A Global Search Algorithm for Determining Water Influx in Naturally Fractured Reservoirs

Abstract

The determination of water influx in naturally fractured reservoirs is always a significant and difficult task in gas reservoir engineering. To improve this situation, this paper presents a new global search algorithm to determine water influx in the naturally fractured gas reservoirs. In the methodology, a dimensionless water influx derivative curve is first introduced in this paper. It is used to identify flow regimes of water invasion by combining with the water influx characteristic curve. Following that, a sensitivity analysis is performed to study the impacts of key factors on flow regimes. Finally, based on the sensitivity study and material balance equation, a global search algorithm is proposed to obtain water influx. Results show that there are two steps in the dimensionless water influx curve and a “V-shape” in the derivative curve. The smaller the aquifer and gas reservoir radius ratio is, the earlier and more obvious the “V-shape” appears. The smaller the storativity ratio is, the earlier the “V-shape” appears. The smaller the interporosity flow coefficient is, the more obvious the “V-shape” is. Results of the field application demonstrate the method applicability, which provide a good reference for further work about determination of water influx.

1. Introduction

Almost every gas reservoir is connected with a water body. When the water body is large and water invasion is strong during gas reservoir exploitation, and water influx becomes an indispensable parameter in the material balance equation of the gas reservoir, the calculation of water influx is particularly important. As it is impossible to monitor the water invasion directly, the determination of water influx in naturally fractured reservoirs has always been a difficult task in gas reservoir engineering. In the current research on water invasion, many scholars have established a steady-state, unsteady-state and pseudo-steady-state water invasion model for homogeneous gas reservoirs, which are mainly solved by the material balance method, statistical formula method and numerical simulation method [1,2,3,4,5,6,7,8,9,10,11,12,13,14].

Hu [3] determined the water saturation of gas reservoirs according to the pseudo-steady state water invasion model, so as to calculate the dynamic reserves. Combined with the principle of material balance, water influx was solved with the relationship between water influx and dynamic reserves. Furthermore, considering the presence of uncertainty in pressures, Mcewen [14] proposed a new and valuable method to calculate water influx based on the material balance equation.

Some scholars have developed some methods to determine water influx [15,16,17,18]. In 2007, Kryuchkov et al. [16] proposed using microscopic computer tomography technology to detect water invasion, which avoids the tedious process of solving mathematical models. The method was simple and reliable, but the cost was high. In 2012, Al-Ghanim [15] proposed a nonparametric optimal transformations method to predict the water influx in the edge water drive reservoirs. This method can directly calculate water influx without looking up tables and interpolation. It not only meets the calculation accuracy, but also greatly improves the calculation speed. In 2015, Yong et al. [17] proposed water invasion diagnosis curves based on well production and pressure data for super-giant aquifer drive gas reservoirs. According to the diagnosis curves, water invasion was divided into three stages: the no aquifer influx stage, early aquifer influx stage, and middle-late aquifer influx stage. Using the same method, in 2017, they [18] proposed the diagnosis curves of water invasion in fractured-vuggy carbonate reservoirs.

Due to the existence of natural fractures in the aquifer, the traditional homogeneous models no longer meet the accuracy requirements. Therefore, many scholars have studied water invasion in dual media gas reservoirs [19,20,21,22,23,24,25,26,27,28]. Due to the more complex percolation mechanism, the statistics and numerical simulation methods are commonly used to calculate water influx. For example, Deng et al. [21] used statistical methods to obtain the nonlinear relationship between fracture height, single well fluid flux and water breakthrough time of gas wells in a fractured bottom-water reservoir, so as to predict the water breakthrough time of gas wells with different fluid flux.

For naturally fractured reservoirs, Roberto Aguilera [29] established an unsteady state water invasion model, taking the transient and pseudo-steady interporosity flow into account. They analyzed the effects of the storativity ratio and interporosity flow coefficient on dimensionless water influx curves. This provides a theoretical basis for studying the water invasion of naturally fractured gas reservoirs with edge water. However, the difference between naturally fractured aquifers and homogenous aquifers is not taken into account, and the characteristics of the dimensionless water influx curve are not obvious enough to effectively identify water invasion in naturally fractured gas reservoirs, let alone to quantify water influx in naturally fractured gas reservoirs and from a complete set of analytical methods.

