1 Introduction

Adsorption is a very important accumulation mechanism in shale gas reservoirs. Statistical studies show that the amount of adsorbed gas in shale gas reservoirs is between 20 and 80% [1,2,3,19,20,21], but there are few studies on the desorption ability of shale adsorbing gas.

In this paper, based on the analysis of the curvature curve of shale gas desorption efficiency, the desorption process is divided into four stages according to the stagnation point and inflection point on the curve, and the ratio formula of desorption ability to gas content in different stages is given. This method is applied to the upper Paleozoic shale strata in Ordos basin, and the ratio of desorption ability to gas content at different stages is analyzed according to the isothermal adsorption curves of different core samples. Shale gas releases the largest amount of gas in Low efficiency desorption stage. Therefore, in the process of shale gas fracturing, the formation pressure can be properly maintained near starting pressure, which is more conducive to improve shale gas recovery.

2 The method

2.1 Derivation of the method

2.1.1 Desorption efficiency

In the exploitation of shale gas, with the different formation pressure, the amount of gas resolved in each ton of shale is also different. In order to obtain the desorption analytical gas volume of shale gas at a certain pressure, the desorption efficiency of shale gas is introduced, expressed in η. When the value of \(\Delta p\) tends to zero, η is the amount of desorption gas per ton of shale at a unit pressure drop and its mathematical representation is \(\eta = \mathop {\lim }\limits_{\Delta P \to 0} \frac{\Delta V}{{\Delta P}} = \frac{dv}{{dp}}\), with Langmuir equation \({\text{V}} = \frac{{{\text{V}}_{L} *P}}{{P_{L} + P}}\), the desorption efficiency formula of shale gas is \(\eta = \frac{dv}{{dp}} = \frac{{{\text{V}}_{L} P_{L} }}{{(P_{L} + P)^{2} }}\).

In this paper, 16 cores from Y88 and Y94 wells in Yulin area of Ordos Basin were selected for isotherm sorption measurement. The instrument was ISO300 isotherm sorption instrument. Take Y88-1 sample as an example (grayish-black mudstone, Benxi formation, sampling depth 2509.45 m), Langmuir volume is 8.92 m3 / t, Langmuir pressure is 2.84 Mpa. It can be seen from the desorption efficiency curve of Y88-1 sample (Fig. 1) that with the decrease of reservoir pressure, the desorption efficiency of shale gas increases gradually.

Fig. 1
figure 1

Desorption efficiency curve of Y88-1 sample

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Curvature is the curvature of a point on the curve. According to the curvature of the desorption curve of shale gas, the desorption stages of shale gas can be divided. According to curvature formula \(K{\mkern 1mu} = {\mkern 1mu} \frac{{y{\mkern 1mu}^{\prime \prime } {\mkern 1mu} }}{{\left( {1{\mkern 1mu} + {\mkern 1mu} y{\mkern 1mu}^{\prime } {\mkern 1mu}^{2} } \right)^{3/2} }}\) [22], The curvature formula of the desorption efficiency curve can be expressed as follows \(K_{\eta } = \frac{{\left| {v^{\prime\prime}} \right|}}{{(1 + v^{{\prime}{2}} )^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} }}\), in the formula \(v^{\prime\prime} = - \frac{{2V_{L} P_{L} }}{{(P + P_{L} )^{3} }}\). Bring \(v^{\prime}\)、\(v^{\prime\prime}\) into the equation \(K_{\eta }\), and we get the formula of \(K_{\eta } = \frac{{{{2V_{L} P_{L} } \mathord{\left/ {\vphantom {{2V_{L} P_{L} } {(P + P_{L} )^{3} }}} \right. \kern-0pt} {(P + P_{L} )^{3} }}}}{{[1 + {{(V_{L} P_{L} )^{2} } \mathord{\left/ {\vphantom {{(V_{L} P_{L} )^{2} } {(P + P_{L} }}} \right. \kern-0pt} {(P + P_{L} }})^{4} ]^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} }}\).

