Determinations of standoff and mud density with gamma density and acoustic tool responses

A system, comprising:

• a drilling rig;
• a drill string;
• a drill bit at a distal end of the drill string;
• a density module, forming at least a portion of the drill string;
• an acoustic module, forming at least a portion of the drill string;
• and
• a controller associated with the drill string, the controller including one or more memories and one or more processors, the one or more memories storing instructions that, when executed by the one or more processors, cause the controller to:
• determine, based at least in part on density module data and acoustic module data, a formation density;
• determine an initial estimation for a mud density;
• determine an initial estimation for a standoff;
• compute, based on the initial estimation for the standoff and a distance-to-spine value, a mud density approximation;
• compute, based on the initial estimation for the mud density and the mud density approximation, a standoff approximation;
• and
• determine, based at least in part on the mud density approximation and the standoff approximation, a computed mud density or a computed standoff.

A method for determining a drilling fluid property based, at least in part, on a formation density, comprising:

• receiving, from a density module, a first density measurement at a first detector position along the density module and a second density measurement from a second detector position along the density module;
• receiving, from an acoustic module, a time of sound propagation between the acoustic module and a borehole wall;
• determining, based at least in part on the first density measurement and the second density measurement, the formation density;
• determining a distance-to-spine value using the formation density and the first density measurement and the second density measurement;
• determining an initial estimation for a mud density;
• determining an initial estimation for a standoff using the acoustic module;
• determining a mud density approximation based on the initial estimation for the standoff and the distance-to-spine value;
• determining a standoff approximation based on the mud density approximation, the initial estimation for the standoff, and the initial estimation for the mud density;
• and
• determining, based at least in part on the standoff approximation, a computed mud density.