From the above, we can see that an efficient method for determining water influx in naturally fractured reservoirs is still lacking. To improve this situation, based on a mathematical model of naturally fractured reservoirs, a new global search algorithm is proposed to calculate water influx. In this paper, the dimensionless water influx derivative is first proposed. With the double logarithmic curves of dimensionless water influx and its derivative, namely the characteristic curves, the flow regimes of water invasion can be identified. Then the water invasion trend can be predicted according to the water invasion curves, which provide reference for designing a reasonable gas well production system and adjusting the development plan. Following that, a sensitivity study of the characteristic curves is performed.

Furthermore, based on the sensitivity study and material balance equation, a new global search algorithm is proposed to calculate water influx. An actual field application is performed to show the accuracy and applicability of the new method, which provides a good reference for further work on the determination of water influx.

2. Conceptual Model

The aquifer of the naturally fractured gas reservoir includes two domains: matrix and natural fractures. In this model, the gas reservoir is regarded as a “well”, and the water influx is the fluid flux of the “well”. The assumptions for the water invasion model are as follows. The specific schematic diagram is shown in Figure 1.

• The formation is equal-thick, circular and finite.
• The fluid in the aquifer is single-phase, slightly compressible and isothermal, observing Darcy’s law. The vertical flow of the fluid is ignored.
• Interflow from matrix to “wellbore” (gas reservoir) is neglected, because matrix permeability in the model is much lower than fracture permeability.
• The initial aquifer pressure is $p_i$, the aquifer thickness is h, the gas reservoir radius is $r_g$, and the aquifer radius is $r_e$.
• The interporosity flow from natural fractures to the matrix is in the transient state in the aquifer.

3. Mathematical Model

Based on the above conceptual model, a mathematical model of water invasion is established [29], which provides the theoretical base for methodology development of the water-influx determination. Taking the transient state interporosity flow in the aquifer into account, the dimensionless radial seepage equation of finite edge water is obtained.

$$\frac{\partial p_{fD}}{\partial r_D^2}+\frac{1}{r_D}\frac{\partial p_{fD}}{\partial r_D}=\mathsf{\omega }\frac{\partial p_{fD}}{\partial t_D}+\frac{\lambda }{3}\frac{\partial p_{mD}}{\partial t_D}$$

(1)

The seepage equation in the matrix system satisfies

$$\frac{{\partial }^2{p}_{mD}}{\partial t_D^2}=3(\frac{1-\mathsf{\omega }}{\lambda })\frac{\partial p_{mD}}{\partial t_D}$$

(2)

Equation (1) is the dimensionless seepage equation of the fracture with transient state interporosity flow, and Equation (2) is the dimensionless seepage equation of the matrix with transient state interporosity flow. $p_{fD}$ is the fracture pressure in the aquifer (dimensionless); $p_{mD}$ is the matrix pressure in the aquifer (dimensionless); $r_D$ is the edge water radius (dimensionless); $\mathsf{\omega }$ is the storativity ratio in the aquifer (dimensionless); $\lambda$ is the interporosity flow coefficient in the aquifer (dimensionless); and $t_D$ is time (dimensionless). Subscript “D” tables dimensionless.

Initial conditions in the edge water:

$$p_{fD}(r_D,0)=p_{mD}(r_D,0)=0$$

(3)

Equation (3) shows that at the initial conditions, the pressure of the fracture and matrix are the same.

The internal boundary conditions in the edge water are given by:

$${\frac{\partial p_{fD}}{\partial r_D}|}_{r_D=1}=-1\hspace{1em}(t_D>0)$$

(4)

Equation (4) shows that at the interface between the gas reservoir and the edge water (internal boundary), edge water flows into the internal gas reservoir at a constant flux.

The external boundary conditions in the edge water are given by:

$${\frac{\partial p_{fD}}{\partial r_D}|}_{r_{eD}}=0\hspace{1em}(t_D>0)$$

(5)

Equation (5) shows that the external boundary of the edge water is a closed boundary and there is no fluid inflow outside.

