Analysis of Production Decline Type Curves Considering Water Influx in Naturally Fractured Gas Reservoirs

Water invasion has always been a significant issue in gas reservoir production [1], with numerous gas reservoirs worldwide encountering this phenomenon during production [2,3]. The topic of water invasion in gas reservoirs has been extensively discussed in various scholarly articles, primarily focusing on control methods of water invasion, determination of the water body size, and prediction of water invasion. For instance, Roozshenas et al. [4] have shed light on the challenges and concepts related to water invasion control. They propose solutions anchored in water sources, water production diagnosis, and actual well data. Zhang et al. [5] introduced a search algorithm leveraging material balance and static geological reserve errors to estimate the size of water invasion in dual-medium gas reservoirs. Feng et al. [6] argue that comprehensive water control hinges on understanding the dynamics of water invasion, predicting the efficacy of control measures, and employing precise, high-quality methods and technical support from various fields.

Therefore, it is essential to diagnose and evaluate gas reservoir water invasion in order to achieve water invasion control. The main methods for diagnostic evaluation are dynamic analysis [7–9] and numerical analysis [10–12]. This article primarily focuses on analyzing and understanding the dynamics of water intrusion. An important preliminary task before dynamic analysis is to determine the approximate geological reserves [13–15]. The reserves of gas reservoirs can be determined through the material balance method [16–18], and the accuracy of this determination has a significant impact on Blasingame production decline analysis [19] and evaluation [20,21]. Scholars have progressively refined mathematical models for water invasion in reservoirs [22–25], which are currently in use despite their limited applicability [26]. This article aims to integrate these mathematical models to develop a dual porosity gas reservoir water invasion model suitable for the Blasingame analysis method.

The advanced Blasingame production decline method is a widely used [27–29] dynamic analysis technique for assessing reservoir physical properties, water invasion volume, and gas storage capacity. Doublet and Blasingame [30] proposed a water invasion Fetkovich decline analysis model [24] for vertical well reservoirs and developed Fetkovich decline type curves. Wei et al. [31] established a water invasion Blasingame curve based on Doublet’s model and analyzed the corresponding flow stages and influencing factors, and the division of flow stages [32] is of great significance in the rate-transient analysis [33] of oil and gas reservoirs. However, they did not consider the impact of dual porosity on water influx in gas reservoirs. In addition to the Blasingame curves, Ilk et al., Idorenyin et al., and Shahamat et al. [34–36] have introduced a $β$-derivative curve that can be utilized for production data analysis/interpretation by diagnosing production and pressure decline. This curve possesses unique identifiable features for each flow state, enhancing its utility in the field.

In this study, a transient flow model for vertical well in gas reservoirs was developed to account for dual-medium characteristics. The real-space production solution was obtained through Laplace transformation, the superposition principle [25], and the Stehfest method [37]. Additionally, modified Blasingame curves were drawn by combining with Blasingame theory, including the $β$ derivative curve to enhance analyzability. By pairing the material balance equation for water invading gas reservoirs with Blasingame theory and actual production data, normalized production curves were drawn for such reservoirs. These normalized production curves illustrate six features: early unsteady flow regime (EUFR), fracture radial flow regime (FRFR), interporosity flow regime (IPFR), primary depletion flow regime (PDFR), second unsteady flow regime (SUFR), and system pseudo-steady flow regime (SPSF). The characteristics of different formation parameters in the curves can be described through model verification and sensitivity analysis of theoretical curves. Ultimately, the dual porosity gas reservoir model was validated with regard to water intrusion, utilizing data from an X2 well that satisfied the stipulated conditions. The model demonstrated excellent applicability to the dynamic analysis of this particular gas well due to the unique production regime, reservoir, and geological attributes of this well, including the assumptions of a homogeneous reservoir with water intrusion boundary conditions, location in the center of the reservoir, and dual media. It is also acknowledged that the model has certain limitations. Further validation using a wider range of gas well data is required to assess the general applicability of the model. In addition, relaxation of some assumptions is necessary to accommodate the analysis of a more diverse range of well scenarios.

