Rate Transient Analysis for Multi-Fractured Wells in Tight Gas Reservoirs Considering Multiple Nonlinear Flow Mechanisms

Abstract

Making rate transient analysis (RTA) and formation evaluation for multi-fractured tight gas wells has always been a difficult problem. This is because the fluid flow in the formation has multiple nonlinear flow mechanisms, including gas-water two-phase flow, gas slippage, low-velocity non-Darcy flow, and stress-dependent permeability. In this paper, a novel RTA method is proposed for multi-fractured wells in tight gas reservoirs incorporating nonlinear flow mechanisms. The RTA method is based on an analytical model, which is modified from the classical trilinear flow model by considering all the nonlinear flow mechanisms. The concept of material balance time and normalized rate is used to process the production data for both water and gas phases. The techniques of approximate solutions in linear/bilinear flow regimes and type curve fitting are combined in the proposed RTA method. After that, the rate transient behaviors and influencing factors of multi-fractured tight gas wells are analyzed. A field case from Northwestern China is used to test the efficiency and practicability of the proposed RTA method. The results show that six flow regimes for both gas and water production performances are exhibited on the log-log plots of normalized production rate against material balance time. The rate transient responses are sensitive to the nonlinear flow mechanisms, and formation and fracture properties. The medium flow regimes are significantly affected by fracture number, fracture conductivity, fracture half-length, stress-dependent permeability, gas-water two-phase flow, and formation permeability, which should be considered in making RTA of fractured tight gas wells. The field case shows that both gas and water production performances can be well-fitted using the proposed RTA method. The major innovation of this paper is that a novel RTA method is proposed for fractured tight gas wells considering multiple nonlinear flow mechanisms, and it can be used to make reasonable formation and fracturing evaluations in the field.

1. Introduction

Tight gas resources have been commercially developed in the past decades with the world’s growing demand for clean energy and the rapid development of fracking technology [1,2]. Making formation evaluations is a critical step for making effective development strategies and optimizing gas engineering schemes [3]. Since the formation properties have significant uncertainties, especially after hydraulic fracturing, it is important to interpret the key formation and fracture parameters. There are several methods for this purpose, including pressure transient analysis (PTA) using well-testing data, rate transient analysis (RTA) based on production data, matching learning method, microseismic data interpretation, and other monitoring techniques [4,5,6]. RTA is widely used in the petroleum industry for formation evaluation for its convenience in data collecting and making long-term production analyses.

The first important issue is to choose an appropriate production model for RTA. Although numerical models can handle heterogeneities and nonlinearities in the model, analytical models are often used for RTA because of the high computational efficiency in history matching. Many analytical models have been proposed so far to capture the fractures generated after fracking and gas flow mechanisms in the formation [7,8,9]. To consider the heterogeneities and hydraulic fractures in the formation, the physical models are often divided into several regions [7,8,9]. In addition, the flow in each region is regarded as linear flow since tight gas wells are reported to have significant linear flow regimes. The linear flow models can be divided into several types according to the number of zones/regions in the model, including single linear flow model, two-linear flow model, trilinear flow model, etc. [7,10,11]. The trilinear flow model is the most used model for modeling fluid flow in unconventional reservoirs, in which the formation is discretized into three zones, including hydraulic fracture, inner stimulated zone, and outer unstimulated zone [10]. The nonlinear flow mechanisms for fluid flow in the tight formation include gas slippage, low-velocity non-Darcy flow, stress-dependent permeability, and two-phase flow [12,13,14,15]. To further capture the nonlinear flow mechanisms in the tight formation, the present linear flow models are modified for this purpose by some researchers [16,17,18,19]. Song et al. (2015) proposed a composite reservoir model to capture the effects of low-velocity non-Darcy flow [16]. Ning et al. (2020) proposed a numerical model to consider the effects of both low-velocity non-Darcy flow and stress-dependent permeability [17]. Wu et al. (2021, 2024) modified the classical trilinear flow model and proposed an analytical model by incorporating all these nonlinear flow mechanisms [18,19].

Another important issue for RTA is to obtain the transient flow behavior of fractured tight gas wells, making production analysis and unknown parameter interpretation [20]. Present studies show that the flow regimes for single-phase gas/oil flow in unconventional formations mainly include linear flow regimes dominated, respectively, by each zone, bilinear/transitional flow regimes dominated by two zones, and boundary-dominated flow regimes [8,10]. Therefore, the characteristic lines and points are often used in production data analysis. First, the slopes of the linear and bilinear flow regimes are used to generate several equations among unknown parameters [21,22,23]. Second, the time of deviation from the LF/BLF to BDF can be used to generate another relationship between the unknowns [24,25]. In this way, the unknown formation and fracture parameters can be interpreted by combining type curve fitting [25]. However, the present RTA method is limited to single-phase flow problems and the nonlinear flow mechanisms are not considered, which makes it difficult to make production data analysis and formation evaluation.

