### Production rate of a vertical well

According to mass conservation, for planar radial flow

$$\frac{\partial (\rho \phi )}{\partial t} + \nabla (\rho v) = 0$$

(1)

where \(\rho\) is gas density, v indicates gas velocity, and \(\phi\) indicates porosity.

Assuming that gas flow rate follows Darcy’s Law,

$$v = - \frac{k}{\mu }\nabla p$$

(2)

where p indicates pressure, k permeability, and \(\mu\) viscosity.

Gas density is dependent on pressure and temperature as follows:

$$\frac{\rho }{{\rho_{sc} }} = \frac{{pz_{sc} T_{sc} }}{{p_{sc} zT}}$$

(3)

where sc is short for stand condition, and z is compressibility coefficient, function of pressure and temperature.

Integrating Eqs. (2) and (3) into (1),

$$\frac{k}{\phi }\nabla \cdot \left( {\frac{p}{\mu Z}\nabla p} \right) = \frac{\partial }{\partial t}\left( {\frac{p}{Z}} \right)$$

(4)

The right hand of Eq. (4) could be expanded as follows:

$$\frac{\partial }{\partial t}\left( {\frac{p}{Z}} \right) = \frac{p}{Z}\frac{\partial p}{\partial t}\left( {\frac{1}{p} - \frac{1}{Z}\frac{\partial Z}{\partial p}} \right)$$

(5)

According to definition of volume compressibility coefficient and combing with Eq. (3),

$$c_{g} = \frac{1}{\rho }\left( {\frac{\partial \rho }{\partial p}} \right) = \frac{1}{p} - \frac{1}{Z}\frac{\partial Z}{\partial p}$$

(6)

Substituting Eq. (6) into Eq. (5),

$$\frac{\partial }{\partial t}\left( {\frac{p}{Z}} \right) = c_{g} \frac{p}{Z}\frac{\partial p}{\partial t}$$

(7)

Introducing pseudo-pressure,

$$m(p) = 2\int_{{p_{m} }}^{p} {\frac{p}{\mu Z}} {\text{d}}p$$

(8)

where \(p_{m}\) is reference pressure with value of 0 or 0.1 MPa.

According to Eq. (8),

$$\nabla m = \frac{2p}{\mu Z}\nabla p$$

(9)

$$\frac{\partial m}{\partial t} = \frac{2p}{\mu Z}\frac{\partial p}{\partial t}$$

(10)

Substituting Eq. (9) into the left hand of Eq. (4) and Eq. (10) into Eq. (7),

$$\frac{k}{\phi }\nabla \cdot \nabla m = c_{g} \mu \frac{\partial m}{\partial t}$$

(11)

Then

$$\nabla^{2} m = \frac{{c_{g} \mu \phi }}{k}\frac{\partial m}{\partial t}$$

(12)

For linearization of Eq. (12), assume that \(\mu \phi\) equals approximately to \(\overline{\mu \phi }\) which is value under average formation pressure \(\bar{p}\).

For steady flow, constant pressure boundary conditions were adopted \(\nabla^{2} m = 0\). And in the polar coordinate system, it can be derived by:

$$\frac{{d^{2} m}}{{dr^{2} }} + \frac{1}{r}\frac{dm}{dr} = 0$$

$$\begin{aligned} m = m_{e} ,r = r_{e} \hfill \\ m = m_{w} ,r = r_{w} \hfill \\ \end{aligned}$$

(13)

where the *r*
_{e} is the drainage radius of the elliptical flow, *r*
_{w} is the radius of the well. And *m*
_{e} is the pseudo-pressure on the boundary of elliptical flow, *m*
_{w} is the pseudo-pressure of the well.

Assume that *k* can be described by \(k_{{\rm max} }\), \(k_{{\rm min} }\) under the polar coordinates as follows:

$$k(\varphi ) = k_{{\rm max} } - (k_{{\rm max} } - k_{{\rm min} } )\left| {\sin (\varphi )} \right|$$

(14)

where \(k_{{\rm max} }\) and \(k_{{\rm min} }\) are maximum and minimum sand permeability oriented in *x* and *y* directions, as shown in Fig. 1.

Solving Eq. (13),

$$m - m_{w} = \frac{{m_{e} - m_{w} }}{{ln\left( {\frac{{r_{e} }}{{r_{w} }}} \right)}}ln\frac{r}{{r_{w} }}$$

(15)

Integrating Eqs. (2), (14), and (15), volumetric flow rate under standard conditions could be obtained:

$$Q_{sc} = \frac{{4\pi z_{sc} T_{sc} h\left[ {k_{{\rm max} } + (\frac{\pi }{2} - 1)k_{{\rm min}}} \right]\left( {p_{e}^{2} - p_{w}^{2} } \right)}}{{p_{sc} zT\mu \ln \frac{{r_{e} }}{{r_{w} }}}}$$

(16)

### Production rate of a fractured vertical well

In the presence of artificial fracture, the radial flow will be replaced by elliptical flow. With the fracture direction as *x* axis direction, as shown in Fig. 1, Cartesian coordinates, (*x*′, *y*′) can be transformed into elliptical coordinates, using the following relationship:

$$\left\{ \begin{array}{l} x = L\cosh \xi \cos \eta \hfill \\ y = L\sinh \xi \sin \eta \hfill \\ \end{array} \right.$$

