RU2172942C1 - Method measuring porosity and method measuring distribution of pores according to sizes - Google Patents

Abstract

FIELD: study of physical characteristics of solid. SUBSTANCE: in accordance with method measuring porosity starting volume of material is measured and material is placed in chamber with liquid. Pressure in chamber is raised to force liquid into material. Volume of liquid forced into material is measured. Correction related to presence of gas in material is accounted for in measured volume of liquid. Porosity of material is found by relation of measured volume of liquid forced into material with due account of correction to starting volume of material. Liquid is poured into chamber before placement of material into it. Pressure in chamber is raised and compressibility of liquid is measured. Time characteristic of pressure in chamber and /or volume of liquid are measured to find point of inflection in it. After total forcing of liquid into material pressure in chamber is raised and deformability of material is measured. Additional corrections related to compressibility of liquid and to deformability of material are taken into account in measured volume of liquid. In correspondence with method measuring distribution of pores according to sizes characteristic of change of relative volume of liquid forced into material by pressure is measured in addition to measurement of volume of liquid and this characteristic is divided into discrete intervals which are used to determine distribution of pores by sizes. Additional corrections related to compressibility of liquid and deformability of material are also taken into account in measured volume of liquid. EFFECT: increased accuracy and reduced time of measurement of porosity and of distribution of pores by sizes in real seam conditions, provision for ecological safety of measurements. 25 cl, 11 dwg, 1 tbl

Description

 The invention relates to the study of the physical characteristics of solids and can be used to measure porosity and to determine the size distribution of pores.

 A known method of measuring core porosity, which consists in placing the material in a closed chamber with a non-wetting liquid, such as mercury, measuring the volume of the displaced liquid equal to the volume of the material, and then reducing the pressure in the chamber to a certain value by increasing the volume of the chamber. In this case, the pore volume of the material is calculated by the volume of gas extracted from the pores and the ratio of the new (reduced) pressure to atmospheric pressure (1).

 A limitation of this method is low accuracy, the impossibility of measuring the pore size distribution and the need to use mercury harmful to the ecology and health of the experimenter in the measurement process.

 A known method for determining porosity by saturating a material with various liquids (Preobrazhensky method), for example kerosene (2). To do this, measure the mass of dry material and material saturated with a liquid of known density over a period of time sufficient to fill the pore space. After that, the mass of the saturated material is measured in air and in the liquid, on the basis of which the volume of the material and the volume of liquid filling the pores are calculated and, accordingly, the porosity of the material is calculated.

 A limitation of this method is its low efficiency, since the process of saturating the test material with liquid requires special equipment and is quite time-consuming (from several hours to several days), as well as low measurement accuracy due to swelling of the material and a change in its volume, incomplete filling of pores.

 The closest in technical essence to the proposed one is a method of measuring porosity, including measuring the initial volume of the material, placing the material in the chamber with the liquid, increasing the pressure in the chamber to press the liquid into the material, measuring the volume of the liquid pressed into the material, taking into account the corrections in the measured volume of the liquid, associated with the presence of gas in the material, the determination of the porosity of the material in relation to the measured volume of liquid pressed into the material, taking into account the amendment, to the initial volume of materials iala (3).

 Closest to the technical nature of the proposed is also a method of measuring the distribution of pore sizes, including measuring the initial volume of the material, placing the material in the chamber with the liquid, increasing the pressure in the chamber to press the liquid into the material, measuring the volume of liquid pressed into the material, taking into account the measured liquid volume corrections associated with the presence of gas in the material, determining the pore size distribution in relation to the measured volume of liquid pressed into the material, taking into account the correction and, to the original volume of material, while in the measurement volume of fluid material forced back into the measured characteristic change of the relative volume of fluid is pressed into the material of the pressure and it is divided into discrete intervals, which determine the pore size distribution (3).

 Typically, the liquid selected is mercury having a high surface tension coefficient and a non-wetting sample at atmospheric pressure. The volume of liquid pressed into the pore space of the sample material under pressure then increases by the volume of gas compressed in the pores. Suppose that when indenting the working fluid during measurement, the pressure will increase to 50 atm, compressing the gas to 1/50 of its volume under normal conditions. In this case, the measured value of the volume of the pressed fluid must be increased by 2% (by the volume of gas compressed in the pores) to obtain the total pore volume.

 The limitation of the method of measuring porosity and the method of measuring the distribution of pores in size is the lack of efficiency and low accuracy of determining the porosity of the material, due to the inability to determine the deformation of the material (skeleton of the sample) and the compressibility of the liquid to introduce a subsequent correction when calculating the final porosity.

 Other limitations of these methods are: the need to use mercury in the measurement process, which is harmful to the environment and the health of the operator, and after which the material sample is difficult to clean for future use; the impossibility of finding the pore size distribution in real reservoir conditions, since the pore size distribution in the method is determined only by the volume of fluid pressed into the material at a certain pressure; the need to use samples of material that are large enough in size; a decrease in the size of material samples leads to a significant decrease in the accuracy of measurement; the need for strict temperature maintenance during the measurement process.

