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

Abstract

Systems and methods for generating a three-dimensional image of a proppant-filled hydraulically-induced fracture in a geologic formation are provided. The image may be generated by capturing electromagnetic fields generated or scattered by the proppant-filled fracture, removing dispersion and/or an attenuation effects from the captured electromagnetic fields, and generating the image based on the dispersion and/or attenuation corrected fields. Removing the dispersion and/or attenuation effects may include back propagating the captured electromagnetic fields in the time domain to a source location. The image may be generated based on locations at which the back propagated fields constructively interfere or may be generated based on a model of the fracture defined using the back propagated fields.

1 . A system, comprising:
 a conductive casing of a well bore that extends from a surface of the Earth to a geologic formation having a fracture;   electrical equipment configured to generate electromagnetic waves using the conductive casing;   a plurality of sensors configured to detect the electromagnetic waves generated by the electrical equipment that propagate from the conductive casing to the plurality of sensors;   a database that stores:
 a propagator, and 
 a window function; and 
   a processor configured to:
 receive measured electromagnetic field data from the plurality of sensors at a plurality of sensor locations, 
 back propagate the measured electromagnetic field data from the sensor locations to a set of source locations using the propagator and the window function by:
 transforming the measured electromagnetic field data to a frequency domain; 
 applying the propagator and the window function to the transformed electromagnetic field data in the frequency domain to form back propagated frequency domain data; and 
 inverse transforming the back propagated frequency domain data, and 
 
 generate an image of a proppant pack in the fracture in the geologic formation using the back propagated measured electromagnetic field data. 
   
     
     
         2 . The system of  claim 1 , wherein the processor is further configured to:
 determine locations at which the back propagated measured electromagnetic field data from the plurality of sensors coalesces; and   generate the image of the proppant pack based on the determined locations.   
     
     
         3 . The system of  claim 2 , where determining the locations at which the back propagated measured electromagnetic field data from the plurality of sensors coalesces comprises determining locations at which the back propagated measured electromagnetic field data includes constructively interfering and destructively interfering electromagnetic waves. 
     
     
         4 . The system of  claim 3 , wherein the determined locations at which the back propagated measured electromagnetic field data from the plurality of sensors coalesces are the locations at which the back propagated measured electromagnetic field data includes constructively interfering electromagnetic waves. 
     
     
         5 . The system of  claim 1 , wherein the propagator comprises a plane wave propagator. 
     
     
         6 . The system of  claim 1 , wherein the propagator comprises a Green's function propagator. 
     
     
         7 . The system of  claim 1 , wherein the propagator includes an exponential function having a sign determined to remove dispersion effects from the received measured electromagnetic field data. 
     
     
         8 . The system of  claim 1 , wherein the propagator includes an imaginary part having a sign determined to restore amplitudes of frequency content lost due to attenuation in the received measured electromagnetic field data. 
     
     
         9 . The system of  claim 1 , wherein the processor is configured to back propagate the measured electromagnetic field data from the sensor locations to the set of source locations using the propagator and the window function, in part, by multiplying the window function by the propagator. 
     
     
         10 . The system of  claim 1 , wherein the processor is further configured to form attenuation and/or dispersion corrected electromagnetic field data based on the back propagating and to adjust a parameter of a fracture model using the attenuation and/or dispersion corrected electromagnetic field data. 
     
     
         11 . The system of  claim 1 , wherein the proppant pack is electrically conductive. 
     
     
         12 . A method, comprising:
 electrically energizing a proppant pack and a casing of a well bore that extends from a surface of the Earth to a geologic formation having a fracture to provide electromagnetic field data;   capturing measured electromagnetic field data with a plurality of sensors at a plurality of sensor locations;   back propagating the measured electromagnetic field data from the sensor locations to a set of source locations using a propagator and a window function, comprising:
 transforming the measured electromagnetic field data to a frequency domain; 
 applying the propagator and the window function to the transformed electromagnetic field data in the frequency domain to form back propagated frequency domain data; and 
 inverse transforming the back propagated frequency domain data; and 
   generating an image of the proppant pack in the fracture in the geologic formation using the back propagated measured electromagnetic field data.   
     
     
         13 . The method of  claim 12 , wherein back propagating the measured electromagnetic field data from the sensor locations to the set of source locations using the propagator and the window function further comprises generating a three-dimensional image of the proppant pack by locating determined locations at which the back propagated measured electromagnetic field data constructively interferes. 
     
     
         14 . The method of  claim 12 , further comprising propagating an input waveform to the set of source locations. 
     
     
         15 . The method of  claim 12 , further comprising:
 forming attenuation and/or dispersion corrected electromagnetic field data based on the back propagating; and   adjusting a parameter of a fracture model using the attenuation and/or dispersion corrected electromagnetic field data.   
     
     
         16 . The method of  claim 12 , wherein transforming the measured electromagnetic field data to the frequency domain comprises Fourier transforming the measured electromagnetic field data. 
     
     
         17 . The method of  claim 12 , further comprising:
 providing a fracture fluid to form the fracture;   injecting electrically conductive proppant into the fracture to form the proppant pack.   
     
     
         18 . 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 to form an image of the proppant pack using a propagator and a window function by:
 transforming the measured electromagnetic field data to a frequency domain; 
 applying the propagator and the window function to the transformed electromagnetic field data in the frequency domain to form back propagated frequency domain data; and 
 inverse transforming the back propagated frequency domain data. 
   
     
     
         19 . The system of  claim 18 , wherein the propagator is selected from the group consisting of a plane wave propagator and a Green's function propagator. 
     
     
         20 . The system of  claim 19 , wherein the computing equipment is configured to back propagate the captured electromagnetic fields to form the image of the proppant pack by multiplying the window function by the propagator.