**8. Constrained 3D grid-based planar hydraulic fracturing simulator**

 Various hydraulic fracturing design scenarios can be analyzed starting from the geometric and engineered designs extracted in Section 3. The 3D reservoir models generated in Section 4 are fed in the hydraulic fracturing design model along with the various stress gradients in the lateral and vertical direction extracted from the differential stress model in Section 6. In the presence of stress gradients created by natural fractures, variable geomechanical properties and depleted reservoirs, the characteristics of the formation would be different on either side of the wellbore. This asymmetry and its correct quantification are important for an optimal well spacing of unconventional reservoirs. Fischer et al. [28] proposed a hydraulic fracture model (Eq. (5)) that explains the relative length change in two opposite wings of the hydraulic fracture.

$$a\_1(\mathbf{t}) = \frac{p\_0^{\text{net}}}{\mathcal{g}} \left( \mathbf{1} - \sqrt{\mathbf{1} - \mathbf{2} \left( \frac{\mathbf{g}}{p\_0^{\text{net}}} \right) a\_2(\mathbf{t}) - \left( \frac{\mathbf{g}}{p\_0^{\text{net}}} \right)^2 a\_2(\mathbf{t})^2} \right) \tag{5}$$

 The model is based on the lateral change in stress that would result in preferential growth of the hydraulic fracture in the direction of the decreasing confining stress. However, the model only explains the relativity of the asymmetric behavior in presence of the stress gradient (g) and does not specify the absolute length or width of the fracture. The relationship between the shorter wing a1 (t) and longer wing a2 (t) is given by Eq. (5) where *p0 net* is the initial net pressure.

 Using this concept, a 3D grid based planar hydraulic fracture model that includes proppant transport is developed. The unique feature of the hydraulic fracturing model is that it combines both analytical and numerical formulations. The effects of stress field change on the relative growth of the fracture is estimated by iterating for the optimum fracture height based on the amount of proppant available. The ability to develop a semi-analytical asymmetric fracture model that solves for the optimum fracture height and lengths is made possible by using the constraints of the geomechanical half lengths derived from the strain map and the estimated asymmetric fracture height derived from the ADaM geomechanical simulation.

The net pressure which is the difference between the fluid pressure and the minimum horizontal stress or the closure stress determines the initiation and propagation of a hydraulic fracture. The effect of stress gradient, along the fracture length is incorporated in the fundamental pressure balance equation at the fracture

**Figure 13.** 

*(A) Pressure match at a stage and resulting (B) complex fracture geometry and conductivity along the wellbore with major lateral and vertical variations due to the variable nature of the rock properties captured by the surface drilling data along the well and at a single stage (C).* 

tip which determines the growth of the fracture. The fluid flow in the fracture is computed numerically. Using the relation of velocity of the fracture fluid and the fracture length, a time dependent solution is achieved which forms the basis of the semi-analytical model. All the rock properties and stresses are input in the hydraulic fracture simulator as 3D models as shown in **Figure 4**.

Using all these constraints as inputs in the 3D planar hydraulic fracture simulator, the pressure monitored during the actual fracturing treatment is easily matched (**Figure 13A**) by altering only the pipe and perf friction and the leak off coefficient which depends on the input porosity or natural fracture model. The resulting frac geometry at one stage (**Figure 13C**) or along the entire wellbore (**Figure 13B**) shows the major lateral and vertical dimension and conductivity variations owing to the variable nature of the rock properties captured by the surface drilling data. With this result at each well, we have all what is needed for the reservoir simulation which is needed to compute the Estimated Ultimate Recovery (EUR) and the resulting asymmetric depletion, a fundamental information needed for the planning of future wells.