 The problem solved by the invention is to increase the efficiency and quality of measurements.

 The technical result that can be obtained by implementing the inventive method for measuring porosity is to increase the accuracy and expressness of measuring the porosity of samples of materials, measure porosity for real reservoir conditions, as well as ensure environmental safety of measurements.

 The technical result that can be obtained by implementing the inventive method for measuring the size distribution is to increase the accuracy, expressness of measurement, measure the distribution of pore sizes in real reservoir conditions, as well as ensure environmental safety of measurements.

 To solve the problem with the achievement of the specified technical result in the known method of measuring porosity, including measuring the initial volume of the material, placing the material in the chamber with the liquid, increasing the pressure in the chamber for pressing the liquid into the material, measuring the volume of the liquid pressed into the material, accounting for the measured volume fluid corrections associated with the presence of gas in the material, determining the porosity of the material in relation to the measured volume of fluid pressed into the material, taking into account the amendment, to the initial volume of material according to the invention before placing the material in the chamber with the liquid, this liquid is placed in it, the pressure in the chamber is increased and the compressibility of the liquid is measured, while measuring the volume of liquid pressed into the material, the temporal characteristic of the pressure in the chamber and / or the volume of liquid is measured for the inflection point in it, which determines the moment when the pores of the material is completely filled with liquid, after the pores of the material are completely filled with liquid, the pressure in the chamber is additionally increased re and measure the deformability of the material, take into account the additional correction associated with its compressibility in the measured volume of the liquid, and take into account the additional correction related to the deformability of the material in the measured initial volume of the material, determine the porosity of the material taking into account additional corrections.

где V_L - объем жидкости при атмосферном давлении P_at,
V^* _L - объем жидкости при величине давления P^* в точке перегиба барической зависимости относительного суммарного объема,
Δ P=P^* - P_at - приращение давления;
- деформируемость β_s материала определяли по формуле

где V_s - объем материала при атмосферном давлении P_at,
V_Σ1,V_Σ2 - суммарные объемы, включающие объем жидкости и объем помещенного в нее материала, при давлениях P₁ и P₂, соответственно, с небольшой разницей давлений из находящихся в диапазоне давлений, больших P^*.When carrying out the invention, additional embodiments of the method of measuring porosity are possible, in which it is advisable that:
- after the complete indentation of the liquid into the material, the pressure in the chamber was additionally increased to a value exceeding the value of the pressure of completely filling the pores by at least 5 times;
- an additional increase in pressure in the chamber was made up to pressures of 300 MPa;
- increase the pressure in the chamber and / or additionally increase the pressure in the chamber at a constant speed;
- increase the pressure in the chamber and / or additionally increase the pressure in the chamber spasmodically;
- increase the pressure in the chamber and / or additionally increase the pressure in the chamber with a speed that provides an isothermal condition for the liquid is pressed into the material;
- the time t ^{* of the} complete filling of the pores of the material with liquid was determined by the inflection point on the pressure dependence of the derivative of the total volume ∂V / ∂t when the liquid was pressed into the material and by the time dependence of the pressure change;
- the time t ^{* of the} complete filling of the pores of the material with liquid was determined by the inflection point on the time dependence of the derivative of the total volume ∂V / ∂t when the liquid is pressed into the material;
- the moment t ^{* of the} complete filling of the pores of the material with liquid was determined by the inflection point on the time dependence of the pressure when the liquid was pressed into the material;
- the time t ^{* of the} complete filling of the pores of the material with liquid was determined by the inflection point on the pressure dependence of the total volume of the liquid when it was pressed into the material and by the time dependence of the pressure change;
- compressibility β _W fluid was determined by the formula

where V _L is the volume of fluid at atmospheric pressure P _at
V ^* _L is the volume of liquid at a pressure value of P ^* at the inflection point of the pressure dependence of the relative total volume,
Δ P = P ^* - P _at - pressure increment;
- deformability β _{s of the} material was determined by the formula

where V _s is the volume of the material at atmospheric pressure P _at ,
V _Σ1 , V _Σ2 - total volumes, including the volume of liquid and the volume of the material placed in it, at pressures P ₁ and P ₂ , respectively, with a small pressure difference from those in the pressure range, large P ^* .

где V $\genfrac{}{}{0ex}{}{\text{*}}{\text{Σ}}$ - суммарный объем при давлении P^*,
V_l = V_L - V_s.V _l ¹ - the volume of liquid at a pressure P _l for the case when liquid and material are placed in the chamber,
V _L1 , V _L2 - fluid volumes for pressures P ₁ and P _2, respectively;
- porosity m ^{* of the} material was determined by the formula
m ^* = m • [1 + β _s • (P ^* -P _at )]
Where

where v $\genfrac{}{}{0ex}{}{\text{*}}{\text{Σ}}$ - total volume at pressure P ^* ,
V _l = V _L - V _s .

