A Dynamic Reserve Evaluation Method for an Ultra-Deep Fractured Tight Sandstone Gas Reservoir

Abstract

Dynamic reserves evaluation is crucial for the development and assessment of gas reservoirs. However, ultra-deep fractured tight sandstone gas reservoirs pose unique challenges due to their abnormally high pressure, tight matrix, and complex fracture distribution. This paper proposes a simplified forward calculation method that incorporates the fracture length for the rapid estimation of dynamic reserves in fractured tight sandstone gas reservoirs. This method was based on the pressure change rate equation and considered the unique characteristics of fractured gas reservoirs. Numerical simulations were conducted to analyze the sensitivity of the proposed method. The proposed method was applied to estimate the dynamic reserves of a fractured gas reservoir, and the results closely approximate the well group method, indicating its accuracy. The main advantage of this method lies in its simplicity, allowing field engineers to perform rapid dynamic reserve evaluations.

1. Introduction

Fractured tight sandstone gas reservoirs play a crucial role in the global oil and gas production and reserves. These reservoirs are characterized by a tight lithology, poor petrophysical properties, and strong rock heterogeneity, with porosities ranging from 2% to 10% and permeabilities in the range of (0.04–1) $\times {10}^{-3}\mathsf{\mu }{\mathrm{m}}^2$. They are extensively fractured, which sets them apart from conventional gas reservoirs in terms of geological characteristics [1,2,3].

Ultra-deep fractured gas reservoirs play a significant role in gas field development in China, for example, the Tarim Kuqa Depression has demonstrated natural gas reserves of $1.47\times {10}^{12}$ m² and a built-up natural gas production capacity of $100\times {10}^8$ m². It serves as a major gas source for China’s “West-East Gas Transmission Project”. These reservoirs are characterized by depths exceeding 6500 m, falling into the category of ultra-deep formations. As shown in Figure 1, the reservoir matrix is tight, but faults and fractures are extensively developed, resulting in permeability that is significantly higher than that of the matrix [4,5].

Despite the significant improvement in reservoir permeability due to fractures, the exploration and development of these gas reservoirs face significant challenges in terms of the reservoir fluid properties, gas reserves, well productivity, and evaluation and formulation of development plans [6].

Reserves are the basis of gas field development, and the remaining reserves are an important indicator for assessing the current development status of a gas field. The remaining reserves numerically equal the difference between the recoverable reserves and cumulative gas production from the reservoir. Estimation of the dynamic reserves is one of the core elements in assessing the remaining gas reserves and potential, evaluating the effectiveness of reservoir development, predicting development dynamics, and adjusting development plans [7,8,9].

Currently, the methods for estimating the dynamic reserves of gas reservoirs can be broadly categorized into three types: material balance methods, modern decline curve methods, and well-testing-related methods [10,11,12]. However, the geological conditions of ultra-deep fractured tight sandstone gas reservoirs differ significantly from conventional gas reservoirs, leading to substantial errors and uncertainties when applying conventional dynamic reserve evaluation methods to the assessment of dynamic reserves in ultra-deep fractured tight sandstone gas reservoirs [13,14,15]. Therefore, this paper proposes an alternative formulation to estimate the dynamic reserves of fractured tight sandstone gas reservoirs.

This paper is organized into four sections. The first part briefly introduces the research status of dynamic reserves in conventional gas reservoirs and highlights the issues in evaluating the dynamic reserves of fractured tight sandstone gas reservoirs. The second part presents a simplified novel method for calculating fracture-controlled dynamic reserves and validates the theoretical model using numerical simulations. Then, the proposed method is applied to estimate the dynamic reserves of a specific fractured tight sandstone gas reservoir, and a comparison is made with the well group method to demonstrate the accuracy and effectiveness of the proposed method. The final part summarizes the research content and presents the relevant conclusions.

2. Literature Review

This section introduces the current research on dynamic reserves evaluation, discussing the research status and the existing issues related to fractured tight sandstone gas reservoirs.

Dynamic reserves are important indicators for technical policy formulation and mid-to-late-stage development adjustments in gas field development. Dynamic reserves refer to the reserves estimated using dynamic methods based on production and pressure data from individual wells or the entire reservoir under the assumption of unchanged process technology and well network development. Compared with the static reserves estimated by conventional volumetric methods, dynamic reserves are typically smaller.