The dimensionless variables are as follows. Dimensionless pressure is

$$p_D(r_D,t_D)=\frac{K_{f}h}{1.842\times {10}^{-3}q{\mu }_w{B}_w}[p_i-p(r,t)]$$

(6)

Dimensionless radius is

$$r_D=\frac{r}{r_g}$$

(7)

Dimensionless time is

$$t_D=\frac{3.6K_{f}t}{[{(\varphi c_t)}_f+{(\varphi c_t)}_m]{\mu }_w{r}_g^2}$$

(8)

The storativity ratio is

$$\mathsf{\omega }=\frac{{\left(\varphi c_t\right)}_f}{{\left(\varphi c_t\right)}_f+{\left(\varphi c_t\right)}_m}$$

(9)

The interporosity flow coefficient is

$$\lambda =\alpha \frac{K_m}{K_f}{r}_g^2$$

(10)

$$\alpha =\frac{4n(n+2)}{L^2}$$

(11)

where q is the fluid flux (m³/d); $B_w$ is the water volume factor (m³/m³); r is the radius (m); $r_e$ is the edge water radius; $r_g$ is the gas reservoir radius (m); K is the permeability (darcy); p is the formation pressure (MPa); t is time (hours); ϕ is porosity (fraction); $c_t$ is comprehensive compressibility (MPa⁻¹); ${\mu }_w$ is water viscosity (mPa·s); $\alpha$ is the form factor (dimensionless); n is the group number of orthogonal fractures, that is, the dimension of the fracture surface (integer); and L is the characteristic length of the rock, i.e., the ratio of the volume of the matrix block to its surface area (m). Subscript “m” tables matrix; subscript “f” tables fractures.

Roberto Aguilera [29] presented an exhaustive compilation of the detailed solutions by using a Laplace transform, and got the results as follows.

$${\overline{p}}_{fD}=\frac{1}{s^{3/2}\sqrt{f\left(s\right)}}\frac{M}{N}$$

(12)

where

$$f(s)=\mathsf{\omega }+\sqrt{\frac{\lambda \left(1-\mathsf{\omega }\right)}{3s}}\tanh \left(\sqrt{\frac{3\left(1-\mathsf{\omega }\right)s}{\lambda }}\right)$$

(13)

$$M=I_1\left(r_{eD}\sqrt{sf(s)}\right)K_0\left(r_D\sqrt{sf(s)}\right)+I_0\left(r_D\sqrt{sf(s)}\right)K_1\left(r_{eD}\sqrt{sf(s)}\right)$$

(14)

$$N=I_1\left(r_{eD}\sqrt{sf(s)}\right)K_1\left(\sqrt{sf(s)}\right)+I_1\left(\sqrt{sf(s)}\right)K_1\left(r_{eD}\sqrt{sf(s)}\right)$$

(15)

where s is the Laplace space operator (dimensionless); $I_0\left(x\right)$ are Bessel functions of the first type of zero order (dimensionless); $I_1\left(x\right)$ are Bessel functions of the first type of one order (dimensionless); $K_0(x)$ are Bessel functions of the second type of zero order (dimensionless); and $K_1(x)$ are Bessel functions of the second type of one order (dimensionless).

According to the study of Everdingen and Hurst in 1949 [1], when $r_D=1$, the relationship between water influx and pressure is:

$$\frac{1}{s^2}=s{\overline{W}}_{eD}\left(r_D=1\right){\overline{p}}_{fD}\left(r_D=1\right)$$

(16)

where $W_{eD}$ is dimensionless water influx in the time domain (dimensionless); and ${\overline{W}}_{eD}$ is dimensionless water influx in the Laplace domain (dimensionless). With the relationship between water influx and pressure (Equation (16)), the dimensionless water influx solutions of a finite aquifer, infinite edge water, and fan-shaped gas reservoir can be obtained as follows:

(1)

The dimensionless water influx in the Laplace domain of a finite aquifer is

$${\overline{W}}_{eD}=\frac{\sqrt{f(s)}}{s^{3/2}}\frac{N}{{M|}_{r_D=1}}$$

(17)

(2)

For infinite edge water, the outer boundary condition is

$$p_{fD}(r_D\to \infty ,t_D)=p_{mD}(r_D\to \infty ,t_D)=0\hspace{1em}(t_D>0)$$

(18)