 To solve the problem with the achievement of the specified technical result in a known method for measuring the distribution of pore sizes, including measuring the initial volume of the material, placing the material in the chamber with the liquid, increasing the pressure in the chamber for pushing the liquid into the material, measuring the volume of the liquid pressed into the material, accounting in the measured liquid volume, corrections associated with the presence of gas in the material, determining the pore size distribution in relation to the measured volume of liquid pressed into taking into account the correction to the initial volume of the material, while measuring the volume of liquid pressed into the material, the characteristic of the change in the relative volume of the pressed liquid into the material is measured from pressure and divided into discrete intervals, according to which the pore size distribution according to the invention is determined before by placing the material in a chamber with a liquid, this liquid is placed in it, the pressure in the chamber is increased and the compressibility of the liquid is measured, when measuring the volume of liquid pressed into the material, one temporarily measure the temporal characteristic of the pressure in the chamber and / or the volume of the liquid to find the inflection point in it, which determines the moment when the pores of the material are completely filled with liquid, after the pores of the material are completely filled with liquid, the pressure in the chamber is additionally increased and the deformability of the material is measured, and the measured volume of liquid additional amendment related to its compressibility, and take into account in the measured initial volume of material an additional amendment related to the deformability of the material la, determine the pore size distribution taking into account the mentioned additional amendments.

где V_L - объем жидкости при атмосферном давлении P_at,
V^* _L - объем жидкости при величине давления P^* в точке перегиба барической зависимости относительного суммарного объема,
Δ P=P^*-P_at - приращение давления;
- деформируемость β_s материала определяли по формуле

где V_s - объем материала при атмосферном давлении P_at,
V_Σ1,V_Σ2 - суммарные объемы, включающие объем жидкости и объем помещенного в нее материала, при давлениях P₁ и P₂ соответственно с небольшой разницей давлений из находящихся в диапазоне давлений, больших P^*.When carrying out the invention, additional embodiments of the method of measuring the distribution of pore sizes are possible, in which it is advisable that:
- after the complete indentation of the liquid into the material, the pressure in the chamber was additionally increased to a value exceeding the value of the pressure of completely filling the pores by at least 5 times;
- an additional increase in pressure in the chamber was made up to pressures of 300 MPa;
- increase the pressure in the chamber and / or additionally increase the pressure in the chamber at a constant speed;
- increase the pressure in the chamber and / or additionally increase the pressure in the chamber spasmodically;
- increase the pressure in the chamber and / or additionally increase the pressure in the chamber with a speed that provides an isothermal condition for the liquid is pressed into the material;
- the time t ^{* of the} complete filling of the pores of the material with liquid was determined by the inflection point on the pressure dependence of the derivative of the total volume ∂V / ∂t when the liquid was pressed into the material and by the time dependence of the pressure change;
- the time t ^{* of the} complete filling of the pores of the material with liquid was determined by the inflection point on the time dependence of the derivative of the total volume ∂V / ∂t when the liquid is pressed into the material;
- the moment t ^{* of the} complete filling of the pores of the material with liquid was determined by the inflection point on the time dependence of the pressure when the liquid was pressed into the material;
- the time t ^{* of the} complete filling of the pores of the material with liquid was determined by the inflection point on the pressure dependence of the total volume of the liquid when it was pressed into the material and by the time dependence of the pressure change;
- compressibility β _W fluid was determined by the formula

where V _L is the volume of fluid at atmospheric pressure P _at
V ^* _L is the volume of liquid at a pressure value of P ^* at the inflection point of the pressure dependence of the relative total volume,
Δ P = P ^* -P _at - pressure increment;
- deformability β _{s of the} material was determined by the formula

where V _s is the volume of the material at atmospheric pressure P _at ,
V _Σ1 , V _Σ2 - total volumes, including the volume of liquid and the volume of the material placed in it, at pressures P ₁ and P _2, respectively, with a small pressure difference from those in the pressure range, large P ^* .

V _l ¹ - the volume of liquid at a pressure P ₁ for the case when the chamber is placed liquid and material,
V _L1 , V _L2 - fluid volumes for pressures P ₁ and P _2, respectively.

 Due to the indentation of the liquid into the pore space of the material in real time, the determination of the compressibility of the liquid and the deformability of the material by the nature of the time and pressure dependences of the change in the total volume of the working fluid with the material placed in it and the time dependences of pressure during indentation, it was possible to solve the problem.

 These advantages and features of the present invention are illustrated by the best option for its implementation with reference to the figures.

