Forecasting of Horizontal Gas Well Production Decline in Unconventional Reservoirs using Productivity, Soft Computing and Swarm Intelligence Models

Abstract

The most widely used production decline forecasting tools are numerical reservoir simulation, material balance estimates and advanced methods of production decline analysis. Besides, these existing production decline evaluation techniques for unconventional reserves estimations have underlying limitations and assumptions incorporated into their formulation which can result into under- and overestimation. This further raises the debate about which decline curve analysis (DCA) is better than the other for unconventional reservoirs predictions. Currently, data-driven artificial neural network (ANN) has emerged as a new paradigm capable of mapping complex functional relationships. In this present study, ANN technology was used to calibrate tight gas carbonate field historical declining trends which exhibit high early peak production rate and quick decline. Hereafter, making reliable predictions posed as a challenge for the complex target field and hard computing protocols. Therefore, this research applied and tested the capability of backpropagation artificial neural network (BPANN), radial basis function neural network (RBNN) and generalized regression neural network (GRNN) as DCA techniques for predicting the historical production decline trends of an ultra-low porosity and permeability tight gas reservoir. The optimum trained ANN DCA models developed were validated with another surrounding well’s dataset, producing acceptable results in agreement with the actual field data. The GRNN DCA model’s testing performance was poor, and its smoothing parameter was optimized with particle swarm optimization (PSO) algorithm which gave satisfactory results comparable to standalone BPANN and RBFNN DCA models. Furthermore, the optimum ANN DCA models’ generalization strength across the entire field dataset revealed that the developed models’ predictions were robust as compared to the data-driven Arps hyperbolic and power law exponential DCA models. This was evident from the statistical performance criteria employed which indicated that BPANN, RBFNN and PSO-GRNN DCA models are plausible better fit models for matching the target field’s historical decline performances. Also, a novel non-Darcy flow horizontal well productivity evaluation model for the target field was developed based on stress sensitivity coefficient (SSC) of permeability, tortuosity factor, Klinkenberg effect, near-wellbore turbulence effect and threshold pressure gradient (TPG) for validating the ANN DCA models predictions. The productivity model was validated with published horizontal well model with closely matched results. For inflow performances, the horizontal well model with turbulence minimizes negative effects on non-Darcy flow rates than without turbulence. Additionally, pressure drawdown influences the tight gas well productivity such that the lower the pressure drawdown, the smaller the tight gas well productivity. The operating points of the tight gas well were determined through inflow performance relation and tubing performance relation at different SSC and TPG for $3{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}$ in tubing size. In another case, synthetic unconventional simulation data with long production history were used for future forecast of 5, 24 and 70 months of production rate and cumulative production which gave pretty close results for all the DCA models, unlike the real field datasets, where the empirical rate-time models under and overestimate. In all, these ANN DCA models and the horizontal well productivity model derived will serve as new computational tools for complementing existing DCA techniques for better understanding of unconventional reservoirs’ production decline performance.

Appendices

Appendix 1: Horizontal Well Performance Model for the Tight Gas Carbonate Field

A horizontal well performance model was derived by considering a box-shaped rectangular reservoir based on SSC of permeability, near-wellbore turbulence effect, tortuosity factor, Klinkenberg effect and TPG. The factors mentioned above have not been comprehensively unified and studied for horizontal wells with ultra-low porosity and permeability formations. In Figure 23 (hydraulically fractured horizontal well), the following assumptions were made to derive a tractable mathematical model of the physical system:

Figure 23

Physical model of the multi-fractured horizontal well in a closed rectangular tight gas reservoir

• 1. A radial flow regime into a centered horizontal wellbore with a closed outer boundary.

• 2. The reservoir has a uniform thickness with initial reservoir pressure equal everywhere.

• 3. Single phase gas fluid flow under isothermal and non-Darcy flow condition.

• 4. Negligible gravity, capillary force and wellbore storage effects.

Forchheimer (1901) modified Darcy’s law to account for near-wellbore turbulence effects as:

$$ \frac{{{\text{d}}p}}{{{\text{d}}r}} - \lambda = \frac{{\mu_{\text{g}} }}{{k_{\text{g}} }}\upsilon + \beta_{\text{g}} \rho_{\text{g}} \upsilon^{2} , $$

(39)

where p is the gas reservoir pressure, $\lambda$ is the threshold pressure gradient, $\mu_{\text{g}}$ is the gas viscosity, r is the radial distance, $k_{\text{g}}$ is the permeability of gas under stress condition with slippage effects, $\upsilon$ is the gas velocity, $\rho_{\text{g}}$ is the density of the gas, $\beta_{\text{g}}$ is the turbulence coefficient.