Existing methods for evaluating dynamic reserves in gas reservoirs mainly include material balance methods, rate transient analysis methods, and well-testing methods [10,11,12,16]. However, these methods have several issues when applied to the estimation of dynamic reserves in ultra-deep fractured tight sandstone gas reservoirs.

First, the material balance method assumes that the reservoir maintains thermodynamic equilibrium during the development process, and the pressure at each point is in equilibrium. Gas reservoirs are generally classified as undersaturated reservoirs, closed reservoirs, or water-driven reservoirs. According to the different types of gas reservoirs, the actual reservoir is simplified as one or more closed or water-invaded containers. During the development process, the volume changes of gas and water follow the principle of material conservation, thus establishing material balance equations. This type of method uses pressure data to estimate geological reserves and treats the cumulative production at zero reservoir pressure as the geological natural gas reserves [11]. The average reservoir pressure is a key parameter for estimating dynamic reserves in the material balance method. Most gas reservoir evaluation methods are based on the material balance method, but these methods require accurate reservoir pressure information. Shut-in pressure tests can provide accurate pressure data for gas reservoirs, but they have high costs and can impact production. Therefore, shut-in pressure tests are rarely conducted in actual production processes. This method requires relatively simple parameters but has certain uncertainties, for example, the measurement errors caused by the formation heterogeneity and fluid composition variations, and data quality and noise errors caused by factors such as the gauge resolution and well-bore instabilities [17,18,19].

Mattar et al. [20,21] proposed the flowing material balance method for water-free gas reservoirs. This method can only be used for constant-rate production cases. For gas wells, due to the neglect of gas property parameters (viscosity, deviation factor, compressibility factor, etc.) changing with pressure, there are theoretical deficiencies and it is difficult to obtain convincing results.

Subsequently, Mattar et al. proposed the dynamic material balance method, which is derived based on the “steady flow” solution of the gas flow model and the material balance equation for undersaturated reservoirs. It introduces pseudo-pressure and pseudo-time and considers the variation in gas properties with pressure. Compared with the previous flowing material balance method, it has a more rigorous theoretical basis. However, this method does not consider the compressibility characteristics of bound water and rock pores [22,23].

In contrast, decline curve analysis methods use daily production data from individual wells to analyze the production decline, including the classical Arps decline method [16], Fetkovich type curve-fitting method [10], modern Blasingame method [24,25], Agarwal–Gardner method [26], and flowing material balance method (FMB) [27]. These methods are based on the unsteady-state flow theory for homogeneous formations, which involves converting and analyzing flowing pressure and production data to establish variable rate–variable pressure plots. Analysis techniques, such as characteristic curves, are employed to determine reservoir flow parameters and estimate dynamic reserves [10,12,25]. These methods possess relatively high accuracy but require sufficient data support. However, the decline curve analysis methods do not account for the abnormal high-pressure characteristics of ultra-deep gas reservoirs, which are typically abnormally high-pressure gas reservoirs where the compressibility of rocks, pores, and bound water cannot be neglected.

The applicability of conventional methods for ultra-deep fractured tight sandstone gas reservoirs with high formation pressure and a tight matrix is limited. The key challenges include the variations in reserve estimation due to different levels of reservoir development, complex fluid flow between the tight matrix and fracture system, and the difficulty in accurately determining the compressibility factor that varies with pressure.

In the early development stage of ultra-deep fractured tight sandstone gas reservoirs, the production time is short and the degree of reservoir pressure depletion is low. The pressure–volume curve $(p/Z)$ does not exhibit inflection points, and there is a large error in the dynamic reserves estimation. According to literature statistics, when the degree of reservoir pressure depletion exceeds 30%, the error in the dynamic reserves estimation is relatively small. However, if the reservoir has a low degree of pressure depletion and the $(p/Z)$ curve does not exhibit inflection points, there is a large error in the dynamic reserves estimation.

