Stress tensor computation using mindlin formulation

A method for performing a borehole-related task on an earth formation having changes in density, the method comprising:

• constructing a geo-cellular model of the earth formation using a processor, the geo-cellular model comprising a plurality of grid cell volumes;
• populating multiple grid cell volumes with corresponding actual density values and actual Poisson's ratio values for the grid cell volumes using the processor;
• calculating with the processor a linear approximation of the density values and a linear approximation of the Poisson's ratio values as a function of spatial coordinates of the individual volume cells to provide linear approximation density values and linear approximation Poisson's ratio values for the plurality of grid cell volumes;
• calculating with the processor a first approximation of components of a stress tensor representing stress in the formation using the linear approximation of the density values and the linear approximation of the Poisson's ratio values for the plurality of grid cell volumes;
• calculating with the processor a difference between the actual density values and the linear approximation density values to provide density difference values for each of the grid cell volumes in the plurality of grid cell volumes;
• calculating with the processor a contribution to stress at each grid cell volume of interest from other grid cell volumes that are at elevations above the grid cell volume of interest by use of the Mindlin Formulation using the density difference values and the Poisson's ratio values associated with each grid cell volume in the plurality of grid cell volumes;
• adding with the processor all contributions to the stresses at each grid cell volume of interest from all other grid cell volumes that are at elevations above the grid cell volume of interest to provide a correction to the stress tensor;
• adding with the processor the first approximation of the components of the stress tensor to the correction to the stress tensor to provide a total stress value that completely defines a state of stress in each grid cell volume of interest;
• constructing with the processor a resulting stress tensor for the earth formation using the total stress values for the grid cell volumes of interest;
• and
• performing the borehole-related task using borehole-related equipment and the resulting stress tensor for the earth formation.