Figure 1 depicts a device for implementing the inventive method;
FIG. 2 - baric dependence of the total fluid volume in the process of pressing into the pores of the material;
FIG. 3 - time dependence of fluid pressure in the process of pressing into the pores of the material;
FIG. 4 - time dependence of the derivative of pressure in the process of pressing into the pores of the material;
FIG. 5 - time dependence of the derivative of the total volume of liquid in the process of pressing into the pores of the material;
FIG. 6 - pressure dependence of the derivative of the total volume in the process of pressing into the pores of the material;
FIG. 7 is a plot of the pressure dependence of the relative total volume in the pressure range at which pore filling occurs;
FIG. 8 - dependence of the pressed volume of the liquid on pressure;
FIG. 9 - dependence of the pressed volume of the liquid on the pore size;
FIG. 10 is a differential pore size distribution curve for a material sample;
FIG. 11 - normalized pore distribution function.

 The device (Fig. 1) for implementing the inventive method comprises a measuring chamber 1 with a sample of material 3 placed in it through a hermetically sealed cover 2, filling the measuring chamber 1 with liquid 4, sensors 5, 6, 7 of volume, pressure and temperature, respectively. The pressure generation system consists of a serially connected actuator 8 with a controlled input, an electric motor 9, a gearbox 10, a high-pressure cylinder 11 with a progressively rotating rod 12, the movement of which through the seal assembly 13 provides a pressure rise in the measuring chamber 1. The device has a data receiving unit 14 the input of which signals from the sensors 5, 6, 7 are supplied, and the first output is connected to the data processing unit 15. The diagram also shows: a differentiating device 16 for differentiating the time dependences of the volume and pressure from block 14, an information display block 17, and block 18, which synchronizes the operation of the entire device and sets the necessary conditions: the liquid compression rate, the frequency of removal of experimental points in real time.

The rotational movement of the stepper motor 9 through the gearbox 10 is converted into the translational motion of the rod 12. As a result of this control of the motor operation, the hydrostatic pressure in the measuring chamber can vary with constant or variable speed from 0.1 MPa / s to 57 MPa / s, volume - from 10 ^-3 mm ³ / s to 30 mm ³ / s, and also carried a compression mode in which the compression provided by isothermal conditions, since the unit 15 are input temperature data and indentation fluid in the pore space of the sample material 3 can be straight plague with automatic adjustment of the compression rate, whereby the isothermal conditions are provided by the compression process (T ^o = const).

 The process of measuring porosity is as follows.

The chamber 1 is filled with liquid 4 and, at a fixed temperature, a measurement is made in real time of the dependence of the volume of liquid V on pressure P in the pressure range of interest. It is convenient to present the data obtained in the form of a dependence of the relative volume V / V _L (V is the current volume at a given pressure P, V _L is the initial volume of liquid at atmospheric pressure equal to the volume V _{k of the} chamber 1 V _L = V _k ) as a function of pressure or time . An example of such a dependence for kerosene in the pressure range up to 50 MPa and a temperature of 20 ^° C is shown in FIG. 2, curve 1. From FIG. 2 shows that with increasing pressure up to 50 MPa, the volume of kerosene in comparison with the volume at atmospheric pressure decreases by 3%.

If in the chamber 1 containing the liquid 4 (Fig. 1), place a sample of material 3 of known volume V _s and measure the pressure dependence of the total volume of the working fluid with the sample placed in it (the initial volume of liquid V _l in this case is equal to the volume of the pressure vessel V _k minus the sample volume V _s : V _l = V _k - V _s ), then information on the porosity of the material sample can be obtained from the obtained data. A typical example of the obtained dependence of the relative total volume of liquid with a sample of material immersed in it on pressure is shown in FIG. 2, curve 2. As a liquid, kerosene was used at a temperature of 20 ^o C.

From FIG. Figure 2 shows that there is a significant difference between curves 1 and 2, namely: for curve 2 there is a sharp drop in the relative volume in the initial section of the curve. At pressures greater than 3 MPa, the dependence of the relative volume reaches a smooth curve similar to curve 1 for a liquid. A sharp drop in the relative volume on curve 2 is due to the indentation of the liquid into the material 3. With a further increase in pressure in excess of 3 MPa, the dependence of the relative total V _Σ volume on pressure goes to a smooth gentle curve, which indicates that the pore space is already filled with liquid and with a further increase pressure occurs only compression of the liquid and the material itself as a whole under the action of hydrostatic pressure forces. Using the dependences presented on curves 1 and 2, it is possible to calculate the porosity of the material with high accuracy.