In this paper, a novel RTA method is proposed for multi-fractured tight gas wells with the consideration of nonlinear flow mechanisms. First, the mathematical model provided in our previous work is applied to theoretical solution generation [19]. The nonlinear flow mechanisms are considered in the model, including gas-water two-phase flow, gas slippage, low-velocity non-Darcy flow, and stress-dependent permeability. Second, the RTA method is proposed based on the mathematical model, including the techniques of production data processing, linear/bilinear flow regime analysis, and type curve fitting. Finally, the rate transient behavior and influencing factors are provided, and a field example from Northwestern China is used to benchmark the proposed RTA method.

2. Mathematical Model

2.1. Physical Model

The microseismic data show that multiple fractures are generated after hydraulic fracturing of the horizontal wells in tight gas formation [26], as shown in Figure 1. In addition to the transverse main fractures, field and laboratory tests show that the multiple induced small-scaled fractures can be generated between the main fractures [27,28]. The stimulated zone between the main fractures is often characterized by the inner reservoir, which has higher permeability than the unstimulated zone (i.e., the outer reservoir in Figure 1 [29,30]). In this way, the reservoir around the horizontal well can be divided into three zones, including the hydraulic fracture, and inner and outer reservoirs. Because many tight gas wells exhibit linear and bilinear flow regimes, the trilinear flow model proposed in our previous work is applied to propose the physical model [20]. Three assumptions are included: (1) the formation is horizontal and with equal thickness; (2) the formation is fully penetrated by the hydraulic fractures; (3) the flow of gas and water in the three zones are regarded as linear, and treating the flux from the outer reservoir as a source of the inner reservoir, and treating the flux from the inner reservoir as source term of the fracture. In this paper, three major modifications are made to the previous model. First, the nonlinear flow mechanisms are addressed, including gas slippage, stress-dependent permeability, and low-velocity non-Darcy flow. Second, the two-phase flow in the formation is considered in the whole reservoir. Third, a rate transient analysis method is proposed in this paper based on the model to make formation and fracturing evaluations by analyzing the production data. One should note that each transverse fracture stage is extracted to model two-phase flow in the formation, so the fractures can be assumed to be with different lengths and unsymmetrical, as shown in Figure 1.

Figure 1 also shows that the interpreted fracture from Instantaneous Shut-in Pressure (ISIP) analysis is beyond the seismic deformation region and even longer than 1000 m, which is not reasonable in the field. Therefore, it is significant to propose a new RTA method to justify fracture parameter interpretations.

2.2. Characterization of the Nonlinear Flow Mechanisms

In this model, the effects of gas slippage, low-velocity non-Darcy flow, and stress-dependent permeability are considered. Following previous works, these flow mechanisms can be characterized with formulations provided in the following.

The effect of gas slippage can be simply characterized by adding a slip factor to the gas permeability. The theoretical model proposed by Klinkenberg (1941) is used in this section [31], as shown in Equation (1).

$$\frac{k_g}{k_{\infty }}=1+\frac{b}{\widehat{p}}$$

(1)

where b is the slip factor; $\widehat{p}$ is the average formation pressure, MPa; $k_g$ and $k_{\infty }$ are, respectively, the permeability with and without the effect of gas slippage, mD.

The effect of low-velocity non-Darcy flow on gas transportation in the formation can be captured with the Pseudo Threshold Pressure Gradient (PTPG) [12], which is applied to modify the velocity formulation of gas flow in the formation and given by

$$v_g=-\frac{k_g}{\mu }\left(\frac{dp}{dx}-\lambda \right)$$

(2)

In which, $k_g$ is the permeability of gas, m²; $v_g$ is the velocity of gas, m/s; $\mu$ is the viscosity of gas, Pa·s; $\frac{dp}{dx}$ is the pressure gradient, Pa/m; $\lambda$ is the PTPG, Pa/m.

The effect of stress-dependent permeability can be characterized using the exponential model proposed by Nur and Yilmaz (1985) [32], given by

$$k=k_i{e}^{-\gamma \left(p_i-p\right)}$$

(3)

In which, $\gamma$ is the permeability modulus, MPa; $p_i$ and $p$ are the formation pressure, respectively at initial and present conditions, MPa.