Its seepage equations, initial conditions, and inner boundary conditions are consistent with the finite aquifer. Thus, the dimensionless water influx in the Laplace domain can be obtained by

$${\overline{W}}_{eD}=\frac{\sqrt{f(s)}}{s^{3/2}}\frac{K_1(\sqrt{sf(s)})}{K_0(\sqrt{sf(s)})}$$

(19)

(3)

The single-pore aquifer is a special case of this model, where f(s) = 1. The dimensionless water influx in the Laplace domain of a fan-shaped gas reservoir is

$${\left({\overline{W}}_{eD}\right)}_{\theta }={\overline{W}}_{eD}\times \frac{\theta }{{360}^{\circ }}$$

(20)

where θ is the angle of water invasion in the gas reservoir in degrees (°).

Finally, with Stehfest numerical inversion [31], the solution of dimensionless water influx $W_{eD}$ in the time domain can be obtained.

4. Results and Discussions

This section involves flow regime identification and the sensitivity study based on the dimensionless water influx solutions, which provide a theoretical base for methodology development of the water influx determination.

4.1. Flow Regimes

As it is difficult to accurately identify flow regimes of water invasion according to the double logarithmic curve of dimensionless water influx, the double logarithmic curve of dimensionless water influx derivative is firstly introduced, and the water influx derivative is defined as:

$$W_{eD}{}^{\prime }=\frac{d{W}_{eD}}{d\ln t_D}=t_D\frac{d{W}_{eD}}{d{t}_D}$$

(21)

Without loss of generality, take $r_{eD}=5$, $\mathsf{\omega }=0.01$, $\lambda ={10}^{-6}$. The double logarithmic curves of dimensionless water influx and their derivatives, as the characteristic curves of the water invasion model, are as follows.

In Figure 2, the flow regimes of water invasion in a naturally fractured gas reservoir with edge water can be identified in detail:

Regime I in Figure 2: Fracture radial flow. In this period, only the fluid in natural fractures flows toward the gas reservoir.

Regime II in Figure 2: Cross flow. “V-shape” caused by matrix supply. During depletion in natural fractures, the fracture pressure reduces over time. Once it reaches a certain value, the fluid within the matrix will enter into natural fractures.

Regime III in Figure 2: Transitional radial flow. The fluid flows from the matrix to natural fractures, and then to the gas reservoir. However, the pressure drop rate is not uniform and the flow does not reach the state of dynamic balance.

Regime IV in Figure 2: Pseudo-steady flow. The dimensionless water influx curve exhibits a horizontal line, and the flow reaches the state of dynamic balance.

In Figure 3, the flow regimes of water invasion in a homogeneous gas reservoir with edge water can be identified in detail:

Regime I in Figure 3: Radial flow. In this period, the fluid in the matrix flows toward the gas reservoir.

Regime IV in Figure 3: Pseudo-steady flow. The dimensionless water influx curve exhibits a horizontal line, and the flow reaches the state of dynamic balance.

Differences in water influx characteristic curves between a homogeneous gas reservoir and dual media gas reservoir: In water influx characteristic curves of the dual media water invasion model, there are four flow regimes of water invasion, the water influx curve is double stepped and the derivative curve has concave features. In water influx characteristic curves of the homogeneous water invasion model, there are only two flow regimes of water invasion, the water influx curve is single step, and the derivative curve has no concave features.

Note that a continuum will occur between the water influx curves of the homogeneous gas reservoir and dual media gas reservoir when the ratio of permeability between the matrix and fractures is 1. In that situation, the properties of naturally fractured gas reservoirs are similar to those of homogeneous gas reservoirs, and the shape of the water influx characteristic curves of the two are the same.

4.2. Sensitivity Study

In this section, the impacts of the aquifer and gas reservoir radius ratio $r_{eD}$, storativity ratio ω and interporosity flow coefficient $\lambda$ on dimensionless water influx transient behaviors are studied.

Without loss of generality, when ω = 0.01, $\lambda ={10}^{-6}$, $r_{eD}=\infty$, $r_{eD}=100$, $r_{eD}=20$, $r_{eD}=5$, the curves of dimensionless water influx and derivative are as follows.

