Introduction

Located in the southeastern Ordos Basin, Yan'an gas field is the largest low-permeability tight sandstone field in China, producing more than 200 × 108m3 for three years from 2019 to 2022, and is also the largest natural gas field in terms of production in China (Clarkson 2013; Ning et al. 2009). The main gas producing layer of Yan'an gas field is He 8 member, with porosity ranging from 2.0 to 12.0% and permeability mainly in the range of 0.1 to 10.0 mD. It is a typical “low pressure, low porosity, low permeability and strong heterogeneity gas field” (Zou et al. 2012). The development of conventional straight wells faces difficulties such as low production, difficulty in stabilizing production and rapid pressure drop (Han et al. 2019). Therefore, it is necessary to analyze the production performance and its controlled reserves.

At present, the main methods for calculating dynamic reserves include MB (material balance method), the production decline method, production accumulation method, elastic two-phase method and so on, which are suitable for conventional reservoirs (Schmoker 2002; Wang et al. 2013; Zhang et al. 2008). Among them, the establishment of the material balance method is relatively easy with less data (Lyu et al. 2020; Muther et al. 2022a; Shults 2020). While there are no data such as bottom hole pressure, the MB cannot calculate the dynamic reserves of gas wells (Li et al. 2015; Zhenhua and Xinwei 2013). The FMB (flowing material balance method) does not take into account the effect of pressure on the viscosity and compressibility of gas, leading to an error in the result.

In order to solve the above problems, a modified FMB method is proposed in this study, in which the influence of pressure on the viscosity and compressibility of gas is considered. Taking the tight reservoir in Yanchang Oilfield in Ordos Basin as an example, the FMB and modified FMB are compared and analyzed.

Geological background

Ordos Basin is a large sedimentary basin with multi-cycle evolution and multi-sedimentary types. At present, the structure is a large syncline with slow width in the east and steep and narrow in the west, and the dip angle is generally less than 1° (Feng et al. 2016). Fault folds in the margin of the basin are well developed and the internal structure is relatively simple. There is no secondary structure in the basin, and the tertiary structure is dominated by nose uplift (Guo et al. 2018). According to the current structural shape, basement properties and structural characteristics of the basin, it can be divided into six units: Yimeng uplift, Weibei uplift, western Shanxi flexure fold belt, Yishan slope, Tianhuan depression and western margin thrust structural belt.

Yan'an gas field is located in the southeast of Yishan slope in Ordos Basin (Fig. 1) (Hu et al. 2008). The comprehensive geological study shows that the Upper Paleozoic in the area has many favorable conditions, such as extensive hydrocarbon generation, development of reservoir rock multi-layer system, wide distribution of regional caprock, which are beneficial to the formation and enrichment of large lithologic gas reservoirs (Hu et al. 2010; Muther et al. 2022b; Sun et al. 2020).

Fig. 1
figure 1

Location map of Yan'an gas field in Ordos Basin

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The sandstones are mainly gray-white, gray-green and dark gray, and the mudstones are mainly black and gray-black, containing a large number of plant fossils. The sedimentary in the study area is delta frontal sedimentation, and the microphases are mainly underwater distributary channel, interdistributary area and bank (Fig. 2). Controlled by the sedimentary, the sand body is north–south trending. The thickness is mainly distributed in 8–12 m, and the average thickness is 8.4 m.

Fig. 2
figure 2

Sedimentary facies and sand thickness of area (A: sedimentary facies; B: sand thickness)

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Rock types are mainly quartz sandstone, followed by lithic quartz sandstone and lithic sandstone (Fig. 3). The fillings are dominated by silica, kaolinite, illite and iron calcite, and contain small amounts of iron chlorite, iron dolomite and hard gypsum. Observed by thin section, the storage space is mainly dominated by residual primary intergranular pores, dissolved intergranular pores, a small amount of intragranular dissolved pores, kaolinite intergranular pores and dissolved microfractures (Fig. 4, Fig. 5).

Fig. 3
figure 3

Petrologic characteristics of reservoirs

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Fig. 4
figure 4

Types of clay minerals

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Fig. 5
figure 5

Microscopic characteristics of the sandstone reservoirs. (A: residual intergranular pore, 2431.23 m; B: residual intergranular pore, 2727.07 m; C: intergranular dissolved pores, 2631.23 m; D: dissolved pores in grains, 2599.45 m; E: intercrystalline pore of kaolinite, 2629.63 m; F: microfracture, 2443.45 m)

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The pore structure can be divided into three types based on the morphology of the capillary pressure curve and the analysis of the characteristics of the samples (Table.1, Fig. 6).