The curvature variation of the desorption curve of sample Y88-1 is shown in Fig. 2. It can be seen from the characteristics of the curve that at pressure point \(P_{tu}\) (Named turning pressure) the curvature of desorption curve is maximized. At pressure point \(P_{st}\) (Named starting pressure) and \(P_{se}\) (Named sensitive pressure) are the nodes where curvature concavity and convexity change. As the pressure increases, the curvature curve can be divided into three parts according to the pressure nodes \(P_{st}\)、\(P_{tu}\)、\(P_{se}\): slow increasing, fast decreasing and rapid reduction. The corresponding desorption efficiency shows three stages: slow increase, fast increase and rapid increase.

Fig. 2
figure 2

Desorption curve curvature curve of Y88-1 sample

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2.1.2 Turning pressure

According to the mathematical definition, when the curvature is maximized, the first derivative of the curvature is equal to zero, that is:

$$K_{\eta }^{\prime } = 6V_{L} P_{L} (P + P_{L} )^{ - 8} [1 + {{(V_{L} P_{L} )^{2} } \mathord{\left/ {\vphantom {{(V_{L} P_{L} )^{2} } {(P + P_{L} )^{4} }}} \right. \kern-0pt} {(P + P_{L} )^{4} }}]^{{ - {5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-0pt} 2}}} [(V_{L} P_{L} )^{2} - (P + P_{L} )^{4} ] = 0$$

In order to make \(K_{\eta }^{\prime }\) equal to zero, only if the value of \((V_{L} P_{L} )^{2} - (P + P_{L} )^{4}\) equals zero. The reservoir pressure at which the curvature is readily obtainable, that is, the reservoir transition pressure \(P_{tu}\), \(P_{tu} = \sqrt {V_{L} P_{L} } - P_{L}\).

When the reservoir pressure at the stagnation point is substituted into Langmuir equation, the corresponding adsorption capacity can be obtained as follows: \(V = V_{L} - \sqrt {V_{L} P_{L} }\).

It can be seen from Fig. 3 that the first derivative of desorption efficiency curve curvature of Y88-1 sample is zero when \(P_{tu}\) is 2.19 MPa, which corresponds to the maximum point of curve curvature.

Fig. 3
figure 3

The first derivative curve of the curvature of the desorption curve of sample Y88-1

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2.1.3 Starting pressure and sensitive pressure

The point where the second derivative of the desorption efficiency curve is zero is the inflection point of the curvature curve, and the formula for calculating the seconswd derivative of the curvature of the desorption curve is as follows.

\(K_{\eta }^{\prime \prime } = 12V_{L} P_{L} (P + P_{L} )^{ - 13} [1 + {{(V_{L} P_{L} )^{2} } \mathord{\left/ {\vphantom {{(V_{L} P_{L} )^{2} } {(P + P_{L} )^{4} }}} \right. \kern-0pt} {(P + P_{L} )^{4} }}]^{{ - {7 \mathord{\left/ {\vphantom {7 2}} \right. \kern-0pt} 2}}} [(V_{L} P_{L} )^{4} - 7(V_{L} P_{L} )^{2} (P + P_{L} )^{4} + 2(P + P_{L} )^{8} ]\). In order for Formula \(K_{\eta }^{\prime \prime } = 0\) to hold, it is only if the formula \((V_{L} P_{L} )^{4} - 7(V_{L} P_{L} )^{2} (P + P_{L} )^{4} + 2(P + P_{L} )^{8} = 0\) when the two inflexion pressures of the curvature curve are obtained:

$$P_{se} \, = \,\left( {\frac{{7 - \sqrt {41} }}{4}} \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} *(V_{L} P_{L} )^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} + P_{L} ,\,P_{st} \, = \,\left( {\frac{{7 + \sqrt {41} }}{4}} \right)^{{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}}} *(V_{L} P_{L} )^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} + P_{L} .$$

The sensitive pressure of Y88-1 sample is 0.29 and the starting pressure is 3.97(Fig. 4).