In Figure 4 and Figure 5, the smaller the aquifer and gas reservoir radius ratio $r_{eD}$ is, the smaller the dimensionless water influx and its derivative are, and the shorter the duration of fracture radial flow (regime I) is, due to less fluid in natural fractures. Furthermore, the “V-shape” in the curve of the water influx derivative is more obvious during the cross flow (regime II). It can be deduced that the aquifer and gas reservoir radius ratio mainly controls the dimensionless water influx value and the depth of the “V-shape” in derivative curves.

Without loss of generality, when $r_{eD}=5$, $\lambda ={10}^{-6}$, ω = 0.01, ω = 0.1, ω = 0.5, the curves of dimensionless water influx and derivative are as follows.

In Figure 6 and Figure 7, at the early stage of water invasion, the smaller the storativity ratio ω, the smaller the value of dimensionless water influx, the shorter the duration of fracture radial flow (regime I) due to less fluid in natural fractures, and the greater the difference in height between the double steps is in the dimensionless water influx curve. For derivative curves, the smaller the storativity ratio ω, the earlier the “V-shape” appears. When it comes to pseudo-steady flow, the water influx is the same and is independent of the ω.

It can be seen that the storativity ratio ω mainly affects the two-step height of the curve of dimensionless water influx and the appearance time of the “V-shape” in the curve of the water influx derivative.

Without loss of generality, when $r_{eD}=5$, ω = 0.01, $\lambda ={10}^{-9}$, $\lambda ={10}^{-6}$, $\lambda ={10}^{-3}$, the curves of dimensionless water influx and its derivative are as follows.

In Figure 8 and Figure 9, the smaller the interporosity flow coefficient $\lambda$ is, the longer the duration of cross flow (regime II). This is because the smaller the interporosity flow coefficient, the smaller the cross fluid flux from the matrix to fractures. As a result, the double steps in the dimensionless water influx curve and the “V-shape” in the derivative curve are more obvious.

Thus, it can be found that the interporosity flow coefficient $\lambda$ mainly influences the depth of the double steps in the dimensionless water influx curve and that of the “V-shape” in the derivative curve.

5. Methodology

Based on the above flow regime identification and sensitivity study, a new global search algorithm is proposed to calculate the water influx, and the calculated process is as follows.

(1) In general, the radius of the gas reservoir and the radius of the aquifer are unknown. Assume the value of the gas reservoir radius $r_g$ and the aquifer and gas reservoir radius ratio $r_{eD}$. In addition, obtain the dimensionless parameters from the field data.

(2) Calculate the dimensionless water influx in the Laplace domain with the dimensionless parameters. With Stehfest numerical inversion [31], the solution of dimensionless water influx $W_{eD}$ in the time domain can be obtained. The calculation details are shown as follows.

$${\overline{W}}_{eD}=\frac{\sqrt{f(s)}}{s^{3/2}}\frac{N}{{M|}_{r_D=1}}$$

(22)

where

$$f(s)=\mathsf{\omega }+\sqrt{\frac{\lambda \left(1-\mathsf{\omega }\right)}{3s}}\tanh \left(\sqrt{\frac{3\left(1-\mathsf{\omega }\right)s}{\lambda }}\right)$$

(23)

$$\{\begin{array}{l}M=I_1\left(r_{eD}\sqrt{sf(s)}\right)K_0\left(r_D\sqrt{sf(s)}\right)+I_0\left(r_D\sqrt{sf(s)}\right)K_1\left(r_{eD}\sqrt{sf(s)}\right)\\ N=I_1\left(r_{eD}\sqrt{sf(s)}\right)K_1\left(\sqrt{sf(s)}\right)+I_1\left(\sqrt{sf(s)}\right)K_1\left(r_{eD}\sqrt{sf(s)}\right)\end{array}$$

(24)

(3) Obtain water influx according to the study of Everdingen and Hursr [1].

$$W_e=A_{g}h{\varphi }_f{c}_{tf}{\displaystyle {\int }_0^t\frac{\partial \Delta p}{\partial t^{\prime }}{W}_{eD}\left(t-t^{\prime }\right)d{t}^{\prime }}\approx A_{g}h{\varphi }_f{c}_{tf}{\displaystyle \sum _{j=0}^t\Delta p_j{W}_{eD}\left(\Delta t_{Dj},r_D\right)}$$