Зависимость (фиг. 2, кривая 2) позволяет определить величину V_p, так как она содержит информацию об объеме жидкости, вдавленной в поры материала. Как отмечалось выше, на кривой 2 имеется точка перегиба (характерный излом), которая указывает на окончание заполнения пор материала жидкостью. Пусть P^* - давление, при котором происходит резкое изменение характера барической зависимости относительного суммарного объема (фиг. 2, кривая 2). Эта точка может быть определена, например, по скачку первой производной временной зависимости давления (фиг. 4) или первой производной барической зависимости суммарного объема (фиг 6). Так, для кривой 2 величина P^* равна приблизительно 3 МПа. Величина относительного суммарного объема

соответствующая давлению P^*, дает информацию об объеме жидкости, вдавленной в поры. Так как произведение начального суммарного объема жидкости V_L на величину изменения относительного суммарного объема [1-V $\genfrac{}{}{0ex}{}{\text{*}}{\text{Σ}}$ /V_L] равно объему жидкости, вдавленной в поры, то есть объему пор V_p, то, зная объем образца материала V_s, с учетом формулы (I) можно рассчитать пористость образца m:

Формула (2) представлена без учета сжимаемости жидкости и деформируемости (сжимаемости скелета образца) материала, а также без учета сдавливаемого в капиллярах газа (воздуха).The porosity of a sample m is the ratio of the pore volume V _p to the sample volume V _s . Thus, to calculate the porosity of a sample of a material of known volume V _s, it is necessary to determine the pore volume in this sample V _p . Then the porosity of the sample can be calculated by the formula:

The dependence (Fig. 2, curve 2) allows you to determine the value of V _p , since it contains information about the volume of fluid pressed into the pores of the material. As noted above, on curve 2 there is an inflection point (characteristic kink), which indicates the end of the filling of the pores of the material with liquid. Let P ^* be the pressure at which a sharp change in the nature of the pressure dependence of the relative total volume occurs (Fig. 2, curve 2). This point can be determined, for example, by a jump in the first derivative of the time dependence of the pressure (Fig. 4) or the first derivative of the pressure dependence of the total volume (Fig. 6). So, for curve 2, the value of P ^* is approximately 3 MPa. The value of the relative total volume

corresponding to the pressure P ^* , gives information about the volume of fluid pressed into the pores. Since the product of the initial total fluid volume V _L by the amount of change in the relative total volume [1-V $\genfrac{}{}{0ex}{}{\text{*}}{\text{Σ}}$ / V _L ] is equal to the volume of liquid pressed into the pores, that is, the pore volume V _p , then, knowing the volume of the material sample V _s , taking into account formula (I), we can calculate the porosity of the sample m:

Formula (2) is presented without taking into account the compressibility of the liquid and the deformability (compressibility of the sample skeleton) of the material, and also without taking into account the gas (air) squeezed in the capillaries.

Если считать, что процесс измерения пористости является изотермическим и что весь объем V_p пор материала перед экспериментом заполнен газом (образец полностью экстрагирован), то отношение объема V_p газа в материале при атмосферном давлении P_at и объема V_b ^* газа при давлении P^* находится по закону Бойля:

Тогда доля V_b ^* газа, зажатого в порах материала при давлении P^*, согласно формуле (4) равна:

Подставляя значение V_b ^* из формулы (5) в уравнение (3), получим уточненное значение V_p:

Окончательная формула расчета пористости после поправки на сдавленный газ согласно (1) будет:

Из формулы (7) следует, что данная поправка может быть значительной даже при высоких давлениях. Так, при давлениях P^* = 3 МПа величина P_at/P^* ≈ 0.03, что составляет приблизительно 3% от измеряемой величины.The correction due to the fact that part of the pores of the material even at pressure P ^* remains filled with compressed gas and therefore cannot be filled with liquid, can be determined as follows. The total volume V _p then equal to the sum of the volume V _L [1-V $\genfrac{}{}{0ex}{}{\text{*}}{\text{Σ}}$ / V _L ] indented fluid and volume of air V _b ^* clamped in pores at pressure P ^* :

If we assume that the process of measuring porosity is isothermal and that the entire volume V _{p of} the material is filled with gas before the experiment (the sample is completely extracted), then the ratio of the volume of gas V _p in the material at atmospheric pressure P _at and the volume V _b ^{* of} gas at pressure P ^* is according to Boyle's law:

Then the fraction V _b ^{* of} gas trapped in the pores of the material at a pressure P ^* , according to formula (4), is equal to:

Substituting the value of V _b ^* from formula (5) in equation (3), we obtain the updated value of V _p :

The final formula for calculating porosity after correction for compressed gas according to (1) will be:

It follows from formula (7) that this correction can be significant even at high pressures. So, at pressures P ^* = 3 MPa, the value of P _at / P ^* ≈ 0.03, which is approximately 3% of the measured value.

The implementation of the proposed method allows you to measure the compressibility β _{W of the} liquid, the correction for which is then taken into account when calculating the porosity of the material. However, in the proposed method, any non-crystallizable working fluid at the indicated high pressures can be used as a liquid.

The decrease in the total volume of liquid (Fig. 2, curve 2) with increasing pressure to P ^* is due not only to the indentation of the liquid into the pores, but also to a decrease in the volume of the liquid itself during its compression due to the compressibility factor. Therefore, this decrease in the volume of liquid due to its compressibility must be taken into account in the final formula for calculating porosity (7). The error due to fluid compressibility can be approximately estimated based on data obtained from curve 1 (Fig. 2). A pressure of 50 MPa corresponds to a change in relative volume of 3%. Therefore, assuming, in a first approximation, a linear dependence of the relative volume of the working fluid on pressure, for pressure P ^* = 3 MPa, we obtain a change in the volume of the fluid of about 0.2%, which must be taken into account in the final calculations of porosity.

