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Abstract
Acid fracturing in tight formations is a representative example of reactive flow coupled with rock deformation and fracture growth, and it remains a widely used stimulation technique[1]. A variety of numerical approaches have been applied to study the dynamic fracture propagation process, including the finite element method (FEM)[2], boundary element method (BEM)[3], discrete element method (DEM)[4], and peridynamics (PD)[5]. Among them, the displacement discontinuity method (DDM), an indirect boundary element formulation, reduces the dimensionality of the problem and provides higher accuracy in estimating fracture apertures and stress fields, while retaining a clear physical meaning for fracture opening. Consequently, DDM has been widely used to simulate complex fracture propagation problems in stimulation processes, and it has supported the development of both commercial and in-house simulators, such as ResFrac[6] and PyFrac[7].
However, in many existing DDM-based simulators, fracture propagation models are largely restricted to 2D[8], planar 3D[9], or pseudo-3D[10] settings. Only a limited number of models can simulate fully 3D, nonplanar propagation of arbitrarily oriented fractures[11]. In addition, a series of studies by Detournay and co-workers[12,13] rigorously established that hydraulic fracture growth exhibits three propagation regimes: toughness-dominated, viscosity-dominated, and transient (mixed). Nevertheless, most existing propagation algorithms can reliably capture only toughness-dominated behavior. Although implicit level-set-based algorithms[14] can model propagation across regimes with high accuracy, extending them to nonplanar growth remains challenging. Motivated by these limitations, this work develops a fracture propagation model capable of simulating fully 3D, nonplanar growth under arbitrary propagation regimes.
In the proposed framework, a generalized 3D-DDM is used to compute fracture aperture, and a finite-volume method (FVM) is employed to solve the in-fracture fluid flow. Fracture growth is governed by the maximum principal stress criterion and an equivalent stress intensity factor, allowing the deflection and twisting angles to be determined. A Paris-type law is adopted to compute the propagation increment. In addition, reaction-controlled dissolution is incorporated to evaluate the irreversible chemical aperture. In the toughness-dominated regime, the global fluid volume balance equation replaces the local mass conservation equation to simplify the computation. In the viscosity-dominated and mixed regimes, the local fluid mass conservation equation is solved in a coupled manner with the nonlocal elastic integral equations. The overall solution workflow is illustrated in Fig. 1. Verification against analytical solutions for a penny-shaped fracture shows excellent agreement in both toughness- and viscosity-dominated regimes.
Finally, the model is applied to simulate viscosity-dominated acid fracturing for a single fracture. The results indicate that the acid-rock reaction rate strongly controls the etching pattern: a higher reaction rate localizes dissolution near the inlet and shortens the penetration depth, whereas a lower reaction rate enlarges the etched zone but promotes acid accumulation behind the tip.
| Country | China |
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