# Modeling Hydraulic Fracture Entering Stress Barrier: Theory and Practical Recommendations
## 2 Theory of Fracture Penetration into Stress Barrier
### General Restrictions of Time Stepping Schemes aggravated by Stress Barrier  
_Explicit Euler scheme_: The general computational difficulties of modeling hydraulic fractures are due to the inherent mathematical difficulty: after spatial discretization, the resulting system of ordinary differential equations (ODEs) is _extremely stiff_[47; 48]. Its Courant time _cou_ (e.g., [49]) is usually very small. The fundamental estimation for the _cou_ is obtained in [47]. They studied the spectrum of the ODE main matrix for the plane-strain KGD problem with fixing the _average_ value \(w\) of the opening at the central part of an HF:  
\[cou=t_{n}\bigg{(}\frac{\Delta x}{w}\bigg{)}^{3} \tag{9}\]  
where  
\[t_{n}=\bigg{(}\frac{\mu^{\prime}}{E^{\prime}}\bigg{)}^{1/n} \tag{10}\]  
The Courant time _cou_ defines the maximal admissible time step \(\Delta t_{E}\) for _explicit Euler_ integration (e.g., [50]). According to (9),  
\[\Delta t_{E}=cou=t_{n}\bigg{(}\frac{\Delta x}{w}\bigg{)}^{3} \tag{11}\]  
Detailed calculations using the explicit Euler scheme for the case of the benchmark plane-strain KGD problem provided the data to check the accuracy of (11) ([34], p. 16, 28, Figure 4). It appears that (11) perfectly agrees with the bounds of stability found in numerical experiments. Specifically, the numerical results reproduce the cubic dependence of _cou_ on the ratio \(\frac{\Delta x}{w}\) and the linear dependence on the intrinsic time \(t_{n}\), defined in (10). The factor (actually, 1.00) in front of \(t_{n}\) also complies with the numerical value to the accuracy of 1%. A non-linear system of ODE, arising after spatial discretization of the system (1)-(3), involves a non-symmetric, fully populated square matrix. In 3D problems, its order \(N_{mat}\) often exceeds first hundreds. This implies that, in each time step of numerical integration of the ODE, the corresponding algebraic system is to be solved iteratively with performing _matrix-to-vector_ multiplications. The time-costly operation of matrix-to-vector multiplication, which requires \(N_{mat}^{2}\)_arithmetic_ multiplications, corresponds to the work unit (_WUI_) of numerical modeling. For the _explicit Euler scheme_, each time step consists of a _single WUI_. This enables us to estimate easily the total number of work units (\(NWUI\)), computational complexity, and the computer time expense for tracing fracture propagation in a typical time interval \(T\) of a hydraulic treatment. Specifically, in view of (11), for the explicit Euler scheme, the number of work units is as follows:  
\[NWUI_{E}=\frac{T}{\Delta t_{E}}=\frac{T}{t_{n}}\Big{(}\frac{w}{\Delta x}\Big{)} ^{3} \tag{12}\]  
With the known \(NWUI\), computational complexity \(C\) and the time expense \(T_{C}\) for tracing HF become known as well: \(C=N_{mat}^{2}NWUI\), \(T_{C}=\Delta t_{a}C\), where \(\Delta t_{a}\) is the processor time for a single arithmetic multiplication with an assigned precision (for a conventional laptop, \(\Delta t_{a}\approx 0.5\cdot 10^{-8}\) s for ordinary, and \(\Delta t_{a}\approx 1.0\cdot 10^{-8}\) s for double precision arithmetic). We may exemplify these estimations by taking typical values: \(t_{n}=10^{-11}\) s, \(\Delta x=2.5\) m, \(w=2\cdot 10^{-3}\) m, \(T=3600\) s, \(N_{mat}=200\), and \(\Delta t_{a}=10^{-8}\) s. Then, \(cou=\Delta t_{E}=0.020\) s, \(NWUI_{E}=184,320\), \(C_{E}=7.4\cdot 10^{9}\), and \(T_{CE}=74\) s. This time is acceptable for numerical modeling with explicit Euler stepping. However, for modeling _propagation through a barrier_, the need for finer grids arises (e.g., [47]). When taking an order finer grid and considering an order thinner fracturing fluid, the Courant time _cou_ becomes four orders lower (\(cou=2\cdot 10^{-6}\) s), while the number of work units \(NWUI_{E}\), complexity \(C_{E}\), and time expense \(T_{CE}\) become four orders higher. Thus, the time cost \(T_{CE}=7.4{\cdot}10^{5}\) s for calculations on a conventional laptop reaches hundreds of hours. The explicit Euler time stepping becomes prohibitively expensive.