Systems and methods for removal of electromagnetic dispersion and attenuation for imaging of proppant in an induced fracture

A system, comprising:

• a well bore having a conductive casing that runs from a surface of the Earth to a geologic formation;
• a proppant pack formed in at least a portion of a fracture that extends from the well bore into the geologic formation;
• a current source configured to electrically energize the conductive casing;
• a plurality of sensors configured to capture electromagnetic fields generated by the energized conductive casing and the proppant pack;
• and
• computing equipment configured to back propagate the captured electromagnetic fields from a set of sensor locations to a set of source locations using a propagator and a window function by:
• applying the propagator and the window function to the captured electromagnetic fields to form back propagated measured electromagnetic field data,
• adjusting parameters of the First Born Approximation model with the back propagated measured electromagnetic field data to provide adjusted model parameters, and
• generating a three-dimensional image of the proppant pack in the fracture in the geologic formation using the adjusted model parameters,
• wherein the computing equipment comprises:
• a data store comprising computer readable storage having an executable code representing the First Born Approximation model stored thereon, the First Born Approximation model having a set of adjustable parameters;
• a primary field modeling engine in communication with the data store, wherein the primary field modeling engine calculates incident electromagnetic field values;
• a secondary field modeling engine in communication with the data store, wherein the secondary field modeling engine calculates scattered electromagnetic field values;
• a measured data processing engine in communication with the plurality of sensors and the data store, the measured data processing engine configured to generate measured change EM data;
• a dispersion and attenuation removal engine in communication with the measured data processing engine, the dispersion and attenuation removal engine configured to back propagate the measured change EM data to generate corrected change EM data;
• a data fitting engine in communication with the dispersion and attenuation removal engine and the primary and secondary field modeling engines, the data fitting engine configured to adjust the adjustable parameters until the calculated scattered electromagnetic field values have been modified to match the measured change EM data, wherein the adjustable parameters are a set of parameters P corresponding to electromagnetic properties of material at Born scatterer locations XB;
• and
• an image generation engine in communication with the data fitting engine and configured to produce the three-dimensional image of the proppant pack.