Comparison of the approaches to assessing the compressibility of the pore space
 1 — Ph.D., Dr.Sci. Chief Researcher Institute of Physics of the Earth RAS ▪ Orcid ▪ Elibrary
 2 — Ph.D., Dr.Sci. Deputy Director Institute of Physics of the Earth RAS ▪ Orcid
Abstract
Integral and differential approaches to determining the volumetric compression of rocks caused by changes in the stress state are considered. Changes in the volume of the pore space of rocks are analyzed with an increase in its allround compression. Estimation of changes in the compressibility coefficients of reservoirs due to the development of fields is an urgent problem, since the spread in the values of compressibility factors reduces the adequacy of estimates of changes in the physical properties and subsidence of the earth's surface of developed fields and underground gas storages. This parameter is key in assessing the geodynamic consequences of the longterm development of hydrocarbon deposits and the operation of underground gas storage facilities. Approaches to the assessment differ in the use of cumulative (integral) or local (differential) changes in porosity with a change in effective pressure. It is shown that the coefficient of volumetric compressibility of pores calculated by the integral approach significantly exceeds its value calculated by the differential approach, which is due to the accumulative nature of pore compression with an increase in effective pressure. It is shown that the differential approach more accurately determines the value of the pore compressibility coefficient, since it takes into account in more detail the features of the change in effective pressure.
Introduction
The physical properties of reservoir rocks are divided into two main categories: porositypermeability and deformationstrength. At the same time, the properties of rocks of the first category, primarily porosity and permeability, are basic in the preparation of projects for the development of hydrocarbon deposits and the calculation of their reserves [13]. Deformationstrength characteristics of rocks [46] (pore volume compressibility factor and tensile strength) are effectively used mainly to assess the deformation effects of longterm development of oil and gas fields and underground gas storages [79]. Changes in the stressstrain state cause the reaction of rocks, which is characterized by the coefficient of volume compressibility [10, 11]. The most wellknown are technogenic changes in the stressstrain state of the subsoil, due to the development of oil and gas fields, which are accompanied by changes in the physical properties of reservoir rocks [8, 12]. Determination of fluidsaturated rocks volumetric compressibility is very important for the development of hydrocarbon deposits [10, 13]. Its role is that oil and gas can be squeezed into wells under the influence of pressure from overlying rocks during the development of deposits for depletion [14, 15]. Also, oil can be displaced into producing wells by special fluids injected into the reservoir. In this case, the actual volumetric compressibility of the pores will affect the production, among other factors. Accurate determination of the volumetric compressibility of the pore space is especially important in layers of weakly consolidated rocks, where a large value of the compressibility of the pores can lead to significant subsidence of the earth's surface [16, 17].
There are many ways to determine the compressibility of rocks [18, 19], but only in rare cases is it indicated under what conditions and at what pressures they were obtained [6, 20]. Large discrepancies in the values of compressibility factors significantly affect the adequacy of estimated changes in physical properties [21, 22] and subsidence of the earth's surface of developed hydrocarbon fields and underground gas storages [23, 24].
The dependence of the compressibility coefficient of the pore volume on the value of the effective pressure has a pronounced nonlinear character [25, 26]. Therefore, it is important to know in what range of effective pressure changes the pore compressibility coefficient was estimated [27, 28]. This prompted the authors to consider the available approaches to assessing compressibility in more details and choose from them the most accurate and adequately reflecting changes in the stress state of rocks that do not exceed their elastic and/or strength limit.
The aim of the work is to analyze and compare the available approaches to assessing the compressibility of rocks and to select from them the most adequately reflecting changes in the effective pressure in the reservoir during the development of oil and gas fields and the operation of underground gas storage facilities.
Methodology
In compressible porous rocks, changes in either pore volume V_{p} or rock sample volume V_{b} are caused by changes in pore pressure or confining pressure [2830]. The compressibility of the pore volume depends on changes in the compression pressure
or changes in pore pressure
in practice, the value of effective pressure is often used
where Р_{с} – allround compression pressure; Р_{р} – pore (reservoir) fluid pressure (water, oil, gas); α – the Biot coefficient, sometimes called the unloading coefficient, characterizes the part of the pore pressure, which counteracts the allround pressure.