The effect of gas-water two-phase flow can be characterized with two-phase relative permeability model. In this paper, the Brooks-Corey model is used for this purpose.

$$k_{rg}=k_{gr}{\left(S^{*}\right)}^{n_g}$$

(4)

$$k_{rw}=k_{wr}{\left(1-S^{*}\right)}^{n_w}$$

(5)

$$S^{*}=\frac{S_g-S_{gc}}{1-S_{wrg}-S_{gc}}$$

(6)

In which, $k_{rw}$ and $k_{rg}$ are, respectively, the water and gas relative permeabilities; $k_{wr}$ and $k_{gr}$ are, respectively, the maximum water and gas relative permeabilities; $S_g$ is the gas saturation; $S_{wrg}$ and $S_{gc}$ are, respectively, the irreducible water saturation and residual gas saturation; $S^{*}$ is an intermediate variable; $n_w$ and $n_g$ are, respectively, the exponents of water and gas relative permeabilities.

2.3. Solution for Flow in the Fractured Tight Formation

The solutions for gas and water flow in the tight formation after hydraulic fracturing can be obtained by solving a trilinear linear flow model. In this paper, the solutions are provided below directly since the highlight of the paper is making rate transient analysis for fractured tight gas wells. The detailed derivations are provided in our previous work [19]. The dimensionless parameters defined in deriving the model are provided in

Table 1. Dimensionless parameters defined in this paper.

Variables	Definition	Variables	Definition
Dimensionless length-X direction	$x_D=\frac{x}{L_r}$	Dimensionless pseudo-pressure (gas)	${\psi }_D=\frac{{\psi }_i-\psi }{{\psi }_i-{\psi }_{wf}}$
Dimensionless length-Y direction	$y_D=\frac{y}{L_r}$	Dimensionless pseudo-time (gas)	$t_{aD}=\frac{{\eta }_r}{L_r^2}{t}_g$
Dimensionless length-Z direction	$z_D=\frac{z}{L_r}$	Dimensionless production rate (gas)	$q_{gD}=\frac{1.291\times {10}^{-3}{q}_{g}T}{\left({\psi }_i-{\psi }_{wf}\right)k_{r}H}$
Dimensionless permeability	$k_D=\frac{k}{k_r}$	Dimensionless pseudo-pressure (water)	$p_D=\frac{{\phi }_i-\phi }{{\phi }_i-{\phi }_{wf}}$
Dimensionless threshold gradient pressure	$G_D=L_r\lambda G_{\lambda }\left(\widehat{p}\right)$	Dimensionless pseudo-time (water)	$t_{wD}=\frac{{\eta }_r}{L_r^2}{t}_w$
Dimensionless pseudo-threshold gradient pressure	${\lambda }_D=\frac{f_p\left(\widehat{p}\right)\lambda L_r}{{\psi }_i-{\psi }_{wf}}$	Dimensionless production rate (water)	$q_{wD}=\frac{1.842q_w{B}_w{\mu }_w{L}_r}{k_{F}H\left({\phi }_i-{\phi }_{wf}\right)}$

.

The solution of dimensionless gas production rate in the Laplace domain is given by

$${\overline{q}}_{gD}=\frac{k_{Frg}{k}_{FD}{w}_{FD}}{\pi }\tanh \left(\sqrt{{\alpha }_F}\cdot x_{FD}\right)\left(\frac{1}{s}+\frac{{\beta }_F}{{\alpha }_F}\right)\sqrt{{\alpha }_F}$$

(7)

In which, ${\beta }_F=\frac{2k_I{k}_{Irg}}{w_{FD}{k}_F{k}_{Frg}}\frac{e^{m_2\left(w_{FD}/2-y_{eD}\right)}-e^{m_1\left(w_{FD}/2-y_{eD}\right)}}{m_1{e}^{m_2\left(w_{FD}/2-y_{eD}\right)}-m_2{e}^{m_1\left(w_{FD}/2-y_{eD}\right)}}{m}_1{m}_2\frac{1}{{\alpha }_I}\frac{k_O{k}_{Org}}{x_{FD}{k}_I{k}_{Irg}}\frac{{\lambda }_D}{s}-\frac{2k_I{k}_{Irg}}{w_{FD}{k}_F{k}_{Frg}}\frac{{\lambda }_D}{s}$, ${\alpha }_F=\frac{s}{k_{Frg}{\eta }_{FD}}-\frac{2k_I{k}_{Irg}}{w_{FD}{k}_F{k}_{Frg}}\frac{e^{m_2\left(w_{FD}/2-y_{eD}\right)}-e^{m_1\left(w_{FD}/2-y_{eD}\right)}}{m_1{e}^{m_2\left(w_{FD}/2-y_{eD}\right)}-m_2{e}^{m_1\left(w_{FD}/2-y_{eD}\right)}}{m}_1{m}_2$, $m_1=\frac{1}{2}\left(G_D+\sqrt{{G_D}^2+4{\alpha }_I}\right)$, $m_2=\frac{1}{2}\left(G_D-\sqrt{{G_D}^2+4{\alpha }_I}\right)$, ${\alpha }_I=\frac{s}{k_{Irg}{\eta }_{ID}}-\frac{k_O{k}_{Org}}{x_{FD}{k}_I{k}_{Irg}}\frac{e^{r_2\left(x_{FD}-x_{eD}\right)}-e^{r_1\left(x_{FD}-x_{eD}\right)}}{r_1{e}^{r_2\left(x_{FD}-x_{eD}\right)}-r_2{e}^{r_1\left(x_{FD}-x_{eD}\right)}}{r}_1{r}_2$, $r_1=\frac{1}{2}\left(G_D+\sqrt{{G_D}^2+\frac{4s}{k_{Org}{\eta }_{OD}}}\right)$, $r_2=\frac{1}{2}\left(G_D-\sqrt{{G_D}^2+\frac{4s}{k_{Org}{\eta }_{OD}}}\right)$.