The exact formula for taking into account the compressibility β _{g of a} liquid when measuring the porosity m of a material using the proposed method can be obtained as follows.

где V - объем жидкости как функция давления P, ΔV,ΔP - приращения объема жидкости и давления соответственно при стремлении ΔP к нулю. Из формулы (8) находим абсолютную деформацию жидкости ΔV_ж Действительно, согласно формуле (8) изменение объема жидкости ΔV_ж для случая увеличения гидростатического давления от P_at до P^* в измерительной камере 1 с помещенным в него образцом материала 3 равно:
ΔV_ж= β_ж•V_l•(P^*-P_at) (9)
Значение β_ж определяется из экспериментальных данных по сжатию для случая, когда измерительная камера 1 заполнена только жидкостью (фиг. 2, кривая 1). Для этого случая в формуле (8) следует приравнять: V = V_L, Δ V = V_L ^* - V_L, Δ P = P^* - P_at, V^* _L - объем жидкости при давлении P^*. В результате получим выражение для сжимаемости жидкости β_ж :

и, следовательно, с учетом (10) формула (9) преобразуется так:

Таким образом, с учетом поправки на сжимаемость жидкости и формулы (6) уточненный объем пор равен:

Тогда, окончательная формула расчета коэффициента пористости образца материала согласно формуле (1) будет выглядеть так:

причем V_l = V_L - V_s.The compressibility of the liquid β _W is determined by the formula:

where V is the fluid volume as a function of pressure P, ΔV, ΔP are the increments of the fluid volume and pressure, respectively, when ΔP tends to zero. From formula (8) we find the absolute deformation of the liquid ΔV _w Indeed, according to formula (8), the change in the volume of the liquid ΔV _w for the case of an increase in hydrostatic pressure from P _at to P ^* in the measuring chamber 1 with a sample of material 3 placed in it is equal to:
ΔV _W = β _W • V _l • (P ^* -P _at ) (9)
The value of β _{g is} determined from the experimental data on compression for the case when the measuring chamber 1 is filled only with liquid (Fig. 2, curve 1). For this case, in formula (8) it should be equated: V = V _L , Δ V = V _L ^* - V _L , Δ P = P ^* - P _at , V ^* _L is the volume of liquid at pressure P ^* . As a result, we obtain the expression for the compressibility of the liquid β _W :

and, therefore, taking into account (10), formula (9) is transformed as follows:

Thus, taking into account the correction for fluid compressibility and formula (6), the specified pore volume is equal to:

Then, the final formula for calculating the porosity coefficient of a material sample according to formula (1) will look like this:

moreover, V _l = V _L - V _s .

If distilled water is used as the liquid, then at P ^* = 5 MPa, the correction for the compressibility of the liquid (2nd term of formula (13)) will be ~ 0.25% of the total porosity m of the material sample. When kerosene is used as a liquid, the compressibility of which is approximately two times greater than that of water, the correction also doubles. It should be noted that with an increase in P ^* for most materials, the correction for the compressibility of liquids naturally increases.

Таким образом, для нахождения объема V_s ^* материала при давлении P^* необходимо знать деформируемость β_s материала. Эта величина может быть получена обработкой результатов измерений (фиг. 2, кривая 2).When hydrostatic pressure is applied to a sample of material, it undergoes compression deformation. Therefore, the volume V _s ^* of the material sample at pressure P ^{* of the} full filling of the pore space of the material will be slightly different from the initial volume V _s measured at atmospheric pressure P _at :
V $\genfrac{}{}{0ex}{}{\text{*}}{\text{s}}$ = V _s (1-β _s • ΔP) (14)
where Δ P = P ^* - P _at , β _s - deformability (compressibility of the skeleton of the sample), determined by the formula:

Thus, to find the volume V _s ^{* of the} material at pressure P ^*, it is necessary to know the deformability β _{s of the} material. This value can be obtained by processing the measurement results (Fig. 2, curve 2).

After filling the pores (at pressures above P ^* ), curve 2 gives the total change in the volume of the liquid (the initial value of the volume V _{l of the} liquid) and the volume of the material (the initial value of the volume V _{s of the} sample material). We use two close pressure values P ₁ and P ₂ with a small pressure difference between them ΔP in the pressure range between P ^* and P ^max (Fig. 2). Let V _L1 , V _L2 be the volume of liquid at pressures P ₁ and P ₂ for the 1st curve, respectively (Fig. 2), the case in which the entire measuring chamber 1 with volume V _{k is} filled with liquid, V _l ¹ , V _l ² - the volume of liquid at pressures P ₁ and P _2, respectively, for the case when the measuring chamber 1 is filled with liquid and contains a sample of material, and V _Σ1 , V _Σ2 is the total volume, including the volume of liquid and the volume of the sample of material placed in it, at pressures P ₁ and P _2, respectively, as shown in FIG. 2, curve 2.