The α coefficient is taken equal to one in the case of permeable rocks with a porosity of more than 35 % [31, 32]. Then C_{pc} and C_{pp} will be equal, since equal changes in the confining and pore pressures cause the same changes in the volume of the pore space, but their signs will be opposite. In this case, a decrease in the volume of the pore space will be caused either by an increase in allround compression, or by a decrease in pore pressure [33]. In our case, the experiments were carried out under conditions when P_{p} = const, and the change in P_{eff }was achieved by increasing the allround compression. In this case, the designation C_{pc} = C_{p} is appropriate. The pore compressibility coefficient exponentially depends on the effective pressure, which can be justified using the concepts of linear poroelasticity.
As is known, the compressibility coefficient of the pore space was determined by the formula
where ΔV_{p} – change in the volume of the pore space (volume of pore fluid squeezed out of the sample), cm^{3}; V_{p}_{0} – the initial volume of the pore space of the sample, cm^{3}; ΔP_{eff} – changes in effective pressure, MPa.
Numerous experiments indicate that with increasing effective pressure, the pore compressibility coefficient decreases. In differential form, this corresponds to the equation
where b – empirical coefficient that varies for different rocks and characterizes the intensity of changes in compressibility with changes in effective pressure, MPa^{–1}.
When integrating equation (5) in the range of changes in the compressibility coefficient from the initial value С_{р}_{0} to the final C_{p} and from the initial effective pressure P_{eff}_{0} to the final P_{eff}, we obtain the equality
When integrating and substituting limits, an exponential dependence С_{р}from ΔP_{eff} is obtained:
or
Typical dependence of the coefficient of volumetric compressibility of the pore space on the change in effective pressure (Fig.1): increase in effective pressure from 0.1 to 37.0 MPa; trend approximation equation С_{р} = 0.0062е^{–0.0259P}^{eff}; coefficient of determination R^{2} = 0.98; intensity of change in compressibility with change in effective pressure b = –0.0259. Naturally, with such a dependence of the compressibility coefficient on the effective pressure, the estimate of the coefficient will differ for different ranges of pressure change.
In this case, 34 sandyargillaceous samples of the Vendian age of the Chayandinskoe field were used under conditions simulating reservoir conditions: porosity in atmospheric conditions is 315.6 %, gas permeability from 2·10^{–15} to 100·10^{–15} m^{2}, the depth from which the core was taken is 16601860 m. The samples are mainly represented by coarse and finegrained sandstones of quartz and quartzfeldspar composition with mixed quartzregeneration, sulfate, clayfilm and partially basal cement [34].
Depending on the chosen range of effective pressure variation, two approaches to estimating the pore compressibility coefficient are possible – integral and differential. This is especially true when the values of the pore compressibility coefficients are used to assess the anomalous deformations of the earth's surface of various oil and gas objects. So, for example, when assessing the subsidence of the earth's surface during the operation of oil fields, it is necessary to take into account the injection of fluid during the reservoir pressure maintenance procedure (RPM), which can restore the initial reservoir pressure and, accordingly, the effective pressure to the initial value after the fluid is withdrawn from the reservoir and reduce reservoir pressure, which can lead to a decrease in drawdowns, down to zero values. When injected into the reservoir, more complex effects are also possible, which go beyond the poroelastic concepts on which this work is based. When developing gas fields in the depletion mode, the increase in effective pressure occurs monotonously over a long period of time. The operation of underground gas storages (UGS) occurs in a cyclic manner, which leads to the need to take into account the intervals of alternating changes in effective pressure. Different modes of change in effective pressure during the operation of oil and gas facilities lead to the need to use different approaches to determine the values of the pore compressibility coefficient.
Integral (averaged) approach implies an estimation of the compressibility factor for the entire range of effective pressure changes. In particular, the change in the coefficient ΔC_{p} is determined in total for the entire interval of effective pressure change from P_{eff 0} to P_{eff}. Usually, the value of the compressibility coefficient at an effective pressure close to atmospheric (0.12.0 MPa) is taken as the initial value, and for the final one – the value of the compressibility coefficient at the effective pressure for conditions simulating reservoir [31, 35], e.g. 37.0 MPa.
Differential approach involves estimating the change in the values of the compressibility coefficients in the selected interval of change in the effective pressure. It should be noted that the differential approach has two forms of representation: cumulative and discrete. In the first variant, the change in the pore volume is counted from the initial value V_{p}_{0}, and its decrease is measured over the
ΔP_{eff} change intervals, which successively increase in the ranges, for example, 0.15.0; 0.110.0; 0.120.0 etc. up to the range of 0.137 MPa. In the second variant, the change in the pore volume is each time counted from the value of V_{p}, which corresponds to the value of the effective pressure from which the interval of its change begins, for example, 0.15.0; 5.010.0; 10.020.0; 20.030.0 and 30.037.0 MPa.