The term $\beta_{\text{g}}$ in Eq. 39 accounts for additional wellbore turbulence pressure drop due to high gas velocities in the near-wellbore region especially in high permeability gas well reservoirs. Turbulence can also reduce the net effect on gas production.

In Eq. 39, the gas seepage velocity (superficial velocity) obtained during radial flow pattern in a horizontal well is given as:

$$ \upsilon = \frac{q}{A} = \frac{1}{r}\frac{{qB_{\text{g}} }}{{2\pi L_{p} }} = \frac{1}{r}\frac{q}{{2\pi L_{p} }}\frac{{p_{\text{sc}} ZT}}{{pZ_{\text{sc}} T_{\text{sc}} }}, $$

(40)

where q is the total gas flow rate into the horizontal wellbore, L_p is the horizontal well producing length (horizontal wellbore length), B_g = gas formation volume factor, $p_{\text{sc}}$ is the standard atmospheric pressure, Z is the gas deviation factor, $Z_{\text{sc}}$ is the gas deviation factor at standard condition, $T_{\text{sc}}$ is the reservoir temperature at standard condition, T is the reservoir temperature, and A is the drainage area.

The gas density and non-Darcy flow coefficient with tortuosity effect (Liu et al. 1995) in porous media are expressed, respectively, as:

$$ \rho_{\text{g}} = \frac{{M_{\text{air}} \gamma_{\text{g}} p}}{ZRT}, $$

(41)

$$ \beta_{\text{g}} = \frac{{1.15 \times 10^{9} \times \tau }}{{k_{\text{g}} \varphi }}, $$

(42)

where $\gamma_{g}$ is the relative density of a natural gas, R is the gas constant, $\varphi$ is the gas reservoir porosity $\tau$ is the tortuosity factor and M_air is the molecular mass of air (M_air = 28.97).

By taking gas slippage effect into account with formation permeability not constant and depends on pressure, then the actual gas permeability is given by Klinkenberg (1941) expressed as nonlinear relationship between absolute permeability and pore pressure as:

$$ k_{\text{g}} = k_{i} \left( {1 + \frac{{b_{k} }}{p}} \right), $$

(43)

where k_g is the apparent gas permeability, k_i is the rock permeability, p is the reservoir pressure, and b_k is the Klinkenberg slippage factor (empirical parameter).

The empirical relationship for Klinkenberg slippage factor by Jones and Owen (1980) for tight gas plays is given as:

$$ b_{k} = 12.639\left( {k_{i} } \right)^{ - 0.33} $$

(44)

The obtained mathematical relations from Eqs. 40, 41, 42 and 43 are substituted into Eq. 39, which yields:

$$ \frac{{{\text{d}}p}}{{{\text{d}}r}} - \lambda = \frac{{Tp_{\text{sc}} \mu_{\text{g}} Zq}}{{2\pi L_{p} T_{\text{sc}} Z_{\text{sc}} }}\frac{1}{{pk_{\text{g}} }}\frac{1}{r} + \frac{{3.33 \times 10^{10} \gamma_{\text{g}} \tau TZp_{\text{sc}}^{2} q^{2} }}{{4\pi^{2} L_{p}^{2} T_{\text{sc}}^{2} Z_{\text{sc}}^{2} R\varphi }}\frac{1}{{pk_{\text{g}} }}\frac{1}{{r^{2} }}. $$

(45)