A vertical well is located at the center of a bounded circular reservoir with thickness $h$⁠, as illustrated in Fig. 1 The physical model includes the following details:

1. The gas reservoir is composed of matrix and fractures, representing a dual porosity and fracture flow system.

2. The gas reservoir has a radius $re$⁠, while the wellbore has a radius $rw$⁠; both the upper and lower boundaries are impermeable.

3. The characteristic parameters of the reservoir (e.g., thickness, permeability, porosity, and initial pressure) remain constant.

4. Fluid flow follows Darcy’s law, with the fluid considered as single phase.

5. Water influx at the outer boundary conforms to the “ramp” rate water influx proposed by Doublet and Blasingame [30], starting from zero and gradually increasing.

6. Thermal transfer and gravity effects are disregarded in the porous media flow.

Doublet and Blasingame proposed that the water influx at the external boundary condition can be categorized into two conditions: the “step” rate condition and the “ramp” rate condition. In most natural water influx scenarios, the energy of water is finite, indicating a more likely occurrence of a “ramp” rate condition. Therefore, as gas well production progresses, the interior of the gas reservoir gradually becomes depleted, leading to a slow increase in boundary gas flowrate from zero to a stable value. The equation expression for the “Ramp” water influx condition is depicted in Fig. 2 as follows:

The impact of varying parameters $qDext,∞$ and $tDstart$ on the curve was investigated accordance with Eq. (9). The parameter $qDext,∞$ was set to values of 0.1, 0.3, 0.5, 0.7, and 0.9, while $tDstart$ was held constant at $103$⁠. As depicted in Fig. 2(a), it is evident that $qDext,∞$ has a significant influence on both the final value of $qDext$ and the rate of increase of the curve. Larger values of $qDext,∞$ lead to a higher final $qDext$ value and a faster rate of increase, although all curves rise over the same duration. In Fig. 2(b), $qDext,∞$ was kept constant, while $tDstart$ varied as $10,102,103$⁠, and $104$⁠. The curves were observed to start rising at different times but increased at the same rate before reaching the same final $qDext$ value. Through this analysis, the distinct impacts of parameters $qDext,∞$ and $tDstart$ on modifications to the curve can now be clearly understood.

Based on the theory mentioned above, modified Blasingame production decline type curves can be obtained by considering dual porosity and water intrusion. In addition, in order to analyze the actual gas well production decline, it is necessary to calculate material balance pseudo time, normalized production rate, normalized cumulative production rate, and normalized integral derivative cumulative production rate using actual production history data (such as time, production rate, and bottom-hole pressure).

Normalized pseudo-pressure is defined as follows:

Normalized production rate is defined as follows:

Production data are processed through normalization and other methods to obtain the normalized production rate, normalized cumulative production rate, and normalized integral derivative cumulative production curves. By matching the Blasingame production decline curves between theoretical and actual curves, it is possible to determine the water flux rate, the beginning time of water flux evaluation, and the original gas in place.

By setting the dimensionless maximum water influx (⁠$qDext,∞$⁠) to 0 (Fig. 3(a)) and 0.7 (Fig. 3(b)), the elastic storativity ratio (⁠$ω$⁠) to 1, the model presented in this article can be simplified to that of Wei’s model, allowing for validation of the current work. When the dimensionless maximum water influx (⁠$qDext,∞$⁠) is set to 0 in Fig. 3(a), the curves are seen to be almost identical. However, in Fig. 3(b) where the dimensionless maximum water influx is set to 0.7, while $qDd$ and $qDdi$ match, $qDdid$ is positioned slightly upward, albeit derived from different algorithms. Consistency with Wei’s model is demonstrated, serving to validate the approach presented in this work.

By the way, the curves with no water influx, water influx, and dual porosity water influx are drawn (Fig. 4). The water influx curves has almost the same part in the early $tcaD$ period, while time is approaching dimensionless beginning time of the water influx, the curves will start to separate. The slope of the water influx curve $qDd$ will gradually increase and then gradually decrease to –1, which is parallel to the no water influx curve $qDd$⁠.