Methodology and research object
The methodology of experimental studies, described in detail in the work [33], allows you to directly measure the volume of the pore fluid ΔV_{p} squeezed out of the sample and calculate not only changes in porosity, but also the volumetric deformation of the sample. Cylindrical sandstone samples with a diameter of 30 mm and a length of 30 mm were studied, which were subjected to allround compression at a controlled pressure of the pore fluid. The use of a technique that makes it possible to measure the volume of the pore fluid squeezed out of the sample with an increase in the allround compression of the sample made it possible to determine changes in the pore volume [34]. The ΔV_{p} value in reservoir conditions is determined taking into account the fact that the compressibility coefficient of the solid rock matrix is several orders of magnitude greater than the compressibility coefficient of the pore space.
When carrying out petrophysical studies using this technique, the pressure and temperature of the pore fluid (formation water model) were maintained constant (13 MPa, 20 °С), only by increasing the allround pressure to 37 MPa. Therefore, taking into account the coefficient of compressibility of the liquid on pressure and temperature was not required. Thus, based on formula (4), the coefficient of volumetric compressibility of the pore space was determined with a change in effective pressure.
In this case, to estimate the compressibility coefficient, an approach can be proposed based on parameters exclusively measured during the experiment: the volume of the pore space ΔV_{p} and variations in the effective pressure ΔP_{eff}. For this, formula (4) can be transformed into a differential form:
Equation (9) implies the differential equation
Then, integrating equation (10) in the range of pore volume changes from the initial value V_{p}_{0} to the final V_{p} and from the initial effective pressure P_{eff}_{0} to the final P_{eff} , we obtain
or
When comparing formulas (7), (8) and (11), (12), it can be seen that formulas (11), (12) for determining the compressibility coefficient contain only quantities measured in this experiment, and formulas (7), (8) include the coefficient b, which must be determined from additional experiments.
Discussion
Integral approach
Determination of the pore volume compressibility coefficient in accordance with the integral approach can be performed graphically (Fig.2), using the linear dependence of changes in the logarithm of the pore volume on the effective pressure. From formula (11) it follows
If we build a semilogarithmic graph of the dependence of ln (V_{p}) on the effective pressure, then we get the equation of an inclined straight line with a constant slope equal to C_{p} (Fig.2). The slope of the graph is the average value of the coefficient of volumetric compressibility of the pore space in accordance with the integral approach. The value of ln(V_{p}_{0}) is determined from the condition ΔP_{eff} = 0. In this case, the value of ln (V_{p}) is dimensionless, which shows the degree to which (~ 2.72) a number equal to the current pore volume (V_{p}) should be raised.
The obtained experimental dependence can be described with high reliability (R^{2} = 0.94) by the logarithmic equation (11):
In this case, the average value of the coefficient is –0.00249 MPa^{–1} (the slope of the graph in Fig.2) in the range of effective pressure changes of 0.137.0 MPa. And the free term of equation (14) 0.631 corresponds to the natural logarithm of the pore volume in the initial state.
Thus, the integral approach can be applied for graphical estimation of the average value of the volumetric compressibility of pores in the development of hydrocarbon fields only according to the data on the change in the volume of the pore space from the initial value V_{p}_{0} to the current V_{p} when the effective pressure changes from the initial value to the corresponding reservoir conditions. However, the nonlinear nature of the change in the compressibility coefficient from the effective pressure (see Fig.1) necessitates a detailed study of this dependence for various intervals of pressure change.
Differential approach
Based on the available experimental data, the entire range of effective pressure changes of 0.137.0 MPa was divided into local segments: 0.15.0; 0.110.0; 0.120.0; 0.130.0 and 0.137.0 MPa. At the same time, the values of the pore volume at an effective pressure of 0.1 MPa were taken as the initial values; the final values are 5.0; 10.0; 20.0; 30.0; 37.0 MPa. Figure 3 shows changes in pore deformation in the differentialaccumulative approach with an increase in effective pressure.
Changes in pore deformation are zero at Р_{eff} = 0.1 MPa and increase with increasing effective pressure, approximated by linear dependences, the slope of which is the volumetric compressibility coefficient of pores С_{р}_{i} (4). The C_{p} value is: –0.005653; –0.004895; –0.003795; –0.003175 and –0.002845 MPa^{–1} with changes in R_{eff }: 0.15.0; 0.110.0; 0.120.0; 0.130.0 and 0.137.0 MPa respectively.