In the middle and later stages of the gas reservoir development, the production time is long, and the dynamic reserves estimation is greatly affected by the matrix gas supply and water invasion. The matrix permeability has a significant impact on the recoverable reserves, especially when the matrix permeability is extremely low (below $0.05$ mD), and the fracture density has a significant impact on the recoverable reserves, while the fracture dip angle has a minor effect [9]. Additionally, the matrix gas supply acts as an energy supplement to the reservoir, causing a deviation in the $p/Z$ curve in the late stage. Numerical simulation analysis shows that when there is a significant disparity in the permeability between the matrix and fractures, a 20–30% error in the dynamic reserves calculation can occur. Thus, applying conventional material balance methods to analyze dynamic reserves in fractured tight sandstone gas reservoirs can result in significant errors.

Furthermore, the determination of the compressibility factor is difficult and the variation in the compressibility factor with pressure poses challenges for the evaluation of dynamic reserves in ultra-deep fractured tight sandstone gas reservoirs. The rock compressibility factor $\left(C_f\right)$ obtained from core testing represents the compressibility of the matrix. When fractures are present, $C_f$ varies significantly, and the overall compressibility of the gas reservoir greatly changes. Therefore, the calculated effective compressibility $\left(C_e\right)$ based on core test data is smaller than the actual Ce of the gas reservoir [28,29,30].

Currently, most dynamic reserve estimation methods are inverse methods, where the dynamic reserves are estimated based on production and pressure data rather than the reservoir parameters. Although these methods are widely applied in estimating dynamic reserves for conventional gas reservoirs, they exhibit significant uncertainties when estimating the dynamic reserves for fractured tight sandstone gas reservoirs, as mentioned above. For example, the material balance method incurs problem when the production time is short and the degree of reservoir pressure depletion is low, and the decline curve analysis methods have large uncertainties for the abnormal high-pressure characteristics of ultra-deep gas reservoirs, where the compressibility of rocks and pores cannot be neglected. Therefore, this paper introduces a forward-modeling approach for estimating dynamic reserves in fractured tight sandstone gas reservoirs. Unlike conventional methods, this approach does not rely on production and pressure data but instead utilizes reservoir properties. The proposed method accounts for the compressibility of rock pores and the pressure-dependent behavior of gas properties, enhancing its accuracy in these complex reservoir systems. Our goal was to provide an alternative formulation that may offer improved accuracy in specific scenarios. While this methodology offers a solution, it is essential to acknowledge its inherent limitations, which is discussed in detail in a later section of this paper. The formula proposed here is not intended to replace the existing methods; rather it serves as an alternative: using the derived formula allows for fast computations of parameters of fractured tight gas reservoirs, as well as a reliable engineering tool for the rapid evaluation of the dynamic reserves in a first attempt to tackle such a problem.

3. Model Introduction

This section first introduces the theory of gas flow in porous media. Second, the pressure change rate function is introduced and combined to derive the equation to estimate the dynamic reserves.

3.1. Gas Flow Theory

The mathematical modeling of gas flow in porous media is typically described by the diffusion equation, where the diffusion coefficient is related to pressure-dependent parameters, such as the viscosity, deviation coefficient, and compressibility factor. Therefore, it is necessary to linearize the governing equations for gas flow in order to simplify the analysis. Previous studies effectively linearized the gas flow equations using pseudo-pressure and pseudo-time [31,32], allowing for the application of solutions developed for slightly compressible liquid flow equations to gas flow. In this paper, we provide a brief overview of the formulation of the comprehensive differential equation for gas flow in porous media and the associated assumptions.

First, the continuity equation for single-phase gas flow in porous media is expressed as follows:

$$-\nabla ·\left(\rho \overrightarrow{v_g}\right)={\rho }_g\varphi \frac{(1-S_{wc})}{\partial t}+1-S_{wc})\frac{{\rho }_g\varphi }{\partial t}$$

(1)

where the variables and parameters are defined as follows: ${\rho }_g$ is the gas density ($\mathrm{kg}/{\mathrm{m}}^3$); $\overrightarrow{v_g}$ is the velocity of the gas in the porous media ($\mathrm{m}/\mathrm{s}$); $\varphi$ is the porosity; $q_g$ is the fluid mass change per unit volume per unit time due to sources or sinks within the control volume ($\mathrm{kg}/({\mathrm{m}}^3·\mathrm{s})$); $S_{wc}$ is the residual water saturation; and t is the time ($\mathrm{s}$).