(25)

The dimensionless water influx is derived following Everdingen and Hursr [1], and the detailed derivations were provided in their work. It is written as

$$W_{eD}={\displaystyle {\int }_0^{t_D}{(r_D\frac{\partial p_D}{\partial r_D})}_{r_D=1}d{t}_D}$$

(26)

Further, one can obtain

$$W_e=2\pi r_g^{2}h{\varphi }_f{c}_{tf}\left[\begin{array}{l}\Delta p_0{W}_{eD}\left({\left(t_j-t_0\right)}_D,r_D\right)+\Delta p_1{W}_{eD}\left({\left(t_j-t_1\right)}_D,r_D\right)\\ +\dots +\Delta p_{t-1}{W}_{eD}\left({\left(t_j-t_{t-1}\right)}_D,r_D\right)\end{array}\right]$$

(27)

$$\{\begin{array}{l}\Delta p_0=p_i-{\overline{p}}_1=p_i-\frac{p_i+p_1}{2}=\frac{p_i-p_1}{2}\\ \Delta p_1={\overline{p}}_1-{\overline{p}}_2=\frac{p_i+p_1}{2}-\frac{p_1+p_2}{2}=\frac{p_i-p_2}{2}\\ \Delta p_2={\overline{p}}_2-{\overline{p}}_3=\frac{p_1+p_2}{2}-\frac{p_2+p_3}{2}=\frac{p_1-p_3}{2}\\ \Delta p_{j-1}={\overline{p}}_{j-1}-{\overline{p}}_j=\frac{p_{j-2}+p_{j-1}}{2}-\frac{p_{j-1}+p_j}{2}=\frac{p_{j-2}-p_j}{2}\end{array}$$

(28)

where $W_e$ is the water influx (m³); $W_{eD}$ is the dimensionless water influx (dimensionless); $\Delta p_0$ represents the pressure difference of the initial segment (MPa); $\Delta p_1$ represents the first pressure difference (MPa); $\Delta p_2$ represents the second pressure difference (MPa); and $\Delta p_{j-1}$ indicates the pressure difference at the (j − 1)th stage (MPa). They are shown in Figure 10.

(4) Verify the results according to the material balance equation and the error between dynamic and static geological reserves. If the linear fitting correlative coefficient of the material balance curve is greater than 0.95, and the error between dynamic and static geological reserves is less than 0.2, the results can be accepted. Otherwise, new values of the gas reservoir radius $r_g$ and aquifer and gas reservoir radius ratio $r_{eD}$ will be reassumed, and recalculate the water influx until the requirements are met. The material balance equation of water-drive gas reservoirs [32] is shown as follows.

$$\frac{p}{Z}(1-c_c\Delta P-e)=\frac{p_i}{Z_i}(1-\frac{G_p}{G})$$

(29)

$$c_c=\frac{c_p+s_{wc}{c}_w}{1-s_{wc}}$$

(30)

$$e=\frac{W}{V_{gi}}=\frac{W_e-W_p{B}_w}{A_{g}h\varphi (1-s_{wc})}$$

(31)

where $Z$ is the compression factor (dimensionless); $\Delta p$ represents the pressure difference (MPa); $G_p$ is the cumulative gas flux (m³); G is the gas reservoir dynamic geological reserves (m³); $c_c$ is defined as gas reservoir volume compressibility (MPa⁻¹); $c_p$ is rock compressibility (MPa⁻¹); $c_w$ is formation water compressibility (MPa⁻¹); $s_{wc}$ is irreducible water saturation (fraction); e is the volume coefficient of stored water, that is, the percentage of water volume left in the volume of gas reservoir (fraction); We is the water influx (m³); Wp is the cumulative water flow flux on the ground (m³); and W is the volume of the water left in the formation (m³). Define the H pressure of the gas reservoir as $p_H$.

$$p_H=\frac{p}{Z}(1-c_c\Delta p-e)$$

(32)

$$p_{Hi}=\frac{p_i}{Z_i}$$

(33)

where $p_{Hi}$ is the H pressure of the gas reservoir under original conditions (MPa). Then the material balance equation for water-drive gas reservoirs becomes:

$$p_H=p_{Hi}(1-\frac{G_p}{G})$$

(34)

The material balance curve of $p_H$ and $G_p$ is used to verify the accuracy of the gas reservoir radius $r_g$ and aquifer and gas reservoir radius ratio $r_{eD}$ and water influx, which is shown in Figure 11.