The solution of dimensionless water production rate in the Laplace domain is given by

$${\overline{q}}_{wD}=\frac{1}{s}\frac{k_{Frw}{k}_{FD}{w}_{FD}}{\pi }\tanh \left(\sqrt{{\alpha }_{Fw}}\cdot x_{FD}\right)\sqrt{{\alpha }_{Fw}}$$

(8)

In which, ${\alpha }_{Iw}=\frac{s}{k_{Irw}{\eta }_{IwD}}-\frac{k_O{k}_{Orw}}{x_{FD}{k}_I{k}_{Irw}}\sqrt{\frac{s}{k_{Orw}{\eta }_{OwD}}}\tanh \left(\sqrt{\frac{s}{k_{Orw}{\eta }_{OwD}}}\cdot \left(x_{FD}-x_{eD}\right)\right)$,${\alpha }_{Fw}=\frac{s}{k_{Frw}{\eta }_{FwD}}-\frac{2k_I{k}_{Irw}}{w_{FD}{k}_F{k}_{Frw}}\sqrt{{\alpha }_{Iw}}\tanh \left(\sqrt{{\alpha }_{Iw}}\cdot \left(w_{FD}/2-y_{eD}\right)\right)$.

The numerical algorithm proposed by Stehfest (1970) should be used to obtain the solution of water and gas production rates in the time domain [33]. One should note that some parameters in Equations (7) and (8) are pressure- and saturation-dependent for the effects of gas pressure-volume-temperature (PVT) properties and nonlinear flow mechanisms. A simple iteration with a material balance method can be used for this purpose by updating the parameters step by step. More details are provided in our previous work [19].

The definitions of some parameters in Table 1 are provided in Appendix A.

2.4. Rate Transient Analysis Method

2.4.1. Production Data Processing

Because variable production rate and bottom-hole pressure (BHP) data should be analyzed for field applications, the primary step is to process the production data before making RTA. In this study, the method proposed by Agarwal et al. (1998) is used for processing the production data by defining the material balance time and normalized rate [34].

For gas production, the material balance time and normalized rate are, respectively, given by

$$t_{c,g}=\frac{{\mu }_i{c}_{ti}}{q_g}{\displaystyle \underset{0}{\overset{t}{\int }}\frac{q_g}{\mu c_t}}dt=-\frac{G_i}{q_g}\frac{{\mu }_i{c}_{ti}{Z}_i}{2p_i}({\psi }_i-\overline{\psi })$$

(9)

$$q_{N,g}=\frac{q_g}{{\psi }_i-{\psi }_{wf}}$$

(10)

For water production, the material balance time and normalized rate are, respectively, given by

$$t_{c,w}=\frac{W_p}{q_w}$$

(11)

$$q_{N,w}=\frac{q_w}{p_i-p_{wf}}$$

(12)

2.4.2. Linear/Bilinear Flow Regime Analysis

Linear and bilinear flow regimes are exhibited by the production data for many tight gas wells, so it is very practical to use approximate solutions to analyze the production data in the linear and bilinear flow regimes. In this section, the results provided in our previous work are modified for gas and water two-phase production in both linear and bilinear flow regimes [25].