By considering Darcy’s law as the governing principle for gas flow, the comprehensive differential equation can be formulated as follows:

$$\overrightarrow{v_g}=-\frac{K}{\mathsf{\mu }}\nabla p$$

(2)

where K is the permeability of the porous media ($\mathsf{\mu }{\mathrm{m}}^2$); $\mathsf{\mu }$ is the viscosity of the gas ($\mathrm{mPa}·\mathrm{s}$); and p is the pressure ($\mathrm{Pa}$).

It should be noted that the application of Darcy’s law to describe gas flow is a widely accepted approach in petroleum engineering textbooks, for example, see George [33]. The only exceptions where non-Darcy flow may occur are in the near-wellbore region, where boundary effects can lead to increased flow velocities, and in shale gas reservoirs, where the extremely small pore sizes can result in non-Darcy flow behavior due to the presence of a threshold pressure gradient.

After a series of transformations, the flow equations can be obtained as follows:

$$\nabla ·(\frac{p}{\mathsf{\mu }Z}\nabla p)=\frac{{\varphi }_i\mathsf{\mu }{C}_t}{K}·\frac{p}{\mathsf{\mu }Z}\frac{\partial p}{\partial t}$$

(3)

where ${\varphi }_i$ represents the porosity under initial formation conditions; Z is the gas compressibility factor; and $C_t$ is the total compressibility factor (${\mathrm{Pa}}^{-1}$).

Neglecting temperature variations, the overall compressibility factor $C_t$ used in this study considers the feature of the ultra-high pressure gas reservoir [34] given by

$$C_t={\mathrm{e}}^{C_{\varphi }(p-p_i)}\left[C_{\varphi }+(1-S_{wc})C_g+S_{wc}{C}_w\right]$$

(4)

where $C_g$ is the gas compressibility factor (${\mathrm{Pa}}^{-1}$); $C_{\varphi }$ is the rock compressibility factor (${\mathrm{Pa}}^{-1}$); $S_{wc}$ is residual water saturation; T is the temperature ($\mathrm{K}$); M is the molar mass of the gas ($\mathrm{kg}/\mathrm{mol}$); and R is the molar gas constant, where $R=$ $8.314$ J/(mol · K).

Zhang et al. [34] developed the above novel equation for the overall compressibility factor. The conventional overall compressibility factor is usually shown as follows:

$$C_t=(1-S_{wi})C_g+C_w{S}_{wi}+C_f$$

(5)

The elastic effects of rock and bound water are often significant, as the formation pressure decreases for ultra-high-pressure gas reservoirs. However, most conventional methods currently used do not take into account the characteristics specific to ultra-high-pressure conditions. Therefore, this study adopted the overall compressibility factor of Zhang et al. [34] that considers the shrinkage of rock porosity, expansion of bound water, and variations in gas properties to address the issue of the compressibility factor, as mentioned in the literature review section.

By introducing pseudo-pressure and pseudo-time, the gas flow equation can be transformed as follows:

$${\nabla }^2{p}_p=\frac{{\varphi }_i\mathsf{\mu }{C}_{ti}}{K}·\frac{\partial p_p}{\partial t_a}$$

(6)

where the pseudo-pressure and pseudo-time are defined as

$$p_p=p_i+\frac{{\mathsf{\mu }}_i}{{\rho }_{gi}}{\int }_{p_i}^p\frac{\rho \left(\xi \right)}{\mathsf{\mu }\left(\xi \right)}\mathrm{d}\xi$$

(7)

$$t_a={\mathsf{\mu }}_i{C}_{ti}{\int }_0^t\frac{1}{\mathsf{\mu }{C}_t}\mathrm{d}t$$

(8)

The gas flow Equation (6) has a similar form as that with liquid, thus it can be solved consequently like the liquid flow problem [33].

3.2. Pressure Change Rate Theorem

A dynamic reserve evaluation model for fracture-controlled gas reservoirs is established by coupling the equation of pressure change rate [35], as well as the solution of the gas diffusivity equation. The pressure change rate theorem shows that the change rate of pressure at the wellbore is always greater than or equal to the change rate of pressure at any other point in the reservoir:

$$\frac{d{p}_w}{dt}\ge \frac{dp(r,t)}{dt}$$

(9)

The theorem was developed based on the assumptions that there are constant pressure boundaries or other fluid sources or sinks, as well as the total compressibility being small, constant, and uniform. However, there is no limit to the type of external boundaries, layering, isotropy, or heterogeneity of the reservoir. The details and the proof of the derivation of the theorem can be found in the appendix of the paper [35].