Величину V_k ¹ можно определить по формуле:
V_l ¹ = (1-V_s/V_L)•V_L1 (18)
При сжатии отношения текущего объема жидкости к начальному объему одинаковы при равных давлениях и не зависят от начального объема жидкости (то есть от того, какая часть камеры 1 заполнена жидкостью). Поэтому отношение V_L1/V_L (для случая, когда весь объем камеры 1 заполнен жидкостью) равно отношению V_l ¹/V₁ (для случая, когда в жидкость камеры 1 погружен образец материала). Тогда с учетом соотношения V₁ = V_L - V_s получаем формулу (18).The absolute deformation of the combined medium (liquid with a sample of material placed in it) is equal to (Fig. 2, curve 2):
ΔV _Σ = | V _Σ2 -V _Σ1 | = ΔV _w + ΔV _s (16)
where Δ V _W = V _l ² - V _l ¹ , Δ V _s = V _s ² - V _s ¹ , V _s ¹ is the volume of the material sample at pressure P ₁ , V _s ² is the volume of the material sample at pressure P _2, respectively. Then, taking into account formulas (8) - (11) for the pressure range Δ P = P ₂ - P ₁ :

The value of V _k ¹ can be determined by the formula:
V _l ¹ = (1-V _s / V _L ) • V _L1 (18)
When compressed, the ratios of the current volume of liquid to the initial volume are the same at equal pressures and do not depend on the initial volume of the liquid (that is, on what part of the chamber 1 is filled with liquid). Therefore, the ratio V _L1 / V _L (for the case when the entire volume of chamber 1 is filled with liquid) is equal to the ratio V _l ¹ / V ₁ (for the case when a sample of material is immersed in the liquid of chamber 1). Then, taking into account the relation V ₁ = V _L - V _s, we obtain formula (18).

Таким образом, формулы (18)-(19) позволяют определить абсолютную деформацию материала в любом диапазоне гидростатического давления и, следовательно, деформация материала (сжимаемость образца) в соответствии с формулой (15) равна:

Таким образом, предлагаемый способ позволяет определить деформацию материала, что используется в дальнейшем для уточнения окончательной пористости образца. Подсчитывая величину β_s по формуле (20) и принимая в первом приближении, что деформация материала постоянна по крайней мере в интервале давлений от P_at до P^*, после подстановки вычисленного значения β_s из формулы (20) в (14) и далее в (13) получим окончательную формулу определения пористости образца материала:
V $\genfrac{}{}{0ex}{}{\text{*}}{\text{s}}$ = V_s[1-β_s•(P^*-P_at)] (21)
и далее

Деформируемость (сжимаемость скелета образцов) материалов согласно справочным данным [Добрынин В. М. , Вендельштейн Б. Ю., Кожевников Д.А. "Петрофизика", М. , 1991, Недра, с. 368] колеблется от 1,17•10^-5 МПа^-1 (доломит) до 2,62•10^-5 МПа^-1 (песчаник кварцевый).Substituting the obtained value of ΔV _W in the formula (16), we obtain for deformation of the sample material ΔV _s

Thus, formulas (18) - (19) make it possible to determine the absolute deformation of the material in any range of hydrostatic pressure and, therefore, the deformation of the material (compressibility of the sample) in accordance with formula (15) is equal to:

Thus, the proposed method allows to determine the deformation of the material, which is used in the future to clarify the final porosity of the sample. Counting the value of β _s according to formula (20) and assuming, to a first approximation, that the deformation of the material is constant at least in the pressure range from P _at to P ^* , after substituting the calculated value of β _s from formula (20) into (14) and then to (13) we obtain the final formula for determining the porosity of a sample of material:
V $\genfrac{}{}{0ex}{}{\text{*}}{\text{s}}$ = V _s [1-β _s • (P ^* -P _at )] (21)
and further

Deformability (compressibility of the skeleton of samples) of materials according to the reference data [Dobrynin V. M., Vendelstein B. Yu., Kozhevnikov D. A. "Petrophysics", M., 1991, Nedra, p. 368] ranges from 1.17 • 10 ^-5 MPa ^-1 (dolomite) to 2.62 • 10 ^-5 MPa ^-1 (quartz sandstone).

With this in mind, formula (22) can be represented as:
m ^* = m • [1 + β _s • (P ^* -P _at )] (23)
where β _s - is determined by the formula (20) from curves 1 and 2 (Fig. 2).

Then, at P ^* = 5 MPa, the correction for the deformability of the material (second term of formula (23)) will be ~ 0.013% (for quartz sandstone) of the total porosity m of the material.

