Mathematical Model of Acid Wormhole Expansion When Combining Diverting Acid with Solid Diverting Agent

Considering characteristics of complex carbonate reservoirs (e.g., high depth, high temperature, and fracture cave development), this paper simulates expansion of the acid wormhole when combining diverting acid and a solid diverting agent for acid fracturing. Using the theory of reaction kinetics, tests of diverting acid reaction kinetics, and flow reaction experiments on the long core and parallel core, this paper presents tests of the acid–rock reaction for a mathematical model of acid diversion. On the basis of a rheological behavior test of diverting acid, we studied the influences of Ca2+ concentration, pH, fiber concentration, and temperature on acid system viscosity. Then, we established a mathematical model of changes in diverting acid viscosity under a multi-factor cooperative control mechanism. On the basis of the kriging method, we established a three-dimensional (3D) geological model involving a random normal distribution and spatial correlation of multi-fracture and pore-permeability properties. We used four models (acid rock reaction rate, viscosity change, 3D acid wormhole expansion, and fluid–solid coupling) of a complex system to study dynamic cooperation characterization of diverting acid and a solid diverting agent under multiple factors. Simulation results show that the temporary plugging of acid and expansion of acid wormholes are mutually restricted. The solid diverting agent blocked the fracture, and a dense filter cake formed at the start of the fracture; thus, the physical flow direction of diverting acid changed, the acid wormhole length increased, and filtration of diverting acid declined to improve the acid's effect. Diverting acid and solid diverting agent work more effectively together. This paper is novel because we consider the respective influences of Ca2+ concentration, pH, flow rate, diverting acid rheological properties, injection parameters, and solid diverting agent concentration on the synergistic steering of steering acid and a solid diverting agent. We then establish a mathematical model to reflect complex stratigraphic conditions and objectively describe the acid flow reaction. We also innovatively solve the problem of predicting acid wormhole expansion given complex fractures and uneven pore distribution. Findings provide a theoretical basis and technical support for the application of acid fracturing in complex carbonate reservoirs.