The minimum reservoir volume is obtained to provide sufficient pressure support to produce a pressure change rate at the wellbore by assuming an identical rate of change of pressure within the investigation volume:

$$\frac{d{p}_D}{d{t}_D}=\frac{d{p}_{wD}}{d{t}_D}$$

(10)

The above equation is also the defining characteristic of pseudosteady-state flow [17], from which the reservoir pore volume is calculated:

$$V_{gpi}=\alpha ·\frac{q_g}{C_t·\left|\frac{d{p}_w}{dt}\right|}$$

(11)

where $\alpha =0.234$ for a field unit system.

The gas in place or dynamic reserve during pseudosteady-state flow in an arbitrarily shaped gas reservoir is therefore obtained:

$$G_{gi}=\alpha ·\frac{\left(1-S_w\right)q_g}{24C_t·\left|\frac{d{p}_w}{dt}\right|}$$

(12)

where $S_w$ is the known water saturation, and the unit of time is in hours. It should be noted that the derivative in the denominator is the primary pressure derivative for the drawdown.

3.3. The Proposed Model

This section elucidates the three typical flow regimes encountered during gas production from fractured reservoirs, along with the corresponding formulae for estimating dynamic reserves. Specifically, the gas flow behavior can be categorized into planar linear flow, fracture linear flow, and planar radial flow stages [18,33,36].

These distinct flow regimes are governed by the intricate interplay between the fracture network and the matrix system, which profoundly influences the gas transport mechanisms within the reservoir. The mathematical formulations developed to quantify the dynamic reserves during each of these flow stages were derived from the fundamental principles of fluid flow in porous media, incorporating the unique characteristics and complexities associated with fractured tight sandstone gas reservoirs. A comprehensive understanding of these flow regimes and their respective governing equations is pivotal for accurately assessing and predicting the dynamic reserve potential of such reservoirs. In fact, the three typical flow equations used in our study were indeed derived from well-established flow theory, which can be found in textbooks or published papers, for example, see Spivey and Lee [18].

The typical planar linear flow is illustrated in Figure 2, and the fluid flow model is represented by Equation (13) and can be found in textbooks on well testing or published papers, for example, see George [33] and Spivey and Lee [18].

$$p_i-p_w=\frac{8.128q_g{B}_g}{HW}\sqrt{\frac{{\mathsf{\mu }}_{g}t}{k\varphi C_t}}+\frac{141.2q_g{B}_g{\mathsf{\mu }}_{g}S}{kH}$$

(13)

where the skin factor affecting the pressure response during the flow regime is reflected in the second term S. W is the width of the fracture and $HW$ is the cross-sectional fracture area.

By applying the pressure rate change theorem, Equation (12) is used to compute the gas volume investigated at time t during the specific flow regime. The pressure derivative is calculated using

$$\frac{d{p}_w}{dt}=\frac{4.046q_g{B}_g}{HW}\sqrt{\frac{{\mathsf{\mu }}_g}{k\varphi C_{t}t}}$$

(14)

The investigated gas dynamic reserves can be consequently calculated as follows by incorporating Equation (14) into Equation (12):

$$G_{gi}=0.0103\frac{\left(1-S_w\right)HW}{B_g}\sqrt{\frac{\varphi kt}{{\mathsf{\mu }}_g{C}_t}}$$

(15)

Similarly, the typical fracture linear flow is illustrated in Figure 3, and the fluid flow model and dynamic reserve are represented by Equations (16) and (17), respectively.

$$p_i-p_w=4.064\frac{q_g{B}_g}{h{L}_f}\sqrt{\frac{{\mathsf{\mu }}_{g}t}{k\varphi C_t}}+141.2\times \frac{q_g{B}_g{\mathsf{\mu }}_{g}S}{kh}$$

(16)

where $L_f$ is the fracture half-length ($\mathrm{m}$) and S is the skin factor to account for the non-Darcy effect around near-wellbore regions of a gas well.

$$G_{gi}=\frac{0.0205\left(1-S_w\right)h{L}_f}{B_g}\sqrt{\frac{k\varphi t}{{\mathsf{\mu }}_g{C}_t}}$$

(17)

This result based on pressure varying with the square root of time is the most important equation for the linear flow. Additionally, a log–log plot of $\Delta P$ versus $\sqrt{t}$ shows a slope of 0.5, indicating a formation fracture linear flow stage [33].