Different intervals of effective pressure variation correspond to different values of the pore compressibility coefficient (Fig.3). This must be taken into account when assessing the final settlements of gas fields developed in the depletion mode [13, 14, 34].
The use of the differentialdiscrete approach is shown in Fig.4, which shows the linear dependences of changes in the volumetric deformation of pores on changes in effective pressure. In contrast to the differentialaccumulative approach, the initial values are the pore volume at the beginning of each individual interval: 0.1; 5; 10; 25; 30; 37 MPa. Changes in pore deformation are equal to zero at Р_{eff} equal to 0.1; 5; 10; 20 and 30 MPa, and increase with increasing effective pressure and are approximated by linear dependencies, the slope of which is the coefficient of pore volume compressibility С_{р}_{i} (4). C_{p} values obtained: –0.005771; –0.003880; –0.002319; –0.001669, –0.001284 MPa^{–1} with the following changes in R_{eff}: 0.15.0; 5.010.0; 10.020.0; 20.030.0 and 30.037.0 MPa, respectively.
The volumetric compressibility coefficient of pores in this case is determined by the slope of the straight lines connecting the initial and final values of the volumetric deformation of pores with increasing effective pressure. In contrast to the differentialaccumulative approach, for normalization (the ratio of changes in the pore volume to the initial value) changes in the volumetric deformation of pores ΔV_{p}/V_{p}, the values of the pore volume at the initial value of the effective pressure of each stage of its growth are used. In Fig.4, both the general decrease in the value of the volumetric compressibility coefficient (from –0.00577 to –0.00128 MPa^{–1}) and the nonmonotonic nature of the decrease with increasing effective pressure are more clearly manifested.
The differentialdiscrete approach can be applied for various changes in the effective pressure, both with an increase and a decrease. However, it is necessary to have information about the volume of pores and their change at each stage of the development process. This is especially relevant when assessing anomalous deformations of the earth's surface during the operation of underground gas storage facilities, where a signalternating change in reservoir pressure is realized.
Pore volume compressibility coefficient estimation comparisson
Values comparison of the pore volume compressibility coefficients, which were obtained experimentally from the volumes of liquid squeezed out of the samples at each stage of effective pressure growth, and calculated using the integral and differential approaches, is shown in Fig.5.
Analyzing the Fig.15, it can be seen that the value of the volumetric compressibility coefficient, which is maximum at the initial (minimum) values of the effective pressure, decreases with its growth and can be approximated with a high degree of reliability (R^{2} = 0.98) by the exponential dependence (8) (see Fig.1). In this case, both integral and differential approaches can be used. And the average value when P_{eff} changes from 0.1 to 37.0 MPa with an integral approach can also be estimated using formula (13) (see Fig.2, 5).
Coefficient of volumetric compressibility of pores values are summarized in the Table.
Comparison of pore compressibility coefficients (C_{p}, 10^{3}·MPa^{–1}) calculated using different approaches
Approach 
Р_{eff}, MPa 

0.1 
5.0 
10.0 
20.0 
30.0 
37.0 

Integral over changes in pore volume ln (V_{p}) 
– 
–2.495 
–2.495 
–2.495 
–2.495 
–2.495 
Differentialcumulative according to the accumulated volumetric deformation of the pores dV_{р}_{i}/V_{р}_{0} 
– 
–5.771 
–4.753 
–3.469 
–2.823 
–2.508 
Differentialdiscrete in steps of change Р_{eff }and volumetric deformation of pores dV_{р}_{i+}_{1}/V_{р}_{i} 
– 
–5.771 
–3.880 
–2.319 
–1.669 
–1.284 
At the initial effective pressure (5.0 MPa), the integral approach significantly underestimates the pore compressibility factor. In this case, the differentialaccumulative and differentialdiscrete approaches coincide. It is obvious that the integral and differentialaccumulative approaches significantly overestimate (up to 95.4 %) the value of the pore compressibility coefficient at effective pressures close to reservoir pressures (3037 MPa) (Fig.5). It is also obvious that the use of the differentialdiscrete approach gives the values of the pore compressibility coefficient, which more adequately correspond to the stress state of the formation.
It is known that the effective pressure in the reservoir under development can vary in the range of ± 1020 MPa [8, 36]. At the same time, the differential (differentialdiscrete) approach proposed by the authors makes it possible to estimate the compressibility of the pore space even in the absence of laboratory studies of the core compressibility. It is easy to show that the change in normalized porosity is equal to the normalized change in pore volume: Δm/m_{0} = ΔV_{p}/V_{p}_{0}. Then, using the values of changes in the porosity coefficient obtained during well logging within the framework of GIS control of the field development process or UGS operation, it is possible to estimate the pore volume compressibility coefficient for known changes in effective pressure. The differential approach makes it possible to more accurately assess the magnitude of possible subsidence of the earth's surface at various stages of field development.