(17)

where \(\xi\) and \(\eta\) separately represents a family of confocal ellipses and a family of confocal hyperbolas with 2*L* (length of fracture) as focal length (Lou et al. 2013). Assume that the production rate in the elliptical area of the fractured vertical well will follow Darcy’s law:

$$dQ_{sc} = \frac{{pz_{sc} T_{sc} }}{{p_{sc} zT}}dA\frac{k}{\mu }\frac{dp}{{d\bar{r}}}$$

(18)

Using Eq. (17) dA can be obtained under elliptical coordinate:

$$dA = hrd\theta = h\sqrt {x^{2} + y^{2} } d\theta = hL\cosh \xi \cos \eta \sqrt {1 + (\tanh \xi \tan \eta )^{2} } d\theta$$

(19)

Applying the relationship between central angle *θ* and eccentric angle \(\eta\) of an ellipse:

$$\tan \eta = \coth \xi \tan \theta$$

(20)

dA can be further derived as a function of \(\xi\) and *θ*

$$dA = hL\cosh \xi \cos \eta \sqrt {1 + (\tan \theta )^{2} } d\theta = \frac{{hL\cosh \xi \sqrt {1 + (\tan \theta )^{2} } }}{{\sqrt {1 + (\coth \xi \tan \theta )^{2} } }}d\theta$$

(21)

Incorporating \(\varphi = \theta + \alpha\) into Eq. (14), we can obtain

$$k(\theta ) = k_{{\rm max} } - (k_{{\rm max} } - k_{{\rm min} } )\left| {\sin (\theta + \alpha )} \right|$$

(22)

Since the elliptical flow is axial symmetrical, assume that the pressure distribution in \(\xi\) does not change with \(\eta\). Under steady state, the pressure obeys Laplace’s equation in \(\xi\) and \(\eta\) plane. Therefore the pressure distribution can be described as:

The constants *a* and *b* in the equation are to be determined from boundary conditions

$$\left\{ \begin{array}{l} p = p_{w} \, \,at \, \,\xi = 0\,\left( {{\text{the}}\,{\text{fracture}}} \right), \\ p = p_{e} \, \, at \,\, \xi = \xi_{e} \left( {{\text{outer}}\,{\text{boundary}}\,{\text{of}}\,{\text{the}}\,{\text{drainage}}\,{\text{area}}} \right) \\ \end{array} \right.$$

(24)

\({\kern 1pt} {\kern 1pt} \xi_{e} {\kern 1pt}\) is related to drainage radius \(r_{e}\) by:

$$\xi_{e} = \ln \frac{{2r_{e} }}{L}$$

(25)

Since *p* is only a function of \(\xi\), we introduce a modified variable \(\bar{r}\) in Darcy’s law, as shown in Eq. (20), where \(\bar{r}\) is defined as:

$$\bar{r}(\xi ) = \frac{1}{2}L(\cosh \xi + \sinh \xi ) = L \cdot \frac{{e^{\xi } }}{2}$$

(26)

Therefore,

$$d\bar{r} = \frac{{d\bar{r}}}{d\xi }d\xi = L \cdot \frac{{e^{\xi } }}{2}d\xi$$

(27)

Incorporating Eqs. (23), (24), (25), (27) into Eq. (20) will derive:

$$dQ_{sc} = \frac{{z_{sc} T_{sc} h}}{{p_{sc} zT}} \cdot \left( {\frac{{p_{e} - p_{w} }}{{\xi_{e} }} \cdot \xi + p_{w} } \right) \cdot \frac{2}{{e^{\xi } }} \cdot \frac{{\cosh \xi \sqrt {1 + (\tan \theta )^{2} } }}{{\sqrt {1 + (\coth \xi \tan \theta )^{2} } }} \cdot \frac{k(\theta )}{\mu }d\theta \cdot \frac{dp}{d\xi }$$

(28)

In steady state, the changes of \(Q\) with \(\xi\) is negligible, thus we can assume \(\xi\) as constant when deriving the final expression of \(Q_{sc}\):

$$\begin{aligned} Q_{sc} = \frac{{z_{sc} T_{sc} h}}{{p_{sc} zT\mu }} \cdot \left(\frac{{p_{e} - p_{w} }}{{\xi_{e} }} \cdot \xi + p_{w} \right) \cdot \frac{2}{{e^{\xi } }} \cdot \left(\frac{{p_{e} - p_{w} }}{{\xi_{e}}}\right) \cdot \cosh \xi \cdot A \hfill \\ where\,A = \int_{0}^{2\pi } {\frac{{\cosh \xi \sqrt {1 + (\tan \theta )^{2} } }}{{\sqrt {1 + (\coth \xi \tan \theta )^{2} } }} \cdot (} k_{1} - (k_{1} - k_{2} )\left| {\sin (\theta + \alpha )} \right|)d\theta \hfill \\ \end{aligned}$$

(29)

### Calculation processes

According to the established productivity equations of vertical wells and vertical fractured wells in channel sand tight gas reservoir, the calculation process is carried out using software Matlab. For fractured vertical wells, the calculation process is shown in Fig. 2.

Based on the known parameters, permeability distribution should be first derived by Eq. (22), different formation permeability ratio and fracture direction will directly influence the distribution of permeability. Pressure distribution can be obtained by Eqs. (23) and (24), the values of drainage area and fracture half-length will influence the parameter of Eq. (23), and thus influence the initial pressure distribution. With permeability and pressure distribution being settled, a loop computing of production rate, following Eq. (29), is introduced to obtain the production rate at each pressure drawdown. Finally, the data are output and drawn as diagrams.