The typical planar radial flow is illustrated in Figure 4, and the fluid flow model and dynamic reserve are represented by Equations (18) and (19), respectively [33].

$$p_i-p_w=\frac{162.6q_g{\mathsf{\mu }}_g{B}_g}{kh}log\left(t\right)+\frac{162.6q_g{\mathsf{\mu }}_g{B}_g}{kh}\left(log\left(\frac{k}{\varphi {\mathsf{\mu }}_g{C}_t{r}_w^2}\right)-3.23+0.869S\right)$$

(18)

$$G_{gi}=\alpha \frac{\left(1-S_w\right)q_{g}t}{24C_t\times \frac{162.6q_g{\mathsf{\mu }}_g{B}_g}{kh}}=\frac{5.9\times {10}^{-4}\left(1-S_w\right)kht}{C_{gt}{\mathsf{\mu }}_g{B}_g}$$

(19)

4. Results and Discussions

This section first introduces the pressure change characteristics of the gas parameters. Some numerical simulation examples are used to demonstrate the applicability of the method proposed in this paper, including the theoretical sensitivity analysis and comparison and validation with numerical simulation results. Then, the model was used to evaluate the dynamic reserves of fracture-controlled gas reservoirs during the unstable pressure stage. Finally, a relationship between dynamic reserves in fracture-controlled reservoirs and the scale of connected fractures in the gas reservoir was established.

4.1. Pressure Change Characteristics of the Gas Parameters

A simulator was employed to simulate the three models (planar linear flow, planar radial flow, and vertical fracture linear flow). It was assumed that the reduction in the pore volume and the expansion of the bound water volume could not be neglected. Other reservoir parameters were set as shown in

Table 1. The parameters of the numerical simulation.

No.	Parameter	Value	No.	Parameter	Value
1	Permeability k	$0.1$ mD	6	Water compressibility $C_w$	$0.0004$ MPa−1
2	Matrix porosity $\varphi$	5%	7	Rock compressibility $C_f$	$0.0004$ MPa−1
3	Water saturation $S_w$	30%	8	Skin factor S	0.1
4	Net pay h	100 m	9	Fracture half-length $L_f$	100 m
5	Width w	100 m	10	Initial reservoir pressure $p_i$	120 MPa

.

The viscosity of natural gas was calculated using the viscosity correlation proposed by Londono et al. [37,38], and the gas compressibility factor was determined [39]. The relationships between the gas viscosity ${\mathsf{\mu }}_g$, compressibility factor $C_g$, formation volume factor $B_g$, and pressure are shown in Figure 5. The simulated pressure and original gas in place changing with time for the three types of gas flow are presented in Figure 6.

The calculated dynamic reserves of the three flow models were compared with the numerical simulation and the results are presented in Figure 7. The reserves within the pressure propagation range at 300 days were in good agreement with numerical simulations. The numerical simulation results had an error of less than 5% compared with the formula calculations.

4.2. Sensitivity Analysis

Based on the above simulation, the sensitivity analysis of the parameter analysis model was conducted. The pressure and fracture-controlled dynamic reserves were calculated for different production rates (5000 m³/day, 10,000 m³/day, 20,000 m³/day) and different permeabilities (0.01 mD, 0.1 mD, 0.5 mD), as shown in Figure 8, Figure 9 and Figure 10, respectively.

The analysis revealed that the production rate had a minor impact on the dynamic reserves estimation within the pressure propagation area, which was mainly caused by the influence of gas petrophysical properties. On the other hand, a larger matrix permeability led to larger dynamic reserves within the pressure propagation area. Fracture density and matrix permeability were the primary factors that influenced the dynamic reserves in the fracture-controlled reservoirs.

