29 . However, because the overburden and adjacent regions are assumed to respond to the volume change within the source region in an elastic manner, the observed deformation at a point x i on the Earth’s surface, denoted by u r ( x i ), is linearly related to the volume change associated with the n -th grid block in the aquifer model, v ( r n ),where Γ in is the elastic Green’s function, the response function for a rectangular grid block, centered at r n , and the summation is over all N of the grid blocks in the aquifer model 18–20 . The components of the indexed vector g i are the response functions for each of the rectangular blocks in the model, while the vector v contains their fractional volume changes.…”

“…The inverse problem is formulated as a least-squares optimization problem in which we seek a spatial distribution of volume changes minimizing the sum of the squares of the InSAR range change residuals 18–20 . In order to stabilize the inversion in the face of the non-uniqueness, one typically introduces penalty or regularization terms that minimize some aspect of the model, such as the size of the volume changes or the overall roughness of the model 30 .…”

“…However, ground deformation itself is an indirect measure of the changes within an aquifer and additional analysis is required in order to extract quantities needed by water managers. Specifically, one must solve an inverse problem in order to relate the InSAR observations to the aquifer volume changes 18–20 . Additional parameters describing the poroelastic or poroplastic response of the medium are required to relate aquifer structural volume changes to changes in the volume or the mass of water in the aquifer 21 .…”