There are applications of hydraulic fracture models, such as real-time fracture evaluation, for which the computational speed is a key selection criterion. To pursue the goal of creating relatively fast and yet accurate fracture propagation solvers, we use the Lagrangian mechanics formulation and obtain a system of a few first-order ordinary differential equations, which are solved relatively fast. The number of equations is determined by the number of generalized coordinates chosen by a modeler depending on the desired complexity of the solution. As an example, we build the Lagrangian mechanics model for a two-dimensional fracture propagation with asymmetric growth in a permeable formation with finite toughness. Verification of the model against exact solutions of the problem in the limiting cases shows how accurately we can solve the problem. Using the model, we study sensitivity of fracture growth asymmetry to non-uniformity of confining rock stress in horizontal direction, Young's modulus, fracture toughness, leakoff coefficient, fracture design parameters such as pumping rate, fracturing fluid viscosity, and fracture height.
For more than 50 years, hydraulic fracturing has been the most effective and popular method of oil and gas reservoir stimulation to recover hydrocarbons at economic rates (King, 2012; Vincent, 2012). Initiation and geometry of fracture propagation at typical reservoir depths can hardly be monitored with desired accuracy and often raises many questions (Mahrer et al., 1996). Nowadays, the only practical way to envision and understand the hydraulic fracturing process in subterranean rock is by modeling matched by bottomhole pressure records (Economides and Nolte, 2000; Nolte, 1979).
Today, there are many developed numerical models of hydraulic fracturing (Adachi et al., 2010; Dontsov and Peirce, 2017, 2015; Rahman and Rahman, 2010). As a rule, most of them require introducing a representative spatial mesh of the fracture surface (Adachi et al., 2007). Consequently, the time-dependent problem must be solved at every point or cell of the introduced spatial mesh. This is the evident reason for noticeable consumption of the computational time depending on the grid size. This is justified for the models making their intentional focus on fracture geometry complexity due to either layered rock property contrasts or interaction with pre-existing rock discontinuities (Sesetty and Ghassemi, 2017; Gao and Ghassemi, 2019). However, often for the needs of fast estimation of fracture extention in length and height simpler models of independent planar fracture propagation are sufficient.