The dual porosity waterflood curves (Fig. 4) can be divided into six regimes: (i) EUFR, which means that pressure begins to spread away from the wellbore; (ii) FRFR has a slope of 0 on the $qDdid$ curve; (iii) IPFR, the curve of $qDdid$ has concave characteristics; (iv) the PDFR; (v) SUFR, water influx, or waterflood support external pressure; and (vi) SPSF.

For the convenience of sensitivity comparison analysis, the default parameters are set for calculating the Blasingame production decline type curves: elastic storativity ratio (⁠$ω$⁠) is 0.1; interporosity flow coefficient (⁠$λ=0.001$⁠); and dimensionless external boundary radius (⁠$reD$⁠) is 100. The water flux type is ramp and dimensionless beginning time of the water influx (⁠$tDstart$⁠) is 50. Dimensionless maximum water influx (⁠$qDext,∞$⁠) is 0.8.

Figure 5 shows the dimensionless external boundary radius $reD$ has a great effect on the I stage. From Fig. 5, as $reD$ increases, the value of the curves decreases during the I period. The physical significance of the $β$ derivative curve is that when the influence of dual media is considered, the formation pressure will decrease faster and more rapidly, and the production will begin to decline earlier.

The influence of $ω$ is mainly reflected in the I, II, and III periods (Fig. 6(a)), but its impact will be more complex during the III period. When $ω$ is smaller, the value of the curves decreases, and the $qDdid$ curve will have more obvious concave characteristics.

The effect of varying $λ$ in Fig. 6(b) appears perplexing. Therefore, the $qDdid$ curve and $β$ derivative curve were separately extracted and analyzed in Figs. 6(c) and 6(d), as they are more illustrative. As $λ$ decreases, Fig. 6(c) shows the $qDdid$ curve becoming concave at progressively later times. Similarly, the concavity onset of the $β$ derivative curve in Fig. 6(d) is seen to be increasingly delayed as $λ$ decreases. By isolating these two key curves, the impact of decreasing $λ$ could be more clearly observed, with both curves exhibiting postponed inflection points at lower $λ$ values.

The effect of $qDext,∞$ is shown in Fig. 7(a), the bigger $qDext,∞$ the slope of curves (⁠$qDd$⁠, $qDdi$⁠, $qDdid$⁠) will take longer to recover to –1 until the slope reaches 0 when $qDext,∞$ equals 1. The $β$ derivative curves will form a concave deeper and deeper until it never recovers.

The effect of $tDstart$ is obvious, and as $tDstart$ increases, all curves will reflect the water influx characteristics later.

X2 is a vertical gas well situated at the approximate center of a circular reservoir comprising sandstone as the reservoir lithology and black shale as the main lithology. According to the results of mud logging and well logging interpretations, this gas formation exhibits fracture porosity. The reservoir gas phase is predominantly methane, with a relative density of 0.58, while the aqueous phase has a pH of 8.23 and a relative density of 1.00. The developed dual porosity water influx model was applied to the gas well X2, with the relevant field data presented in Table 2 and the production data with water included in Fig. 8.

First, the gas field was assumed to be homogeneous, yielding the matched curves shown in Fig. 9(a). Considering dual porosity characteristics, matched curves were subsequently obtained as depicted in Fig. 9(b). As evident from poor fits achieved using homogeneous models at I, II, and III periods, the dual porosity approach presented here was anticipated to perform better. The results, shown in Table 3, confirm that modeling the field as a dual porosity system improved history matching.

While the water invasion model proposed in this study demonstrates satisfactory performance in the validation of a single actual well, it is important to acknowledge the existence of certain limitations.

1. The model was validated using a single-well dataset, which limits its applicability. To evaluate the broad applicability of the model, it is necessary to expand the coverage of the validation data to include different types of reservoirs and geological conditions.

2. The model is based on a number of assumptions, including the assumption of reservoir homogeneity, the specification of boundary conditions, and the identification of well locations. These assumptions may have an impact on the accuracy of the model. In future work, we will attempt to relax these assumptions in order to enhance the model’s applicability. For instance, further work is required for the models of fractured vertical wells and fractured horizontal wells, as well as for the case where the reservoir boundary is rectangular.