  1. (1)

    Type-I

    The curve is platform-shaped and slightly concave to the left. The discharge pressure is very low, mainly distributed in the range of 0.03 to 0.1 MPa, with an average pressure of 0.224 MPa. The average of pore throat radius is large, mainly distributed in the range of 1.41 to 4.12 μm, with an average radius of 3.35 μm; the throat distribution is coarse skewness, and the sorting is medium.

  2. (2)

    Type-II

    The curve is gently sloping and also slightly concave to the left, with higher discharge pressure and median pressure than Type-I. The discharge pressure is mainly distributed in the range of 0.2 to 1.5 MPa, with an average of 0.52 MPa. The average radius of the pore throat is smaller than Type-I, mainly distributed in the range of 0.17 to 1.09 μm, with an average of 0.69 μm.

  3. (3)

    Type-III

    The discharge pressure of these reservoirs is medium, distributed in the range of 0.4 to 1.5 MPa. The capillary pressure curve is steeply sloping and the platform is not obvious. The pore throat radius is 0.14–0.44 μm and the sorting is poor.

Table 1 Classification of pore parameter
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Fig.6
figure 6

Classification of pore structure (A: Type-I; B: Type-II; C: Type-III)

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Method

Due to the different conditions of methods, the results vary widely. According to the research on the geology of Yan'an gas field, the applicable methods for dynamic reserves in low-permeability tight sandstone reservoirs are the MB (material balance method) and the ETM (elastic two-phase method) (He et al. 2021; Luo et al. 2019; Zou et al. 2012).

Dynamic reserves calculation method

Material balance method

The MB uses a “pressure drop diagram” consisting of apparent formation pressure (P/Z) and cumulative production (Gp) to determine the dynamic reserves. However, it is not applicable to water-driven reservoirs and abnormally high pressure reservoirs (Andersen and Hyman 2001; Ruilan et al. 2010; Wei et al. 2016).

From the perspective of seepage mechanics, for closed reservoirs, the pressure wave is transmitted to the boundary of the formation after a certain period of relatively stable production, and the seepage enters the quasi-steady stage (Fig. 7). Mattar et al. proposed that formation pressure can be replaced by wellhead casing pressure and bottom hole pressure in MB (Fig. 8) (Mattar et al. 2006).

Fig. 7
figure 7

Pressure depletion curve of reservoir

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Fig. 8
figure 8

Casing pressure depletion curve

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The elastic two-phase method

For a gas reservoir with a finite confinement of constant productivity, the pressure drop curve can be divided into three parts: unstable seepage early stage, unstable seepage late stage and pseudo steady (Fig. 9). At pseudo steady, the flow pressure at the bottom is linear with time (Fig. 10), which is used to determine the reserves (Kanianska et al. 2011; Long et al. 2011).

Fig. 9
figure 9

Elastic two-phase theoretical curve

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Fig. 10
figure 10

Pressure profile of the well

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The ETM for evaluating control reserves requires the well to reach the pseudo steady stage (Han et al. 2019). If the production is too large, a “false steady state” is likely to occur, and the slope of the linear is larger than the true value, resulting in a smaller calculation. In addition, for tight reservoirs with low permeability, the pressure wave propagation is slow and it takes a long time, which results in high testing cost.

FMB method

Principle

When there are no data such as bottom hole pressure, the MB cannot calculate the dynamic reserves (Hirst et al. 2001; Nie et al. 2018; Ning et al. 2009). In order to solve this problem, Mattar put forward the FMB, which is analyzed from the point of view of percolation mechanics (Mattar et al. 2006). It is considered that the viscosity and compressibility of natural gas remain unchanged (Ning et al. 2009). However, when the formation pressure of the reservoir is low, the assumption is not valid.

In order to solve the problems, a modified FMB is proposed in this study, in which the influence of pressure on the viscosity and compressibility is considered. Taking the tight reservoir of Yan'an gas field in Ordos Basin as an example, the FMB and modified FMB are compared and analyzed.

Property of natural gas

  1. (1)

    Viscosity

    Through 30 gas samples (Tables.2, 3, 4) under the temperature of 352 K and pressure of 30 MPa, the viscosity of natural gas increases with the temperature under the condition of low pressure, and the viscosity increases with pressure (Fig. 11).