4.3. Applications to Real Field Cases

4.3.1. Brief Introduction to the Gas Reservoirs

Two gas reservoirs were studied in this research. The gas reservoirs were mainly buried at ultra-deep depths (6400–8200 m). The reservoir was characterized by significant thickness (250–310 m) and represented the subaqueous distributary channel deposits of a braided river delta in a wide and shallow lake basin. The predominant lithology was medium-to-fine-grained lithic feldspar sandstone, with the reservoir space primarily composed of residual intergranular and interparticle dissolution pores consisting of micropores and nanometer-scale throats. The reservoir had poor petrophysical properties, with a matrix densities and measured matrix porosities ranging from 4% to 7% and matrix permeabilities of $(0.01-0.5)\times {10}^{-3}\mathsf{\mu }{\mathrm{m}}^2$. Various hierarchical structural fractures had developed, including sub-micron fractures, microscopic fractures, small-scale fractures, and large-scale fractures. The sub-micron and microscopic fractures acted as micro-scale fractures, which primarily improved the connectivity of matrix pore networks, while small-and-large-scale fractures served as macro-scale fractures, providing dominant pathways for natural gas migration and reservoir water invasion.

The measured fracture porosity in the core was relatively low, accounting for only 0.15% to 2.9% of the total porosity. However, the well-testing permeability of the reservoir was in the range of $(1.3-116.5)\times {10}^{-3}\mathsf{\mu }{\mathrm{m}}^2$, indicating that fractures contributed relatively little to the porosity but significantly enhanced the matrix permeability by 2 to 4 orders of magnitude as the main fluid transport channels.

In the gas reservoir A, the overall development of the fracture patterns was low, predominantly consisting of isolated and parallel fractures with non-uniform spatial distribution and poor continuity. Conversely, in the gas reservoir B, the fracture pattern exhibited a higher development degree, which was mainly characterized by a combination of network and oblique intersecting fractures. The fracture network showed a uniform spatial distribution and strong continuity.

4.3.2. Results and Discussions of the Applications

Fracture-controlled dynamic reserves refer to the dynamic reserves within the fractures and matrix, representing the reserves within the pressure propagation range. The main influencing factors include the spatial distribution area of the fractures (fracture area), gas high-pressure properties, and matrix properties. Based on simulation calculations, the models and pressure propagation conditions at different fracture lengths (100 m, 500 m, 1000 m, and 2900 m) at the same time were analyzed, as shown in Figure 11.

According to Equation (20), factors that influence the fracture-controlled dynamic reserves include the total connected fracture area, matrix properties, gas high-pressure properties, and time:

$$G_{gi}=\frac{0.0205\left(1-S_w\right)h{L}_f}{B_g}\sqrt{\frac{k\varphi t}{{\mathsf{\mu }}_g{C}_t}}$$

(20)

The parameter $h{L}_f$ represents the connected fracture area in the reservoir, and it is defined as F. Based on this definition, the relationship between the dynamic reserves and the distribution of fractures in the reservoir was established, as shown below:

$$G_{gi}=f\left(F\right)$$

(21)

Based on the simulation analysis, it can be observed that the connected fracture scale was positively correlated with the fracture-controlled dynamic reserves, as depicted in Figure 12.

The well test experiment was conducted for a well in reservoir A, as shown in Figure 13. Three flow stages were observed and explained. Stage I showed a dual porosity model where the curve exhibited a “concave downward” feature, reflecting the characteristics of the dual-porosity model. Stage II was where the derivative curve showed an overall linear upward trend with a slowing down at the end, with a slope of 0.5, indicating linear flow characteristics. Stage III showed an upward trend with flow obstruction, characterized by a boundary flow with a slope of 1. No significant radial flow characteristics were observed.

By applying Equation (20), the dynamic reserves in the gas reservoirs A and B were calculated and are shown in

Table 2. The relationship table between dynamic reserves and fracture parameters for Blocks 2 and 8 for the reservoirs A and B.