3. In this study, only the effect of water influx on the production decay curve is considered. However, there may be other complex factors in actual reservoirs, such as non-Darcy flow, capillary force, and stress sensitivity of the reservoir, which may lead to changes in the reservoir characteristic parameters during the production process. Further investigation of these factors is required.

4. The determination of the model parameters is contingent upon the availability of actual well data, and the predictive capacity of the model may be constrained in the absence of such data. Further improvements to the model and an expansion of the validation range will be made in future research, with the aim of enhancing its applicability and reliability.

In this study, a typical curve model of declining production for water influx in vertical wells was established. This was done by applying Darcy’s law, Duhamel’s principle, the material balance equation of water intrusion, and Blasingame’s theory of diminishing production analysis. The model was validated through a comparison process, and a sensitivity analysis was conducted. To assess the model’s applicability, a gas well that met the model’s assumptions was selected for analysis. The principal conclusions may be summarized as follows:

1. New Blasingame production declining type curves for water influx into naturally fractured reservoirs are plotted by combining beta inverse curves and conventional Blasingame curves. The flow characteristics of the curves were used to classify them into six distinct flow regimes: EUFR, FRFR, IPFR, PDFR, SUFR, and SPSF.

2. The influence of the dimensionless external boundary radius is primarily evident in the early flow regime. As the dimensionless external boundary radius increases, the early curve exhibits a more pronounced downward curvature. The effect of the elastic storativity ratio is also predominantly observed in the early stage. In general, a larger elastic storativity ratio results in a more pronounced downward concave trend of the early curve. The exponentially decreasing interporosity flow coefficient causes the concave curve, which is characteristic of dual media, to emerge later in the process.

3. For the water influx related parameters, the impact of maximum water influx is more pronounced in the subsequent phase, as evidenced by the lower gradient of the curve when the maximum water influx is substantial and the duration of the SUFR phase is prolonged. The onset of water influx primarily influences the emergence of the SUFR stage, with a longer initial influx delaying the onset of the SUFR stage.

4. The new Blasingame yield decrement type curve for fractured reservoirs with water inflow, proposed in this study, has been validated using data from a single field. This validation demonstrates that the curve can be used to assess the parameters of dual porosity and fracture flow regime, water flux, and water inflow time for that particular field. Further validation using additional field data is required to assess the wider applicability of this type curve. In the future, the model will need to consider corrections for additional influences.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

$h$ =

thickness of reservoir (m)

$p$ =

pressure (MPa)

$r$ =

radius (m)

$t$ =

time (day)

$K$ =

permeability (mD)

$V$ =

volume of natural gas reservoir (⁠$m3$⁠)

$p0$ =

atmospheric pressure (MPa)

$qDext,∞$ =

maximum water flux, which does not exceed the well production (⁠$m3/d$⁠)

$qext(t)$ =

water flux at the external boundary (⁠$m3/d$⁠)

$re$ =

reservoir radius (m)

$rw$ =

well radius (m)

$swi$ =

the saturation of formation bound water (%)

$Cf$ =

the rock compression coefficient (⁠$MPa−1$⁠)

$Ct$ =

the total compression coefficient (⁠$MPa−1$⁠)

$Cw$ =

the water compression coefficient (⁠$MPa−1$⁠)

$Gp$ =

the cumulative gas production (⁠$m3$⁠)

$We$ =

the amount of water invasion (⁠$m3$⁠)

$Wp$ =

the cumulative water production (⁠$m3$⁠)

$α$ =

shape factor

$λ$ =

interporosity flow coefficient

  $μ$ =

fluid viscosity (mPa s)

$ϕ$ =

porosity

$ψ$ =

reservoir pseudo-pressure

$ω$ =

elastic storativity ratio

$s$ =

laplace space operator

$z$ =

deviation factor

$B$ =

volume coefficient

$G$ =

the geological reserves of gas (⁠$m3$⁠)

$tDstart$ =

beginning time of the water flux

f =

fracture

g =

gas

i =

formation initial conditions

m =

matrix

 w =

formation water

D =

dimensionless

- =

Laplace transform