  2. (2)

    Z coefficient

    The Z at different temperatures and pressure can be gotten, as shown in Fig. 11. It decreases at first (P < 15 MPa) and then increases (P > 15 MPa).

  3. (3)

    Compressibility

    The relationship of P–Cg at different temperatures can be obtained (Fig. 11): the compressibility decreases with temperature and pressure.

  4. (4)

    Volume coefficient

Table 2 Composition of 30 groups of natural gas samples-a
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Table 3 Composition of 30 groups of natural gas samples-b
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Table 4 Composition of 30 groups of natural gas samples-c
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Fig. 11
figure 11

Property curve of natural gas (A1: P–μ curve of natural gas and P–Cg curve of natural gas; A2: P–Z curve of natural gas and P–Bg curve of natural gas)

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The volume coefficient decreases with pressure and increases with temperature (Fig. 11).

Modified FMB

In the FMB, it is assumed that the pressure has no effect on the viscosity and compressibility (Clarkson 2013; Li et al. 2015):

$$\partial \left( {\overline{{u_{{\text{g}}} }} \overline{{C_{{\text{g}}} }} } \right) = \partial \left( {u_{{{\text{gwf}}}} C_{{{\text{gwf}}}} } \right)$$
(1)

However, the viscosity and compressibility of natural gas vary with pressure (Fig. 12); therefore, Eq. (1) is not valid.

Fig. 12
figure 12

P–ugCg curve of natural gas

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By deforming Eq. (1):

$$\frac{{\partial \left( {{{\overline{P} } \mathord{\left/ {\vphantom {{\overline{P} } {\overline{Z} }}} \right. \kern-\nulldelimiterspace} {\overline{Z} }}} \right)}}{{\partial G_{{\text{P}}} }} = \frac{{\partial \left( {\overline{{u_{g} }} \overline{{C_{g} }} } \right)}}{{\partial \left( {u_{{{\text{gwf}}}} C_{{{\text{gwf}}}} } \right)}} \cdot \frac{{\partial \left( {{{P_{{{\text{wf}}}} } \mathord{\left/ {\vphantom {{P_{{{\text{wf}}}} } {Z_{{{\text{wf}}}} }}} \right. \kern-\nulldelimiterspace} {Z_{{{\text{wf}}}} }}} \right)}}{{\partial G_{{\text{P}}} }}$$
(2)
$$\frac{{\partial \left( {\overline{{u_{{\text{g}}} }} \overline{{C_{g} }} } \right)}}{{\partial \left( {u_{{{\text{gwf}}}} C_{{{\text{gwf}}}} } \right)}} \approx \frac{{\left. {\left( {u_{g} C_{g} } \right)} \right|_{{\overline{{P_{{{\text{pss}}}} }} }} }}{{\left. {\left( {u_{g} C_{g} } \right)} \right|_{{P_{{{\text{wf}}}} - pss}} }} \approx \frac{{\left. {\left( {u_{g} C_{g} } \right)} \right|_{{p_{{\text{i}}} }} }}{{\left. {\left( {u_{g} C_{g} } \right)} \right|_{{P_{{{\text{wf}}}} - pss}} }} = \lambda$$
(3)

The steps of the modified FMB are as follows (Fig. 13):

  1. (1)

    According to the relationship between p and μgcg, \(\left. {\left( {u_{g} c_{g} } \right)} \right|_{pi}\) and \(\left. {\left( {u_{g} c_{g} } \right)} \right|_{{p_{wf - pss} }}\) can be gotten.

  2. (2)

    The -m can be got with the bottom hole pressure and cumulative production data (Pwf ~ Zwf ~ Gp).

  3. (3)

    Make a straight line over Pi/Zi with the slope of -λm, and the intercept is the reserves determined by the modified FMB (modified Gi).

  4. (4)

    Similarly, the wellhead casing pressure Pc is used to replace the bottom hole pressure Pwf.

Fig. 13
figure 13

Process of modified FMB method

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Result

According to reservoir characteristics and pore structure, the gas wells in the study area can be divided into three types: I, II and III. The production law of the three types of gas wells is analyzed. At the same time, the controlled reserves are calculated by using the established modified FMB.

Type-I

The initial production of Type-I wells in Yan'an gas field is high, and the pressure drops slowly, so it has a good productivity under the condition of low pressure (Fig. 14). The OFR of Type-I well is the largest, with average of 30.57 × 104m3/d. The average monthly production is 95 × 104m3/m and the decline pattern is at a low level with 17.9% at the initial stage. The casing pressure and the production decrease slowly in the second stage. The production was kept at a middle level and casing pressure is about 8.6 MPa in the third stage.