Fig. 4
figure 4

The curve diagram of second derivative of curvature of desorption curve

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The desorption efficiency of shale is \(2.59{{m^{3} } \mathord{\left/ {\vphantom {{m^{3} } {(t \cdot Mpa)}}} \right. \kern-0pt} {(t \cdot Mpa)}}\) under sensitive pressure and \(0.55{{m^{3} } \mathord{\left/ {\vphantom {{m^{3} } {(t \cdot Mpa)}}} \right. \kern-0pt} {(t \cdot Mpa)}}\) under starting pressure.

2.2 Method establishment

According to the three key pressure points of starting pressure, turning pressure and sensitive pressure, the isotherm adsorption curve can be divided into three stages: low efficiency analysis stage, slow resolution stage, rapid resolution stage and sensitive resolution stage (Fig. 5).According to the different pressure of shale reservoir, the ratio of desorption gas content to gas content in different desorption stages can be obtained. The calculation formula is shown in Table 1.

Fig. 5
figure 5

Schematic diagram of stage division of isotherm adsorption curve

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Table 1 the ratio of desorption gas content of gas content in different stages of shale interpretation under different reservoir pressure conditions
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In the process of pressure-relief exploitation of shale gas reservoir, it can be divided into four desorption stages:

  1. (1)

    Low efficiency desorption stage: formation pressure is greater than starting pressure, desorption efficiency is less than 0.55 \(m^{3} /(t \cdot MPa)\), shale gas content in this stage of the ratio of gas content is high, productivity contribution is the largest.

  2. (2)

    Slow desorption stage: formation pressure is between transition pressure and starting pressure, desorption efficiency is in the range of 0.55–1.0 \(m^{3} /(t \cdot MPa)\), shale gas content in this stage accounted for a higher proportion of gas content, which contributes a lot to shale gas well productivity.

  3. (3)

    Rapid desorption stage: formation pressure is between sensitive pressure and turning pressure, analytic efficiency is between 1.0–2.59 \(m^{3} /(t \cdot MPa)\), the ratio of shale analytic gas content of gas content is low in this stage, and it does not contribute much to shale gas well productivity.

  4. (4)

    Sensitive desorption stage: formation pressure is between the transition pressure and start-up pressure, desorption efficiency is greater than 2.59 \(m^{3} /(t \cdot MPa)\), shale gas content in this stage of gas content is low, and productivity contribution is small.

Taking Y88-1 sample as an example, the sampling depth is 2509.45 m, the average pressure gradient in the southeast edge of Ordos Basin is \(0.6MPa/100m\) [23], the sample location layer pressure is 15.1 \(MPa\). The desorption efficiency under formation pressure is 0.079. With the decrease of pressure, the average pressure gradient of Y88-1 layer in the southeastern edge of Ordos Basin is 15.1. With the decrease of pressure, the resolution efficiency increases gradually. The ratio of gas content in the stage of sensitive, rapid, slow and low efficiency is 10.1%, 40.8%, 17.5% and 30.7%, respectively.

The buried depth of shale samples tested in this study is 2300–2600 m, the formation pressure is 14.1–15.1 \(MPa\), and the formation pressure is higher than the starting pressure. Theoretically, there should be three pressure nodes in any desorption curve: starting, transition and sensitivity. Because of the difference of adsorption capacity, reservoir pressure and gas saturation, most shale cannot go through four desorption stages completely. With the exception of the Y88-1, which has gone through four complete desorption stages, others have no sensitive pressure points, and some samples have no transition pressure points and starting pressure points. The ratio of \(V_{L} /P_{L}\) is concentrated in the range of 0.5–2.0, indicating that in the process of depressurized production of Upper Paleozoic shale reservoirs in Ordos Basin, the desorption gas is dominated by low efficiency analysis, supplemented by slow analysis, and The exploitation stage does not go through the sensitive analysis stage or rarely go through the rapid analysis stage. The desorption efficiency of shale reservoirs is low, most of them are less than 0.55, a few of them are in the range of 0.55–1.0.The results are shown in Table 2.