For the linear flow regime dominated by the flow in the inner reservoir, the approximate solution is given by

$$\frac{1}{q_D}=\frac{1}{1+s_c}\frac{\pi }{2k_{ID}{k}_{Irp}{x}_{FD}}\sqrt{\pi {\eta }_{ID}{k}_{Irp}{t}_D}$$

(13)

Substituting the dimensionless parameters for gas and water in Table 1, the approximate solution for gas and water can be obtained, respectively, given by

$$\frac{{\psi }_i-{\psi }_{wf}}{q_{gsc}}=\sqrt{t_{c,g}}\frac{1.291\times {10}^{-3}\pi T\sqrt{\pi {\eta }_I{k}_{Irg}}}{2\left(1+s_c\right)H{\displaystyle \sum \left(k_I{k}_{Irg}{x}_F\right)}}$$

(14)

$$\frac{{\phi }_i-{\phi }_{wf}}{q_w}=\sqrt{t_{c,w}}\frac{1.842\pi B_w{\mu }_w}{2\left(1+s_c\right)H}\frac{\sqrt{\pi {\eta }_I{k}_{Irw}}}{{\displaystyle \sum \left(k_I{k}_{Irw}{x}_F\right)}}$$

(15)

For the bilinear flow regime dominated by flow in the fracture and inner reservoir, the approximate solution is given by

$$\frac{1}{q_D}=\frac{1}{1+s_c}\frac{1.2254\pi }{\sqrt{2k_D{k}_{Irp}{k}_{FD}{k}_{Frp}{w}_{FD}}}\sqrt[4]{{\eta }_{ID}{k}_{Irp}{t}_D}$$

(16)

Substituting the dimensionless parameters for gas and water in Table 1, we obtain

$$\frac{{\psi }_i-{\psi }_{wf}}{q_{sc}}=\sqrt[4]{t_{c,g}}\frac{T\sqrt[4]{{\eta }_I{k}_{Irg}}}{284.8\sqrt{k_I{k}_{Irg}{k}_F{k}_{Frg}{w}_F}\left(1+s_c\right)H{N}_F}$$

(17)

$$\frac{{\phi }_i-{\phi }_{wf}}{q_w}=\sqrt[4]{t_{c,w}}\frac{B_w{\mu }_w\sqrt[4]{{\eta }_I{k}_{Irw}}}{0.2\sqrt{k_I{k}_{Irw}{k}_F{k}_{Frw}{w}_F}\left(1+s_c\right)H{N}_F}$$

(18)

It should be noted that the interpreted parameters using Equations (14), (15), (17) and (18) should be averaged since there are two sets of equations obtained, respectively, from gas and water equations.

The end time of the linear and bilinear flow regimes can also be used to generate a formulation between fracture space and formation permeability, given by

$$y_F=0.5836\sqrt{\frac{k_I{k}_{Irp}{t}_{elf}}{{\left(\varphi {\mu }_i{\mathrm{c}}_{ti}\right)}_p}}$$

(19)

Then, the fracture number can be obtained

$$N_F=\frac{y_{Well}}{2y_F}$$

(20)

Equations (13)–(16) can be used to generate several equations in production data analysis, which are efficient in reducing the uncertainties of unknown parameters interpretation in history matching.

2.4.3. Type Curve Fitting

After linear/bilinear flow regime analysis, type curve fitting should be used to interpret the unknown parameters. In this way, an integrated workflow should be proposed to combine the process of linear/bilinear flow regime analysis and type curve fitting. As shown in Figure 2, several steps are included in the workflow, including data input, production data processing, flow regimes diagnosis, linear/bilinear flow regime analysis, and type curve fitting. Finally, multiple parameters can be obtained with the proposed workflow RTA for fractured tight gas wells, some of which are associated with nonlinear flow mechanisms, including the initial water saturation S_wi, permeability modulus γ, and PTPG λ. The effects of multiple factors on the rate transient behavior of tight gas wells will be provided in Section 3, and the application of the RTA method will be further illustrated in Section 4.

3. Results and Discussion

3.1. Flow Regimes of Gas and Water Phase

In this section, the flow regimes exhibited by the production performance of both gas and water phases are analyzed for multi-fractured horizontal wells in tight gas formation. The formation properties of a tight gas field in Northwestern China are applied in the analysis. The formation is saturated with both water and gas, and the initial water saturation is 0.5. The tight gas resource is developed with multi-fractured horizontal wells, and the well spaces are about 600 m. In this section, the production performances of a single fractured horizontal well are analyzed. The length of the horizontal well is 1100 m, and the well is hydraulically fractured with 11 stages. In this way, the proposed RTA method is used to analyze the flow regimes. The basic formation properties are provided in

Table 2. Model parameters for rate transient behavior analysis.

Model Parameter	Value	Units
Initial formation pressure	23.4	MPa
Formation temperature	345	K
Formation thickness	10	m
Well length	1100	m
Number of fractures	11	Dimensionless
Fracture conductivity	20	D·cm
Half-length of the fracture	100	m
Well space	600	m
Rock compressibility	5 × 10−5	MPa−1
Porosity of the hydraulic fracture	0.3	Dimensionless
Bottom hole pressure of the wellbore	5	MPa
The permeability of the matrix	0.01	mD
The porosity of the matrix	0.1	Dimensionless
kI/kO	2	Dimensionless
Viscosity of water	0.3	mPa·s
Water compressibility	5 × 10−4	MPa−1
Initial water saturation	0.5	Dimensionless
PTPG	1 × 10−3	MPa/m
Permeability modulus	0.01	MPa−1
Slippage factor	1	MPa

. The relative permeability curves and gas PVT properties are provided in Figure 3. The permeability curves 1w and 1g are used for this case, which is generated using the Brooks-Corey model.