Assuming external pressure (p_e) at the outer boundary at r_e in Figure 24 is equal to the initial reservoir pressure (p_i), since it takes long time for pseudosteady state to be established in ultra-low porosity and permeability tight gas reservoir. Hence, integrating both sides of Eq. 45:

$$ \int_{{p_{wf} }}^{{p_{i} }} {pk_{\text{g}} {\text{d}}p - \int_{{r_{w} }}^{{r_{e} }} {\lambda pk_{\text{g}} dr = } } \frac{{Tp_{\text{sc}} \mu_{\text{g}} Zq}}{{2\pi L_{p} T_{\text{sc}} Z_{\text{sc}} }}\int_{{r_{w} }}^{{r_{e} }} {\frac{1}{r}{\text{d}}r + \frac{{3.33 \times 10^{10} \gamma_{\text{g}} \tau TZp_{\text{sc}}^{2} q^{2} }}{{4\pi^{2} L_{p}^{2} T_{\text{sc}}^{2} Z_{\text{sc}}^{2} R\varphi }}\int_{{r_{w} }}^{{r_{e} }} {\frac{1}{{r^{2} }}} } {\text{d}}r, $$

(46)

where p_wf is the bottomhole well flowing pressure, p_i is the initial reservoir pressure, r_e is the horizontal well drainage radius, and r_w is the horizontal wellbore radius.

Figure 24

Plane view of a radial flow pattern

For horizontal wells, Joshi (1991) introduced the concept of effective wellbore radius ($r^{\prime}_{\text{w}}$) expressed as:

$$ r^{\prime}_{w} = \frac{{r_{{{\text{e}}h}} \left( {\frac{{L_{p} }}{2}} \right)}}{{a\left[ {1 + \sqrt {1 - \left( {\frac{{L_{p} }}{2a}} \right)^{2} } } \right]\left[ {\frac{h}{{\left( {2r_{\text{w}} } \right)}}} \right]^{{\frac{h}{{L_{p} }}}} }}, $$

(47)

For an isotropic reservoir, $k_{h} = k_{v} = k$; but, for anisotropic reservoir, the effective wellbore radius is given as:

$$ r^{\prime}_{w} = \frac{{r_{eh} \left( {\frac{{L_{p} }}{2}} \right)}}{{a\left[ {1 + \sqrt {1 - \left( {\frac{{L_{p} }}{2a}} \right)^{2} } } \right]\left[ {\frac{\beta h}{{\left( {2r_{w} } \right)}}} \right]^{{\frac{\beta h}{{L_{p} }}}} }}, $$

(48)

where L_p is the length of the horizontal well (major axis), h is the reservoir formation thickness (minor axis), anisotropic permeability factor $\beta = \sqrt {{\raise0.7ex\hbox{${k_{\text{h}} }$} \!\mathord{\left/ {\vphantom {{k_{\text{h}} } {k_{\text{v}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${k_{\text{v}} }$}}}$, effective permeability of the formation $k = \sqrt {k_{\text{h}} k_{\text{v}} }$, k_h is the average horizontal permeability, and k_v is the vertical permeability, r_eh is the radius of drainage area of a horizontal well $\left( {r_{eh} = \sqrt {\frac{A \times 43560}{\pi }} } \right)$, and A is the drainage area.

The a in Eqs. 47 and 48 is the half major axis of the elliptical drainage area given as:

$$ a = \left( {\frac{{L_{p} }}{2}} \right)\left[ {0.5 + \sqrt {0.25 + \left( {\frac{{2r_{{{\text{e}}h}} }}{{L_{p} }}} \right)^{4} } } \right]^{0.5} $$

(49)

The effective wellbore radius ($r^{\prime}_{\text{w}}$) is introduced into Eq. 46 which is rewritten as:

$$ \int_{{p_{wf} }}^{{p_{i} }} {pk_{\text{g}} {\text{d}}p - \int_{{r^{\prime}_{\text{w}} }}^{{r_{\text{e}} }} {\lambda pk_{\text{g}} {\text{d}}r = } } \frac{{Tp_{\text{sc}} \mu_{\text{g}} Zq}}{{2\pi L_{p} T_{\text{sc}} Z_{\text{sc}} }}\int_{{r^{\prime}_{\text{w}} }}^{{r_{\text{e}} }} {\frac{1}{r}{\text{d}}r + \frac{{3.33 \times 10^{10} \gamma_{\text{g}} \tau TZp_{\text{sc}}^{2} q^{2} }}{{4\pi^{2} L_{p}^{2} T_{\text{sc}}^{2} Z_{\text{sc}}^{2} R\varphi }}\int_{{r^{\prime}_{w} }}^{{r_{\text{e}} }} {\frac{1}{{r^{2} }}} } {\text{d}}r, $$

(50)

The relationship between SSC of permeability and effective stress is given as:

$$ k_{\text{g}} = k_{i} {\text{e}}^{{ - \int\limits_{{p_{{}} }}^{{p_{i} }} {\alpha_{k} {\text{d}}p} }} , $$

$$ k_{\text{g}} = k_{i} {\text{e}}^{{ - \alpha_{k} \left[ {\left( {p_{i} - p_{{}} } \right)} \right]}} , $$

(51)

where (p_i − p) is the effective stress, $k_{i}$ is the original rock permeability of gas, $\alpha_{k}$ is the stress sensitivity coefficient of permeability (permeability modulus), p is the reservoir pressure.

