**2.3 Calculation of fracture conductivity**

CFD-CFX is used to calculate pressure drop and velocity across the inlets as shown in **Figure 3**. The finite volume method is adopted to solve the threedimensional Navier–Stokes equations. Consistent with the experimental conditions for conductivity measurements, the flow was the steady state at 25°C. The continuity and momentum balance equations for the steady-state flow are shown below.

Continuity equation

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \tag{14}$$

Momentum equations

$$
\rho \left( u \frac{\partial u}{\partial \mathbf{x}} + v \frac{\partial u}{\partial y} w \frac{\partial u}{\partial z} \right) = \frac{\partial P}{\partial \mathbf{x}} + \mu \left( \frac{\partial^2 u}{\partial \mathbf{x}^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) \tag{15}
$$

$$
\rho \left( u \frac{\partial v}{\partial \mathbf{x}} + v \frac{\partial v}{\partial \mathbf{y}} w \frac{\partial v}{\partial \mathbf{z}} \right) = \frac{\partial P}{\partial \mathbf{y}} + \mu \left( \frac{\partial^2 v}{\partial \mathbf{x}^2} + \frac{\partial^2 v}{\partial \mathbf{y}^2} + \frac{\partial^2 v}{\partial \mathbf{z}^2} \right) \tag{16}
$$

$$
\rho \left( u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y} w \frac{\partial w}{\partial z} \right) = \frac{\partial P}{\partial z} + \mu \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2} \right) \tag{17}
$$

where *x*, *y*, *z* are the dimensions; *u*, *v*, *w* are the velocity directions in *x*, *y*, *z* directions; *p* is pressure inside fracture. Fracture permeability was determined according to Darcy's law provided in Eq. (18).

$$K = \frac{Q \,\mu L}{A \Delta P} \tag{18}$$

where *K* is the permeability, *Q* is the flow rate of the injected fracturing fluid, *μ* is the viscosity, *L* is the length of the fracture around the proppant, A is the crosssectional area of the fracture zone, and *ΔP* is the differential pressure between inlet and outlet across the proppant. The conductivity of hydraulic fracture is generally

defined as the maximum ability of the fracture to transmit a reservoir fluid through it. The conductivity is measured in μm<sup>2</sup> .cm based on the propped fracture width (cm) and permeability (μm<sup>2</sup> ).

$$\text{Fracture conductivity} = Kf \ast Wf \tag{19}$$

where *Kf* is the proppant permeability, μm2 and *Wf* is the fracture width, cm.