The log-log plots of normalized production rate ($q_{N,g}$ and $q_{N,w}$) against pseudo-time ($t_g$ and $t_w$) and material balance time ($t_{c,g}$ and $t_{c,w}$) are used to analyze the flow regimes. Figure 4 shows the results of rate transient behavior exhibited by fractured tight gas wells with the two-phase flow in the formation. Six flow regimes can be diagnosed from the plots based on the slope of the curves, including linear flow dominated by the fractures (I), bilinear flow dominated by both the fractures and inner reservoir (II), linear flow dominated by the inner reservoir (III), transition flow dominated by the inner and outer reservoirs (IV), linear flow dominated by the outer reservoir (V), and boundary dominated flow (VI). The slopes of the curves are close to −1/4 for bilinear flow regimes and −1/2 for linear flow regimes. In this case, the bilinear flow regime II is dominant and easy to observe on the plots. The flow regimes are similar to that of the single-phase flow case [20], so the approximate solutions derived for the single-phase flow case can be modified to two-phase flow problems, as shown in Section 2.4.2. The results also show that flow regimes I~III are almost the same in the pseudo-time plot and material balance time plot, but significant differences are exhibited by the type curves in flow regimes IV~VI. As gas and water production rates and BHP are variable in the field, the material balance time plot should be used for type curve fitting in field applications. It should be noted that the time intervals of the flow regimes for gas and water phases may not be the same. This is because the definitions of modified material balance time and normalized rate are different for water and gas cases.

3.2. The Effects of Nonlinear Flow Mechanisms

In this section, the effects of nonlinear flow mechanisms on the log-log plots of normalized production rate $q_{N,g}$ and $q_{N,w}$ against material balance time $t_{c,g}$ and $t_{c,w}$ are analyzed, including the stress-dependent permeability, gas slippage, low-velocity non-Darcy flow, and two-phase flow. The model parameters are the same as the case shown in Section 3.1, except for the analyzed parameter. $q_{N,g}$ and $q_{N,w}$ against $t_{c,g}$ and $t_{c,w}$ plots are used in the analysis in this section, which should be used to process the variable BHP/rates production data and make analysis in field applications.

3.2.1. Stress-Dependent Permeability

Figure 5 shows the effects of stress-dependent permeability on the RTA responses. The results show that almost all the flow regimes are affected by the stress-dependent permeability. In addition, both the RTA responses of gas and water phases are significantly influenced. The normalized production rates of gas and water phases will decrease dramatically when the permeability modulus is increased. Therefore, we cannot neglect the effects of stress-dependent permeability in production data analysis for tight gas wells.

3.2.2. Gas Slippage

Gas slippage will increase the apparent permeability of gas in the tight formation, which further enhances the production of tight gas. In this section, we compared the results of cases with and without the effects of gas slippage, as shown in Figure 6. It is shown that a slight influence is observed on the type of curve. Therefore, the effects of gas slippage can be neglected in the RTA of fractured wells in tight gas reservoirs.

3.2.3. Low-Velocity Non-Darcy Flow

The effects of low-velocity non-Darcy flow are often considered for gas flow in tight formations. In this section, the PTPG is, respectively, given as 0.1, 0.01, and 0.001 MPa/m for gas flow in the formation, while the effects of low-velocity non-Darcy flow are not considered for the water case. The log-log plot of the normalized production rate $q_{N,g}$ and $q_{N,w}$ against material balance time $t_{c,g}$ and $t_{c,w}$ is shown in Figure 7. It is shown that the late flow regimes of gas production performance are affected by low-velocity non-Darcy flow, including flow regimes IV~VI. However, the influences on these flow regimes are very limited and hard to observe in the history matching of filed data.

3.2.4. Gas–Water Two-Phase Flow

Two factors can be considered to study the effects of gas-water two-phase flow on the RTA responses, including initial water saturation and gas-water relative permeability curves as shown in Figure 3a. In this part, the effects of the initial water saturation on the RTA responses of gas and water phases are provided, and the effects of relative permeability curves are similar. The results show that the rate transient behaviors of both water and gas phases are significantly affected by gas-water two-phase flow, as shown in Figure 8. The rate transient curves of water will move upward on the log-log plot with the increase in initial water saturation, while the curves for gas will move downward. Therefore, the effects of two-phase flow in the formation should be considered in the RTA of fractured tight gas wells.