Introducing Eq. 51, which accounts for the permeability change during the tight gas reservoir production, into Eq. 50 will yield:

$$ \begin{aligned} \int_{{p_{wf} }}^{{p_{i} }} {pk_{i} {\text{e}}^{{ - \alpha_{k} \left( {p_{i} - p_{{}} } \right)}} {\text{d}}p - \int_{{r^{\prime}_{\text{w}} }}^{{r_{e} }} {\lambda pk_{i} {\text{e}}^{{ - \alpha_{k} \left( {p_{i} - p_{{}} } \right)}} {\text{d}}r = } } \frac{{Tp_{\text{sc}} \mu_{\text{g}} Zq}}{{2\pi L_{p} T_{\text{sc}} Z_{\text{sc}} }}\int_{{r^{\prime}_{\text{w}} }}^{{r_{e} }} {\frac{1}{r}{\text{d}}r + } \hfill \\ \frac{{3.33 \times 10^{10} \gamma_{\text{g}} \tau TZp_{\text{sc}}^{2} q^{2} }}{{4\pi^{2} L_{p}^{2} T_{\text{sc}}^{2} Z_{\text{sc}}^{2} R\varphi }}\int_{{r^{\prime}_{\text{w}} }}^{{r_{e} }} {\frac{1}{{r^{2} }}} {\text{d}}r \hfill \\ \end{aligned} $$

(52)

Dividing Eq. 52 by the original rock permeability of gas (k_i) produces:

$$ \begin{aligned} \int_{{p_{wf} }}^{{p_{i} }} {p{\text{e}}^{{ - \alpha_{k} \left( {p_{i} - p_{{}} } \right)}} {\text{d}}p - \int_{{r^{\prime}_{w} }}^{{r_{e} }} {\lambda p{\text{e}}^{{ - \alpha_{k} \left( {p_{i} - p_{{}} } \right)}} {\text{d}}r = } } \frac{{Tp_{\text{sc}} \mu_{\text{g}} Zq}}{{2\pi L_{p} T_{\text{sc}} Z_{\text{sc}} k_{i} }}\int_{{r^{\prime}_{\text{w}} }}^{{r_{e} }} {\frac{1}{r}{\text{d}}r + } \hfill \\ \frac{{3.33 \times 10^{10} \gamma_{\text{g}} \tau TZp_{\text{sc}}^{2} q^{2} }}{{4\pi^{2} L_{p}^{2} T_{\text{sc}}^{2} Z_{\text{sc}}^{2} R\varphi k_{i} }}\int_{{r^{\prime}_{\text{w}} }}^{{r_{e} }} {\frac{1}{{r^{2} }}} {\text{d}}r \hfill \\ \end{aligned} $$

(53)

Integration on Eq. 53 results into:

$$ \begin{aligned} \left[ {p_{i} \left( {\frac{1}{{ - \alpha_{k} }}} \right) - p_{wf} \left( {\frac{1}{{ - \alpha_{k} }}{\text{e}}^{{ - \alpha_{k} \left( {p_{i} - p_{wf} } \right)}} } \right) - \left( {\frac{1}{{\alpha_{k}^{2} }} - \frac{1}{{\alpha_{k}^{2} }}{\text{e}}^{{ - \alpha_{k} \left( {p_{i} - p_{wf} } \right)}} } \right)} \right] - \left[ {\lambda p{\text{e}}^{{ - \alpha_{k} \left( {p_{i} - p_{{}} } \right)}} \left( {r_{e} - r^{\prime}_{w} } \right)} \right] = \hfill \\ \frac{{Tp_{\text{sc}} \mu_{\text{g}} Zq}}{{2\pi L_{p} T_{\text{sc}} Z_{\text{sc}} k_{i} }}\ln \left( {\frac{{r_{e} }}{{r^{\prime}_{w} }}} \right) + \frac{{3.33 \times 10^{10} \gamma_{\text{g}} \tau TZp_{sc}^{2} q^{2} }}{{4\pi^{2} L_{p}^{2} T_{\text{sc}}^{2} Z_{\text{sc}}^{2} R\varphi k_{i} }}\left( {\frac{1}{{r^{\prime}_{w} }} - \frac{1}{{r_{e} }}} \right) \hfill \\ \end{aligned} $$