If the linear fitting correlative coefficient of the material balance curve of $p_H$ and $G_p$ is greater than 0.95, the linear regression equation of the $p_H$ and $G_p$ can be obtained.

$$p_H=a+b{G}_p$$

(35)

where a and b are constants (dimensionless). The dynamic geological reserves of gas reservoir G are obtained.

$$G=\left|\frac{a}{b}\right|$$

(36)

where G is the dynamic geological reserve of the gas reservoir (m³).

The static geological reserves $G_s$ are calculated according to the volume method.

$$B_g=\frac{p_{sc}}{Z_{sc}{T}_{sc}}\frac{ZT}{p}$$

(37)

$$G_s=\frac{2\mathsf{\pi }{r}_g^{2}h\varphi (1-s_{wc})}{B_g}$$

(38)

where $B_g$ is the gas volume factor (m³/m³); $G_s$ is the static geological reserve of the gas reservoir, (m³); T is the gas reservoir temperature (K). Subscript “sc” tables ground conditions.

The error between dynamic and static geological reserves is

$$\Delta \delta =\frac{G_s-G}{G}$$

(39)

where $\Delta \delta$ is the error between dynamic and static geological reserves (decimal).

(6) Obtain the dimensionless water invasion characteristic curves according to the value of the gas reservoir radius $r_g$ and aquifer and gas reservoir radius ratio $r_{eD}$. Then, combined with dimensionless time, the flow regime of water invasion can be identified.

6. Field Application

In this section, a naturally fractured edge water gas reservoir Z1 was chosen for field application by using the proposed methodology.

The data of the reservoir and fluid are collected in

Table 1. Data parameters of gas reservoir Z1.

Parameters	Values	Units
Reservoir thickness	27	m
Irreducible water saturation	0.32	decimal
Formation water viscosity	0.45	mPa·s
Formation water volume coefficient	1.01	m3/m3
Storativity ratio	0.1	dimensionless
Interporosity flow coefficient	10−6	dimensionless
Pseudo-critical pressure	4.67	MPa
Pseudo-critical temperature	195.15	K
Initial formation pressure	29.7	MPa
Initial formation temperature	364.15	K
Porosity	0.1	decimal
Natural fracture permeability	1	mD
Matrix permeability	0.01	mD
Formation water compressibility	0.0046	MPa−1
Rock compressibility	0.000435	MPa−1

, and its production dynamic data is shown in

Table 2. Production dynamic data of gas reservoir Z1.

t/mon	p/MPa	Gp/108 m3	Wp/104 m3
0	29.7	0	0
12	27.4	6.88	0
20	25	17	120
28	23.7	27.5	180
36	21.3	37.3	260.7
48	19.4	51.5	500
60	17.6	61.6	1050

. In Table 1, the reservoir thickness, irreducible water saturation, and porosity are obtained by well logging. The formation water viscosity, formation water volume coefficient, pseudo-critical pressure, pseudo-critical temperature, and formation water compressibility are measured by PVT (pressure vs. temperature) experiments. Rock compressibility can be obtained by the rock compression test. The storativity ratio, initial formation pressure, initial formation temperature, natural fracture permeability, and matrix permeability are acquired by the well test.

Where bot $G_p$ is the cumulative gas flux (10⁸ m³); and $W_p$ is the cumulative water flux (10⁴ m³). According to the statistical formula, the compression factor Z is calculated.

$$Z=1-\frac{3.52p_r}{{10}^{0.9813T_r}}+\frac{0.274p_r^2}{{10}^{0.8157T_r}}$$

(40)

where Z is the compression factor (dimensionless); $p_r$ is pseudo reduced pressure (dimensionless); and $T_r$ is the pseudo reduced temperature (dimensionless).