3.3. The Effects of Fracture and Formation Parameters

In this section, we analyzed the effects of fracture and formation parameters on the rate transient behavior of both gas and water phases. The log-log plots of normalized production rate $q_{N,g}$ and $q_{N,w}$ against material balance time $t_{c,g}$ and $t_{c,w}$ are also applied in the analysis. The analyzed factors include the half-length of the fractures, fracture number, fracture conductivity, and formation permeability. The model parameters are the same as the case shown in Section 3.1, excepting the analyzed parameter.

3.3.1. Half-Length of the Fractures

The half-length of the fracture is, respectively, given as 50 m, 100 m, and 150 m in this case to study the effects on rate transient behaviors of water and gas phases. Figure 9 shows that the medium flow regimes of both water and gas phases are affected by the fracture half-length. The results also show that similar influences for water and gas phases are observed on the log-log plot. It should be noted that the production time of the well is mainly during the affected flow regimes, so this should be considered in PDA in field applications.

3.3.2. Fracture Number

The fracture number determines the fracture space y_f in the model, as shown in Figure 1. The modeling results are shown in Figure 10. It is shown that the influences on the early and medium flow regimes can be easily observed on the log-log plot, while the late flow regimes are almost not affected by the change in the fracture number. In addition, the effects of fracture number on the rate transient responses of water and gas phases are similar, and larger normalized rates are obtained with more fracture stages in the model.

3.3.3. Fracture Conductivity

Figure 11 shows the effects of fracture conductivity on the rate of transient responses of water and gas phases. It is shown that the influence of fracture conductivity is mainly in the early and medium flow regimes on the log-log plot, which is like that of the fracture number. The influences on water and gas rate transient responses are similar, and fractures with larger conductivity will result in a larger normalized rate. However, the increase in normalized rate is limited when the fracture conductivity is gradually close to infinite conductivity.

3.3.4. Formation Permeability

The effects of formation permeability on the responses of rate transient behaviors are mainly in two aspects, as shown in Figure 12. On the one hand, formation with larger permeability will result in a larger normalized rate, which shows that the fractured horizontal well will have higher productivity. On the other hand, the flow regimes for cases with lower formation permeability will be easily recognized on the type of curve, while it is hard to diagnose linear flow regimes dominated by the fracture and inner zone for cases with larger formation permeability.

4. Field Case

In this section, a field case from Northwestern China is used to benchmark the proposed RTA method for fractured tight gas wells exhibiting gas-water two-phase flow. The length of the horizontal well is 1215 m, and the wellbore is fractured with 12 stages. Two or three clusters are included in each stage. However, the fracture number is unknown because of fracture interference in hydraulic fracturing. In addition, some other parameters are also unknown, including the fracture length, fracture conductivity, formation permeability, permeability modulus, etc.

The production performance of the well is presented in Figure 13, which shows that the wellbore exhibits an apparent two-phase flow. The relative permeability curves and gas PVT properties are provided in Figure 3. The permeability curve 2 is used for the field case, and the initial water saturation is about 0.65 in the formation. Other parameters are provided in

Table 3. Model parameters for the field case.

Model Parameter	Value	Units
Initial formation pressure	23.4	MPa
Formation temperature	345	K
Formation thickness	10	m
Well length	1215	m
Well space	600	m
Rock compressibility	5 × 10−5	MPa−1
Porosity of the hydraulic fracture	0.3	Dimensionless
Bottom hole pressure of the wellbore	5	MPa
The porosity of the matrix	0.1	Dimensionless
Initial water saturation	0.65	Dimensionless
Viscosity of water	0.3	mPa·s
Water compressibility	5 × 10−4	MPa−1
PTPG	5 × 10−4	MPa/m
Permeability modulus	0.01	MPa−1
* kI/kO	1	Dimensionless
* Fracture space	35.7	m
* Number of fractures	17	Dimensionless
* Fracture conductivity	1.3	D.cm
* Half-length of the fracture	100	m
* The permeability of the matrix	0.07	mD

Note: Parameters marked with “*” are obtained after the RTA of the production data.

. The nonlinear flow mechanisms are also considered in the field case. The effect of gas slippage is neglected since it cannot be recognized on the type of curve. Core displacement experiments show that the PTPG and permeability modulus are, respectively, given as 5 × 10⁻⁴ MPa and 0.01 MPa.