(54)

Since gas deviation factor and gas viscosity are strong functions of pressure, they were removed from the integral to simplify the final form of the horizontal well deliverability equation. Thus the average reservoir pressure ($\overline{p}$) of the gas well used to estimate gas deviation factor and average gas viscosity is specified as:

$$ \overline{p} = \sqrt {\frac{{\left( {p_{i}^{2} + p_{wf}^{2} } \right)}}{2}} $$

(55)

However, Eq. 55 will only be valid for applications where pressure is less than 2000 psi. The approximation method is called the pressure squared method. If pressure is greater than 3000 psi, then the gas properties are evaluated at an average pressure known as the pressure approximation method:

$$ \overline{p} = \frac{{\left( {p_{i} + p_{wf} } \right)}}{2} $$

(56)

Similarly, the m(p) (real gas pseudopressure) solution method (exact solution) is designated as:

$$ m(p) = \int_{0}^{p} {\frac{2p}{\mu z}} {\text{d}}p $$

(57)

From Eq. 54, bar symbol is introduced over gas deviation factor (Z) and gas viscosity $\left( {\mu_{g} } \right)$ to represent average values evaluated at an average pressure. Therefore, Eq. 54 is recast as:

$$ \begin{aligned} \left[ {p_{i} \left( {\frac{1}{{ - \alpha_{k} }}} \right) - p_{wf} \left( {\frac{1}{{ - \alpha_{k} }}{\text{e}}^{{ - \alpha_{k} \left( {p_{i} - p_{wf} } \right)}} } \right) - \left( {\frac{1}{{\alpha_{k}^{2} }} - \frac{1}{{\alpha_{k}^{2} }}{\text{e}}^{{ - \alpha_{k} \left( {p_{i} - p_{wf} } \right)}} } \right)} \right] - \left[ {\lambda \bar{p}{\text{e}}^{{ - \alpha_{k} \left( {p_{i} - \bar{p}} \right)}} \left( {r_{e} - r^{\prime}_{w} } \right)} \right] = \hfill \\ \frac{{Tp_{\text{sc}} \bar{\mu }_{\text{g}} \bar{Z}q}}{{2\pi L_{p} T_{\text{sc}} Z_{\text{sc}} k_{i} }}\ln \left( {\frac{{r_{e} }}{{r^{\prime}_{w} }}} \right) + \frac{{3.33 \times 10^{10} \gamma_{\text{g}} \tau T\bar{Z}p_{\text{sc}}^{2} q^{2} }}{{4\pi^{2} L_{p}^{2} T_{\text{sc}}^{2} Z_{\text{sc}}^{2} R\varphi k_{i} }}\left( {\frac{1}{{r^{\prime}_{w} }} - \frac{1}{{r_{e} }}} \right) \hfill \\ \end{aligned} $$

(58)

To convert Eq. 58 into practical SI unit systems, it is rewritten in Eq. 59 as the deliverability equation for the tight gas carbonate horizontal well productivity model in Figure 23.