Table 2 Isotherm adsorption parameter table of shale in Yulin area
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3 Results

The production of shale gas is related to the desorption efficiency of shale. The desorption efficiency is determined by Langmuir volume, Langmuir pressure and reservoir pressure. The desorption efficiency is positively correlated with Langmuir volume and Langmuir pressure. There is a negative correlation between reservoir pressure and Langmuir pressure. By studying the curvature of the desorption efficiency curve, it is found that the Langmuir isotherm adsorption line can be divided into four stages: sensitive resolution, fast resolution, slow resolution and low efficiency resolution, according to the three pressure nodes of the starting pressure, the turning pressure and the sensitive pressure. Slow desorption and low efficiency desorption contribute greatly to the productivity of shale gas wells. When the radio of \(V_{L} /P_{L}\) is greater than 2.59, Isothermal adsorption curve can also be divided into the four stages, and the larger the ratio is, the longer the sensitive efficiency desorption stage is. When the radio of \(V_{L} /P_{L}\) is in the range of 1.0–2.59, it can be divided into three stages: rapid, slow and low, and when the radio is 0.54–1, only slow and low efficiency desorption exist. When the radio is less than 0.54, there's only an inefficient parsing phase.

The buried depth of Upper Paleozoic shale is 2300–2600 m and the formation pressure is 14.1–15.1 in the eastern Ordos Basin. The formation pressure is obviously higher than the starting pressure. In the isothermal adsorption experiment, most of the samples were tested for starting pressure and turning pressure, but the sensitive pressure was rarely measured. The ratio of \(V_{L} /P_{L}\) is concentrated in 0.5–2.0. So, the isotherm adsorption curve almost has no sensitive resolution stage, rarely appears rapid resolution stage, with low efficiency desorption as the main, slow resolution as the auxiliary.

4 Discussion

  1. (1)

    The integrality of shale gas desorption stage is used to judge the quality of shale reservoir. The more complete the stage of shale gas desorption, the higher the efficiency of shale gas desorption, which is more conducive to shale gas exploitation. Organic carbon content and adsorbed gas content are important parameters of shale reservoir evaluation [24, 25]. It can be seen from Fig. 6 that there is a good positive correlation between shale adsorption capacity, organic carbon content and desorption efficiency. The desorption efficiency of shale gas can be used as an index to judge shale reservoir.

  2. (2)

    In the process of shale exploitation, with the decrease of pressure, the desorption efficiency of shale gas increases. According to Table 1, the ratio of desorption gas to gas content in different desorption stages under formation pressure can be calculated. It can guide the development of shale gas to a certain extent. If the formation pressure is low, the pressure can also be raised near the starting pressure to improve oil recovery. The average starting pressure of the Upper Paleozoic in the Ordos basin is 1.2, and the formation pressure of the Ordovician should be higher. The formation pressure coefficient of low pressure marine shale gas reservoir in Ordos Basin is increased from 0.7 to 1.88 by injecting 30,668 m3 fracturing fluid and 800 m3 liquid nitrogen and the test well ZP1 realizes 94 days of continuous gas–liquid two-phase flow [26, 27].Gas energy-enhanced fracturing tests have been carried out in 4 shale gas dessert areas in the central-northern part of the western Ordos Basin, which has effectively increased the formation pressure and achieved a breakthrough in the exploration of marine shale gas. Because the literature does not specifically mention the formation pressure data of each well after pressurization, it can not be compared with the data calculated in this paper, but the development idea used is consistent [27].

Fig. 6
figure 6

Diagram of the relationship between shale resolution efficiency and isotherm adsorption gas content & TOC

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5 Conclusion

The curvature of shale gas desorption efficiency curve of Upper Paleozoic shale samples in Ordos Basin is studied, and the shale gas desorption stages are divided according to start-up pressure, sensitive pressure and steering pressure. the ratio of desorption content to gas content in different stages is calculated. In the process of hydraulic fracturing of shale gas, kee** the formation pressure near the starting pressure is beneficial to improve the oil recovery of shale gas, which has a certain guiding significance for the exploration and development of shale gas. These are the theoretical results of our study of isothermal adsorption curves, and in the next step, we will compare shale gas development data of Wulalik formation in the central-northern part of western Ordos Basin with the isothermal adsorption curve of cores in the same formation, in order to obtain more practical data.