The workflow shown in Figure 2 is used to analyze the production data. First, the production data are processed with the material balance method, and the results are provided in Figure 14. One can note that a linear flow regime (the slope is about −1/2) is dominated in Figure 14. This shows that Equations (14) and (15) can be used to interpret the production data. For simplicity, the square root of time plot is used in analysis since $\frac{{\psi }_i-{\psi }_{wf}}{q_{gsc}}$ and $\sqrt{t_{c,g}}$ have a linear relation. The slope of the linear flow regime can be used to calculate ${\displaystyle \sum \left(x_F\sqrt{k_I}\right)}$. For gas and water production performances, we can obtain two equations, respectively, given by

$${\displaystyle \sum \left(x_F\sqrt{k_I}\right)}=\frac{1.291\times {10}^{-3}\pi T\sqrt{\frac{0.0864\pi }{{\left(\varphi c_t\mu \right)}_I}}}{2H\sqrt{k_{Irg}}{m}_g}$$

(21)

$${\displaystyle \sum \left(x_F\sqrt{k_I}\right)}=\frac{1.842\pi B_w{\mu }_w\sqrt{\frac{0.0864\pi }{{\left(\varphi c_t\mu \right)}_I}}}{2H\sqrt{k_{Irw}}{m}_w}$$

(22)

where $m_g$ and $m_w$ are, respectively, the slopes obtained using gas and water production data in linear flow regime. It should be noted that two ${\displaystyle \sum \left(x_F\sqrt{k_I}\right)}$ values can be obtained using Equations (21) and (22), and the geometrical mean is used in the analysis.

Using Equation (19), $y_F/\sqrt{k_I}$ can be obtained and given by

$$y_F/\sqrt{k_I}=0.5836\sqrt{\frac{k_{Irp}{t}_{elf}}{{\left(\varphi {\mu }_i{\mathrm{c}}_{ti}\right)}_p}}$$

(23)

In this case, the square root of time plots for gas and water production data are shown in Figure 15. Using the slopes, the calculated $x_F\sqrt{k_I}$ using gas and water production data are, respectively, 24 m² and 30.4 m², and the geometrical mean is 27 m². In addition, the calculated $y_F/\sqrt{k_I}$ using the $t_{elf}$ in Figure 14 are, respectively, 1.12 × 10³ and 1.15 × 10³, and the geometrical mean is 1.14 × 10³. It should be noted that the $t_{elf}$ can be determined using the point where the trend of production data deviated from the linear flow regime. In this way, two independent equations can be obtained and used for type curving fitting to reduce the uncertainties in RTA. The type of curve fitting results are shown in Figure 14. We can find that good matches are obtained for both gas and water production data. This shows that the proposed RTA method is suitable for multi-fractured wells in tight gas reservoirs with multiple nonlinear flow mechanisms. The interpreted model parameters are provided in Table 3, which are marked with “*”.

The shortcomings of this study and future work for RTA of fractured tight gas wells mainly include three aspects. First, in addition to the fractures and nonlinear flow mechanisms in the formation, some tight gas reservoirs have bottom and edge water, so high-performance numerical models should be proposed in the future to capture these factors. Second, multi-source data can be used to evaluate the formation and fracture properties to reduce uncertainties, including seismic, well-log, core, hydraulic fracturing, etc. Third, an efficient history-matching method should be used to absorb multi-source data and quantify the uncertainties of the interpreted formation and fracture parameters.

5. Conclusions

This paper presents a new RTA method for multi-fractured wells in tight gas reservoirs with multiple nonlinear flow mechanisms. From this study, the following conclusions are guaranteed:

• The flow regimes of fractured tight gas wells can be recognized using the log-log plots of normalized production rate against material balance time. According to the slopes of the type curves, six flow regimes can be observed, including linear flow dominated by the fractures (I), bilinear flow dominated by both the fractures and inner reservoir (II), linear flow dominated by the inner reservoir (III), transition flow dominated by the inner and outer reservoirs (IV), linear flow dominated by the outer reservoir (V), and boundary dominated flow (VI).
• The nonlinear flow mechanisms, and formation and fracture properties can have significant influences on the rate transient responses of fractured tight gas wells. Fracture number, fracture conductivity, and half-length mainly influence the early and medium flow regimes. Low-velocity non-Darcy flow mainly influences the late flow regimes. Stress-dependent permeability, gas-water two-phase flow, and formation permeability can have a significant influence on almost all the flow regimes.
• The RTA method for fractured tight gas wells should consider the effects of nonlinear flow mechanisms, especially gas-water two-phase flow in the formation. The nonlinear flow mechanisms should be considered in the mathematical model for RTA to obtain a reasonable theoretical solution. In addition, gas and water production data can be processed and analyzed separately with the techniques of the material balance method.
• In making field applications, reasonable history-matching results can be obtained using either gas or water production data of tight gas wells, and the parameter inversion results can be averaged in analysis. In addition, the typical square root of time and log-log plots can be incorporated into the RTA method for uncertainty reduction.