Fig. 14
figure 14

Decline pattern of wells (A1: production; A2: surface casing pressure; A3: production decline rate; A4: decline rate of surface casing pressure)

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The results show that the dynamic reserve of Type-I well are 1.398 × 108m3 with FMB and 1.906 × 108m3 with modified FMB (Fig. 15).

Fig. 15
figure 15

Calculation results of dynamic reserves by modified FMB method (A1: decline characteristic; A2: Type-I wells; A3: Type-II wells; A4: Type-III wells)

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Type-II

The OFR of Type-II is middle. The average production is 79 × 104m3/d and the decline rate is 37.32% from the production curve (Fig. 14). After the production is reduced by 1 × 104m3/d, the casing pressure decreases from 14.41 to 8.36 MPa. Up to June 2017, the cumulative gas production is 2273.123 × 104m3.

The results show that the dynamic reserve of Type-II is 0.807 × 108m3 with FMB and 1.233 × 108m3 with modified FMB (Fig. 15).

Type-III

The OFR of Type-III well is the lowest, with average 6.2 × 104m3/d from the production curve (Fig. 14). It has been in production since August 2015 and the average monthly production is 27 × 104m3/m at the initial stage. The casing pressure decreases rapidly and the monthly production remained unchanged in the second stage. Up to now, the cumulative production of Type-III is 352.896 × 104m3.

The results show that the dynamic reserve of Type-III well is 0.787 × 108m3 with FMB and 11,186.93 × 108m3 with modified FMB (Fig. 15).

Discussion

Compared with the FMB, the MB uses the average formation pressure measured after shut-in for a long time, so it is more reliable.

Method verification

In order to verify the accuracy of the calculation of the modified FMB, the diagram between the cumulative production and the P/Z is drawn with the formation pressure at different stages, as shown in Table.5, Fig. 16.

Table 5 Measured pressure of three wells
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Fig. 16
figure 16

Calculation of dynamic reserves by MB

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The dynamic reserves calculated of three wells by MB can be obtained. ①The reserve of Type-I is 1.87 × 108m3. The error of FMB is 25.33% and the modified FMB is 1.79% (Table.6, Fig. 16).②The reserve of Type-II is 1.18 × 108m3. The error of FMB and the error of modified FMB are 32.02% and 3.82%.③The reserve of Type-III is 0.20 × 108m3, with the error of FMB of 36.73% and the error of modified FMB of 6.69%.

Table 6 Calculation results of FMB method and modified FMB method
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Through the above calculation, compared with the MB, the result of the FMB is generally small, with an average error of 31.36%; the error of the modified FMB is small, with an average of 4.10%. Therefore, the method established in this paper can provide scientific guidance for the determination of single-well dynamic reserves in tight sandstone reservoirs.

Application

Three dynamic reserve methods are used to calculate 88 wells, and the results are shown in Table.7 (More calculation are in the Supplementary online). The average reserves calculated by the MB and the FMB are 0.523 × 108m3, 0.372 × 108m3, respectively, and the average error of FMB is 28.87%. The result of modified FMB is 0.516 × 108m3, with minimum error of 0.04%, the maximum error of 4.97% and the average error of 2.45%.

Table 7 Calculation results of three dynamic reserve methods
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Conclusion

  1. (1)

    Controlled by sedimentary, the pore structure in the study area can be divided into three types based on the morphology of the capillary pressure and the analysis of the reservoir characteristics: Type-I, Type-II and Type-III.

  2. (2)

    The viscosity of natural gas increases rapidly with pressure through theoretical calculation and numerical simulation. The compressibility decreases when pressure is less than 15 MPa and then increases when pressure is great than 15 MPa, and increases with temperature monotonically. Then, the results calculated by the FMB are smaller when the effect of pressure on viscosity and compressibility is not considered.

  3. (3)

    Considering the viscosity and compressibility, the modified FMB and the calculation steps are given. The results of three typical wells show that compared with the results of the MB, the average error of the FMB is 31.36%, and the average error of the modified FMB is 4.10%.

  4. (4)

    The calculation of 88 wells shows that the error of FMB is 28.87% and the error of modified FMB is only 2.45%. Therefore, the proposed method in the paper can quickly and accurately calculate the dynamic reserves of single well in tight reservoir.