Similarly, during the operation of underground storage facilities, to assess the pore compressibility factor, it is recommended to use a differentialdiscrete approach in order to more correctly (differentially) take into account changes in effective pressure occurring in the reservoir during gas injection and withdrawal periods. It is of fundamental importance that the values of the pore volume compressibility factors will be different for the period of gas injection and withdrawal even with the same change in reservoir (effective) pressure. This happens because the same (taking into account the sign) changes in the pore volume are normalized (divided) into different values of the pore volume corresponding to different values of the effective pressure. For example, with an increase in effective pressure from 10 to 20 MPa, the change in pore volume ΔV_{p}_{0} is divided by the value of the pore volume V_{p}_{0}, which corresponds to a value of 10 MPa. When the effective pressure decreases from 20 to 10 MPa, the identical change in the pore volume ΔV_{p} (in this case, the increase) is normalized to the value V_{p}_{0}, which corresponds to the value of the effective pressure of 20 MPa. Naturally, in the second case, the compressibility factor will be greater than in the first, since the denominator of the fraction ΔV_{p}/V_{p}_{0} in the first case will be greater than in the second. This must be taken into account in a detailed analysis of the results of geodynamic monitoring of underground gas storage facilities.
The results of the experimental data discussion make it possible to recommend laboratory differential approaches (cumulative and discrete) for the practical determination of the pore volume compressibility coefficient, since other external factors and uncertainties in pressure values that are present in field determinations during field development are excluded. The GIScontrol method using changes in porosity can be recommended as a refinement method aimed at monitoring changes in the volumetric compressibility of the pore space and confirming laboratory results. For a detailed analysis of changes in the stress state of underground gas storage facilities, a differentialdiscrete approach to estimating the pore volume compressibility coefficient can be recommended, as it allows taking into account alternating changes in reservoir pressure.
Consistently taking into account the differences in compressibility coefficients also makes it possible to study such an important empirical fact as weak subsidence of the earth's surface in longterm developed gas fields in the depletion mode, which was noted based on the results of longterm monitoring at a field in Turkmenistan [8, 37]. To explain it, a genetic model was used for the formation of subsidence in the territory of deposits, which used the relationship between relative changes in the pore compressibility coefficient and relative changes in the volumetric modulus K of the host medium in an environment of permanent (quasistatic) stresses that formed the structure of the anticlinal uplift over geological time, to which the deposit is assigned. These forces (stresses) contribute to the uplift of the earth's surface, compensating for subsidence, under conditions of a drop in the effective stiffness (volumetric modulus) of the reservoir. It is known from geomechanics that the compressibility coefficient is inversely proportional to K [6, 8]. Relative (normalized) changes in the compressibility coefficient С_{р} are related to relative changes in the bulk modulus by the following formula:
The minus sign in equation (15) means that as the compressibility coefficient increases, the value of the bulk modulus decreases, and vice versa. Relation (15) makes it possible to make the transition from petrophysical models to models of poroelastic heterogeneity, on the basis of which it is possible to evaluate the change in the stressstrain state of the developed reservoir (layers) of oil and gas fields and underground gas storages.
The scientific novelty of the work lies in the analytical substantiation of the exponential dependence of the pore compressibility coefficient on the effective pressure, which was previously assumed in [34] and confirmed by the results of experimental studies on rock samples.
Conclusion
Comparison of the integral and differential approaches to determining the coefficient of volumetric compressibility of the rock samples pore space emphasizes the damping nature of its decrease with an increase in effective pressure, which can be approximated with a high degree of reliability by an exponential dependence, which has been proven using the basic relations of geomechanics of poroelastic media. It was found that the coefficient of pore volume compressibility calculated by the integral approach is significantly higher (up to 130 %) than the coefficient calculated using the differential approach. This difference is due to the cumulative nature of the decrease in pore volume with an increase in effective pressure. Obviously, the differential approach makes it possible to more accurately determine the compressibility of the pore space due to a more appropriate consideration of changes in the effective pressure in the reservoir and can be recommended for assessing the dynamics of the physical properties of rocks in the process of developing oil and gas fields and operation of underground storage facilities, as well as for studying anomalous deformations of the earth's surface within these objects.
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