$$ \begin{aligned} \left[ {p_{i} \left( {\frac{1}{{ - \alpha_{k} }}} \right) - p_{wf} \left( {\frac{1}{{ - \alpha_{k} }}{\text{e}}^{{ - \alpha_{k} \left( {p_{i} - p_{wf} } \right)}} } \right) - \left( {\frac{1}{{\alpha_{k}^{2} }} - \frac{1}{{\alpha_{k}^{2} }}{\text{e}}^{{ - \alpha_{k} \left( {p_{i} - p_{wf} } \right)}} } \right)} \right] - \left[ {\lambda \bar{p}{\text{e}}^{{ - \alpha_{k} \left( {p_{i} - \bar{p}} \right)}} \left( {r_{e} - r^{\prime}_{w} } \right)} \right] = \hfill \\ \frac{{1.84 \times 10^{4} Tp_{\text{sc}} \bar{\mu }_{\text{g}} \bar{Z}q}}{{L_{p} T_{\text{sc}} Z_{\text{sc}} k_{i} }}\ln \left( {\frac{{r_{e} }}{{r^{\prime}_{w} }}} \right) + \frac{{45.218\gamma_{\text{g}} \tau T\bar{Z}p_{\text{sc}}^{2} q^{2} }}{{L_{p}^{2} T_{\text{sc}}^{2} Z_{\text{sc}}^{2} R\varphi k_{i} }}\left( {\frac{1}{{r^{\prime}_{w} }} - \frac{1}{{r_{e} }}} \right) \hfill \\ \end{aligned} $$

(59)

where $\overline{p}$ average reservoir pressure, T is the reservoir temperature, $\overline{\mu }_{g}$ is the average viscosity of gas, $\overline{Z}$ is the average deviation factor.

Skin factor is a dimensionless pressure drop due to near-wellbore damage or stimulation, perforation and partial penetration effects. A positive skin value indicates damage in the near-wellbore region while a negative skin value indicates improved conditions around the wellbore. Introducing skin factor into Eq. 59 results in:

$$ \begin{aligned} \left[ {p_{i} \left( {\frac{1}{{ - \alpha_{k} }}} \right) - p_{wf} \left( {\frac{1}{{ - \alpha_{k} }}{\text{e}}^{{ - \alpha_{k} \left( {p_{i} - p_{wf} } \right)}} } \right) - \left( {\frac{1}{{\alpha_{k}^{2} }} - \frac{1}{{\alpha_{k}^{2} }}{\text{e}}^{{ - \alpha_{k} \left( {p_{i} - p_{wf} } \right)}} } \right)} \right] - \left[ {\lambda \bar{p}{\text{e}}^{{ - \alpha_{k} \left( {p_{i} - \bar{p}} \right)}} \left( {r_{e} - r^{\prime}_{w} } \right)} \right] = \hfill \\ \frac{{1.84 \times 10^{4} Tp_{\text{sc}} \bar{\mu }_{\text{g}} \bar{Z}q}}{{L_{p} T_{\text{sc}} Z_{\text{sc}} k_{i} }}\ln \left( {\frac{{r_{e} }}{{r^{\prime}_{w} }} - F + s} \right) + \frac{{45.218\gamma_{\text{g}} \tau T\bar{Z}p_{\text{sc}}^{2} q^{2} }}{{L_{p}^{2} T_{\text{sc}}^{2} Z_{\text{sc}}^{2} R\varphi k_{i} }}\left( {\frac{1}{{r^{\prime}_{w} }} - \frac{1}{{r_{e} }}} \right) \hfill \\ \end{aligned} $$

(60)

whenever F in Eq. 60 is ${\raise0.7ex\hbox{$3$} \!\mathord{\left/ {\vphantom {3 4}}\right.\kern-0pt} \!\lower0.7ex\hbox{$4$}}$, then pseudosteady-state flow condition is attained, while if F is ${\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}$, then, steady-state condition exists. Also, skin factor (s) in Eq. 60 can be given by $s = s_{\text{m}} + s_{p}$ where s_m is the mechanical skin and s_p is the skin resulting from partial penetration. However, if the horizontal well fully penetrates the reservoir system, then s_p= 0. The skin factor for radial convergence effect is read as:

$$ s = \frac{{k_{o} h}}{{k_{f} w_{f} }}\left[ {\ln \left( {\frac{h}{{2r_{w} }}} \right) - \frac{\pi }{2}} \right] $$

(61)

where k_f is the fracture permeability and w_f is the fracture width.