Reservoir	Dynamic Reserve	k	$\mathit{\varphi }$	${\mathit{S}}_{\mathit{wi}}$	${\mathsf{\mu }}_{\mathit{g}}$	${\mathit{C}}_{\mathit{t}}$	${\mathit{B}}_{\mathit{g}}$	$\mathit{\beta }={\mathit{hL}}_{\mathit{f}}$
A	820	0.1	0.06	0.3	0.0474	0.0058	0.0030	34,010,554
B	560	0.1	0.06	0.3	0.0460	0.0063	0.0032	23,671,205

. The volume of fractures accounted for approximately 0.15–0.2% of the underground volume of dynamic reserves. The application of the aforementioned model showed that the dynamic reserves in the gas reservoirs A and B exhibited a basic linear relationship with fracture parameters, as illustrated in Figure 14.

To validate the dynamic reserves calculated by the method mentioned in this paper, the single-well cumulative method and the well group method were employed [25,26]. The single-well cumulative method calculates the dynamic reserves of each individual well, and then arithmetically sums them to obtain the block dynamic reserves. The well group method uses the total production rate of the entire block for each well, but the pressure used is the individual well’s bottom hole pressure [25]. The basic principle of the well group method is as follows. For multiple wells in a block, after the flow reaches boundary-dominated flow, the flow equation can be written in the following form:

$$\frac{q_g\left(t\right)}{(p_i-p_{wf}\left(t\right))}=\frac{1}{\frac{1}{N_{C_t}\overrightarrow{t_{tot}}}+b_{pss}}$$

(22)

$$t_{tot}=\frac{1}{q_g\left(t\right)}{\int }_0^t\sum _{i=1}^{n_{well}}{q}_t\left(\tau \right)\mathrm{d}\tau =\frac{N_{p,tot}}{q_g\left(t\right)}$$

(23)

where $t_{tot}$ is the material balance time, q is the flow rate, and p is the pressure.

The above two equations are the general form of Arps’ harmonic decline curve, which is also a form of the material balance equation, but it is applicable to cases with variable flow rates or pressures. This equation indicates that by plotting the scatter points curve between $\frac{q_g\left(t\right)}{(p_i-p_{wf}\left(t\right))}$ and $t_{tot}$, the total reserves of the entire block can be estimated.

The dynamic reserves of the gas reservoirs A and B were estimated using both the single-well cumulative method and the well group method, with both methods utilizing IHS Harmony software for the calculations. The results are shown in Figure 15.

Based on the comparative analysis of dynamic reserves in the gas reservoirs A and B, the dynamic reserves calculated by the single-well cumulative method was significantly higher than the reservoir dynamic reserves calculated by the other two methods (15–30% difference). This, on the one hand, indicates the good overall connectivity of the reservoir that has significant interference between some wells, suggesting the presence of overlap** and recombination issues. On the other hand, it reflects that the results obtained by the single-well cumulative method had redundancy, which was caused by the interference phenomenon between wells. Notably, the dynamic reserves evaluated using the well group method were slightly higher than those evaluated using the method proposed in this paper, where the relative differences between these two methods were 7% and 17% for reservoirs A and B, respectively, as shown in Figure 15.

5. Conclusions

This paper proposes a simplified forward-modeling approach for dynamic reserve estimation that leverages the pressure change rate function and gas diffusivity equation. The proposed method explicitly incorporates the influence of the fracture length, enabling rapid evaluation of dynamic reserves in fractured tight sandstone gas reservoirs.

This method first establishes a theoretical model, and through numerical simulation calculations and sensitivity analysis, unveils the intrinsic correlation between dynamic reserves under fracture-controlled flow and the scale of the connected fractures in the reservoir. The validity and applicability of the method were confirmed by comparing its results with the numerical simulation results.

Furthermore, the method was applied to a real-world case study that involved fractured tight sandstone gas reservoirs. The dynamic reserve estimates obtained using our approach exhibited reasonable agreement with those derived from the well-established well group method, thereby confirming the accuracy and reliability of our methodology.

A key advantage of our proposed method lies in its concise formulation, which enables field engineers to conduct dynamic reserve assessments without the need for complex computations or extensive historical production data.

While our research presents an effective method for dynamic reserve evaluation in fractured tight sandstone gas reservoirs, we acknowledge the potential limitations for further improvement and extension. Future research efforts can focus on refining the method to account for additional reservoir complexities, investigating the applicability to other unconventional reservoir types, and integrating advanced data analytics techniques to enhance the accuracy and robustness of the dynamic reserve estimates.