The new global search algorithm is used to calculate water influx. Set the value of $r_g$ to 100, 200, ..., 5000, and the value of $r_{eD}$ is 1, 2, ..., 40. After 2000 iterations, the optimal gas reservoir radius $r_g$ is found to be 3600 m, and the optimal aquifer and gas reservoir radius ratio $r_{eD}$ is 2. The material balance curve of $p_H$ and $G_p$ is used to verify the results, which are shown in Figure 12.

The correlative coefficient of linear fitting is 0.9838, which meets the requirement. Combined with the linear regression equation, the dynamic geological reserves of gas reservoir G are obtained.

$$G=\frac{28.87}{0.1742}=165.7({10}^8{\mathrm{m}}^3)$$

(41)

The static geological reserves $G_s$ are calculated according to the volume method.

$$B_g=\frac{p_{sc}}{Z_{sc}{T}_{sc}}\frac{ZT}{p}=0.004236({\mathrm{m}}^3{/\mathrm{m}}^3)$$

(42)

$$G_s=\frac{2\mathsf{\pi }{r}_g^{2}h\varphi (1-s_{wc})}{B_g}=174.1({10}^8{\mathrm{m}}^3)$$

(43)

The error between dynamic and static geological reserves is

$$\Delta \delta =\frac{G_s-G}{G}=0.0507$$

(44)

where $\Delta \delta$ is the error between dynamic and static geological reserves (decimal). The error meets the engineering requirement, and the results are accepted. Therefore, the gas reservoir radius $r_g$ is 3600 m, and the aquifer and gas reservoir radius ratio $r_{eD}$ is 2. Furthermore, water influx and corresponding dimensionless time can be obtained as shown in

Table 3. Water influx of gas reservoir Z1.

We/104 m3	${\mathit{t}}_{\mathit{D}}$	${\mathit{p}}_{\mathit{H}}$
0.00	0.000	29.62
0.39	0.011	27.51
1.17	0.018	25.37
2.09	0.025	24.14
3.18	0.032	21.71
4.99	0.042	19.96
6.88	0.053	18.63

.

The water invasion characteristic curves can be obtained, which are shown in Figure 13.

According to the dimensionless time and water invasion characteristic curves of gas reservoirs, it can be judged that the flow regime of the gas reservoir is the early state of cross flow and the water influx is small. If the gas flow flux is too large to make water influx increase sharply, it is probable that the gas well will be flooded. At this time, a reasonable gas well production working system is needed to extend the dry production time and improve gas reservoir recovery.

7. Conclusions

In this paper, a dimensionless water influx derivative curve is first introduced, which is used to identify flow regimes of water invasion. Then, a sensitivity analysis is performed to study the impacts of key factors on flow regimes, including formation pore type, the aquifer and gas reservoir radius ratio, the storativity ratio, and the interporosity flow coefficient. Based on the sensitivity study and material balance equation, a new method for determining water influx in naturally fractured reservoirs is developed. Some important conclusions are drawn:

• After analyzing the water influx characteristic curves of naturally fractured reservoirs, there are four flow regimes, the water influx curves are double stepped and a “V-shape” appears in the derivative curves. For the water influx characteristic curves of homogeneous reservoirs, there are only two flow regimes, the water influx curves are single stepped and the derivative curves have no “V-shape”.
• With the decrease in the storativity ratio ω, the “V-shape” in the derivative curve appears earlier, and the difference in height between the double steps in the dimensionless water influx curve is greater.
• The smaller the interporosity flow coefficient $\lambda$ is, the more obvious the double steps in the dimensionless water influx curve and the more apparent the “V-shape” in the derivative curve are.
• The smaller the aquifer and gas reservoir radius ratio $r_{eD}$ is, the more obvious the “V-shape” in the curve of the water influx derivative during the cross flow is, and the smaller the dimensionless water influx and its derivative are. These features provide meaningful tools to quantify the water influx.
• Water influx, gas radius, aquifer radius, dynamic geological reserves, and the regime of water invasion can be obtained by the global search algorithm. The results from the case study were verified by analyzing the error between dynamic and static geological reserves, and the error met the engineering accuracy, which indicates that the accuracy of the proposed method is acceptable.

This is our primary work about water influx, and more efforts will be made to focus on (1) incorporation of other approaches, like automatic inversion, and (2) global sensitivity analysis [33] to avoid a risk of loss of generality, and (3) type-curve matching by using the water influx curves.