Putting the following standard condition values of T_sc = 293 K, R = 0.008314 MPa m³/kmol. K and $p_{sc}$ = 0.101325 MPa into Eq. 60 yields:

$$ \begin{aligned} \left[ {p_{i} \left( {\frac{1}{{ - \alpha_{k} }}} \right) - p_{wf} \left( {\frac{1}{{ - \alpha_{k} }}{\text{e}}^{{ - \alpha_{k} \left( {p_{i} - p_{wf} } \right)}} } \right) - \left( {\frac{1}{{\alpha_{k}^{2} }} - \frac{1}{{\alpha_{k}^{2} }}{\text{e}}^{{ - \alpha_{k} \left( {p_{i} - p_{wf} } \right)}} } \right)} \right] - \left[ {\lambda \bar{p}{\text{e}}^{{ - \alpha_{k} \left( {p_{i} - \bar{p}} \right)}} \left( {r_{e} - r^{\prime}_{w} } \right)} \right] = \hfill \\ \frac{{6.343T\bar{\mu }_{\text{g}} \bar{Z}}}{{L_{p} k_{i} }}\ln \left( {\frac{{r_{e} }}{{r^{\prime}_{w} }} - F + s} \right) + \frac{{6.463 \times 10^{ - 4} \gamma_{\text{g}} \tau T\bar{Z}}}{{L_{p}^{2} \varphi k_{i} }}\left( {\frac{1}{{r^{\prime}_{w} }} - \frac{1}{{r_{e} }}} \right)q^{2} \hfill \\ \end{aligned} $$

(62)

Therefore, Eq. 62 is recast into quadratic equation form known as binomial deliverability equation as:

$$ \begin{aligned} \left[ {p_{i} \left( {\frac{1}{{ - \alpha_{k} }}} \right) - p_{wf} \left( {\frac{1}{{ - \alpha_{k} }}{\text{e}}^{{ - \alpha_{k} \left( {p_{i} - p_{wf} } \right)}} } \right) - \left( {\frac{1}{{\alpha_{k}^{2} }} - \frac{1}{{\alpha_{k}^{2} }}{\text{e}}^{{ - \alpha_{k} \left( {p_{i} - p_{wf} } \right)}} } \right)} \right] - \left[ {\lambda \bar{p}{\text{e}}^{{ - \alpha_{k} \left( {p_{i} - \bar{p}} \right)}} \left( {r_{e} - r^{\prime}_{w} } \right)} \right] = \hfill \\ Aq^{2} + Bq. \hfill \\ \end{aligned} $$

(63)

The B (pressure loss induced by viscosity) in Eq. 63 represents laminar coefficient, while A represents turbulence coefficient. The Aq in Eq. 63 represents pressure loss induced by inertia. The sum of the two terms forms the total pressure drop of inflow. In addition, whenever the fluid flow velocity is very low, then the flow is considered linear and Aq can be negligible in Eq. 63. However, with increment in flow velocity or multiphase flow, Aq (inertia resistance) in Eq. 63 should be taken into an account.

The solution to Eq. 63 is solved using the quadratic formula as:

$$ q = \frac{{ - B + \sqrt {B^{2} + 4AC} }}{2A}, $$

(64)

where $A = \frac{{6.463 \times 10^{ - 4} \gamma_{\text{g}} \tau T\bar{Z}}}{{L_{p}^{2} \varphi k_{i} }}\left( {\frac{1}{{r^{\prime}_{w} }} - \frac{1}{{r_{e} }}} \right),$$B = \frac{{6.343T\bar{\mu }_{\text{g}} \bar{Z}}}{{L_{p} k_{i} }}\ln \left( {\frac{{r_{e} }}{{r^{\prime}_{w} }} - F + s} \right),$

$$ C = \left[ {p_{i} \left( {\frac{1}{{ - \alpha_{k} }}} \right) - p_{wf} \left( {\frac{1}{{ - \alpha_{k} }}{\text{e}}^{{ - \alpha_{k} \left( {p_{i} - p_{wf} } \right)}} } \right) - \left( {\frac{1}{{\alpha_{k}^{2} }} - \frac{1}{{\alpha_{k}^{2} }}{\text{e}}^{{ - \alpha_{k} \left( {p_{i} - p_{wf} } \right)}} } \right)} \right] - \left[ {\lambda \bar{p}{\text{e}}^{{ - \alpha_{k} \left( {p_{i} - \bar{p}} \right)}} \left( {r_{e} - r^{\prime}_{w} } \right)} \right]. $$

The absolute open flow potential (AOF) is the rate at which a well would deliver against zero sandface back pressure. Thus, $p_{wf} = 0\;{\text{MPa}}$ (maximum pressure drawdown). Hence, taking AOF as the maximum rate of the well in Eq. 24 gives the quadratic equation form in Eq. 65 as:

$$ q_{\text{AOF}}^{{}} = \frac{{ - B + \sqrt {B^{2} + 4AC_{o} } }}{2A}, $$

(65)

where $A = \frac{{6.463 \times 10^{ - 4} \gamma_{\text{g}} \tau T\bar{Z}}}{{L_{p}^{2} \varphi k_{i} }}\left( {\frac{1}{{r^{\prime}_{w} }} - \frac{1}{{r_{e} }}} \right),$$B = \frac{{6.343T\bar{\mu }_{\text{g}} \bar{Z}}}{{L_{p} k_{i} }}\ln \left( {\frac{{r_{e} }}{{r^{\prime}_{w} }} - F + s} \right),$ and

$$ C_{\text{o}} = \left[ {p_{i} \left( {\frac{1}{{ - \alpha_{k} }}} \right) - \left( {\frac{1}{{\alpha_{k}^{2} }} - \frac{1}{{\alpha_{k}^{2} }}{\text{e}}^{{ - \alpha_{k} p_{i} }} } \right)} \right] - \left[ {\lambda \bar{p}{\text{e}}^{{ - \alpha_{k} \left( {p_{i} - \bar{p}} \right)}} \left( {r_{e} - r^{\prime}_{w} } \right)} \right]. $$

Also, it is very important to represent the degree of TPG and SSC of permeability influence on the tight gas carbonate horizontal well productivity. Hence, production decline rate (D_gr) was introduced to estimate production decline for the tight gas carbonate reservoir with reference to an ideal situation (AOF) expressed as:

$$ D_{\text{gr}} = \left( {1 - \frac{q}{{q_{\text{AOF}} }}} \right) \times 100\% , $$

(66)

where D_gr is the production decline rate, q_AOF is the tight gas well yield of productivity under an ideal condition of maximum rate, q is the tight gas horizontal well productivity considering influential parameters (SSC of permeability, near-wellbore turbulence effect, tortuosity factor, Klinkenberg effect and TPG).

Then, putting quadratic formulae of Eqs. 64 and 65 into Eq. 66 produces Eq. 67 as the tight gas carbonate horizontal well productivity reduction rate model for the influential parameters on the tight gas well productivity after eliminating like terms and simplifying:

$$ D_{\text{gr}} = \frac{{\sqrt {B^{2} + 4AC} - \sqrt {B^{2} + 4AC_{\text{o}} } }}{{ - B + \sqrt {B^{2} + 4AC_{\text{o}} } }} $$

(67)

Finally, for a hydraulic fractured gas well with radial flow as shown in Figure 23, the fold of increase in well productivity is written as:

$$ \frac{J}{{J_{\text{o}} }} = \frac{{\ln \frac{{r_{e} }}{{r_{w} }}}}{{\ln \frac{{r_{e} }}{{r_{w} }} + s}}, $$

(68)

where J and J_o are the productivity of fractured and non-fractured gas well, respectively.

Appendix 2: Turbulence Effects on Horizontal Well Performance

Figure 25 demonstrates the effect of IPR for horizontal well with or without turbulence by solving Eq. 62 explicitly for various values of p_wf. From Figure 25, it can be seen that horizontal wells have the capability of minimizing turbulence effects.

Figure 25

Effect of turbulence on inflow performance relation for horizontal well

Appendix 3: Units Conversion Factors

$$ \begin{aligned} &1\;{\text{m}}^{3} / {\text{D}} = 1.1574 \times 10^{ - 5} \;{\text{m}}^{ 3} / {\text{s}} \hfill \\ &1\;\left( {\text{MPa}} \right)^{2} /\left( {{\text{mPa}}\;{\text{s}}} \right) = 10^{15} \;{\text{Pa/s}} \hfill \\ &1\;{\text{mPa}}\;{\text{s}} = 10^{ - 3} \;{\text{Pa}}\;{\text{s}} \hfill \\ &1\;\upmu{\text{m}}^{2} = 10^{ - 12} \;{\text{m}}^{ 2} \hfill \\ &1\;{\text{ft}} = 0.3048\;{\text{m}} \hfill \\ &1\;{\text{MPa}} = 10^{6} \;{\text{Pa}} \hfill \\ \end{aligned} $$