As follows from formulas (2), (7), (13), (23), the accuracy of determining the porosity of a substance is mainly determined by the accuracy of determining the volume of liquid V ^* pressed into the pores and the pressure P ^* at which the filling of the pores ends. The accuracy of determining V ^* is ~ 10 ^-3 mm ³ and is ensured by the correspondingly selected high resolution of the volume sensor 5 and the high accuracy of the data processing unit 15.

The main error in the measurement of porosity is due to the uncertainty in determining the pressure P ^* at which the pores of the material are completely filled with the working fluid. The proposed method allows to determine the parameter P ^* with a high degree of accuracy by the characteristic break in the baric dependence of the total volume (Fig. 2) or the baric dependence of the derivative on the total volume ∂V / ∂t (Fig. 6), as well as by the time moment t ^{* of the} total pore filling (the time t ^* uniquely corresponds to the pressure P ^{* of the} full filling of pores in the time dependence of pressure (Fig. 3)) recorded by the inflection point (sharp break) in the time dependence of the first derivative of pressure ∂P / ∂t (Fig. 4) , or at a temporary factory the variability of the first derivative of the total volume ∂V / ∂t (Fig. 5). The differentiating device 16 introduced to the input of the data processing unit 15 makes it possible to determine the moment t ^* with high accuracy from a characteristic break in the time dependence. The value of t ^{* is} sent to the data processing unit 15, in which V $\genfrac{}{}{0ex}{}{\text{*}}{\text{Σ}}$ and P ^* and then the calculation of the volume V _p pressed into the pores of the liquid material according to the formula (3) and (12), the correction for compressed gas according to the formula (5), the compressibility of the liquid according to the formula (10) and the correction due to it (formulas (11) and (12)), the deformation of the material according to formulas (18) - (21) and the correction due to the deformation of the material, as well as the final calculation of the porosity of the sample material according to formula (22) - (23) and the output of the data in tabular and graphic form to block 17 display. Block 18 synchronizes the operation of the device and sets the necessary conditions for the experiment: the compression rate of the liquid, the frequency of removal of experimental points in real time.

 The compressibility of the working fluid according to the formula (10) is determined only once for a given fluid, and then it is stored and used by the data block 15, as for the fluid used.

The pressure in the chamber is increased and / or an additional increase in pressure in the chamber 1 for quantities larger than P ^* can be produced at a constant speed, which allows working in real time. However, to increase the rate of determining porosity, it is advisable to increase the pressure in the chamber 1 stepwise for a more complete and quick filling of pores.

 As noted, due to automatic control, it is possible to increase the pressure in the chamber 1 and / or additionally increase the pressure in the chamber 1 at a speed that provides an isothermal condition for the liquid is pressed into the material, since the device (Fig. 1) is equipped with a temperature sensor 7.

 The table presents comparative data on the measurement of porosity of the claimed method and the traditional method in accordance with a known source of information (2).

 The comparison results indicate the high accuracy of the proposed method. It should be noted that to improve the accuracy of determining porosity and reduce the experiment time, it is better to use a liquid that does not wet the sample. When using a wetting liquid, preliminary impregnation of the sample and finding the initial saturation are required. So, if kerosene is used as the working fluid, then to increase the accuracy of porosity measurement, the sample is pre-impregnated in kerosene for several hours, weighing determines the initial saturation with kerosene, and then the porosity of the sample is determined using the measurement method proposed above and according to formulas (22) - (23). The experiment time in this case is several hours. When using liquid that does not wet the sample, the experiment time is significantly reduced and is only a few minutes.

 Thus, the claimed method of measuring porosity allows you to measure quickly, to take into account the corrections due to the compressibility of the fluid and the deformability of the material.

 The process of measuring the distribution of pore size is as follows. A fluid is selected that does not wet the material sample. The porous medium of the sample is considered as a combination of capillary cylindrical channels of various diameters.

Для вычисления распределения пор по размерам можно использовать зависимость относительного суммарного объема от давления, но только в диапазоне давлений, при которых происходит насыщение образца жидкостью. В качестве жидкости в этом случае желательно применять жидкость, которая не смачивает образец и для которой известен угол смачивания θ. В качестве примера кривой насыщения пор материала (фиг. 7) приведен участок барической зависимости относительного суммарного объема в диапазоне давлений, при которых происходит заполнение пор. С ростом давления происходит заполнение все новых и новых пор, причем с ростом давления в соответствии с формулой (24) заполняются поры все меньшего диаметра. Кривая насыщения образца горной породы (фиг. 7) и формула (24) в предположении, что поры имеют цилиндрическую форму и известен угол смачивания θ позволяет проводить расчет распределения пор по размерам.When a liquid is pressed into a porous medium with a non-wetting liquid, it is necessary to overcome the capillary pressure P _i , upon reaching which the liquid is filled with capillaries with a radius r _i . The capillary pressure P _i depends on the magnitude of the surface tension σ of the liquid and the wetting angle θ by it of the surface of the material sample and is expressed by the Laplace formula (24):

To calculate the pore size distribution, one can use the dependence of the relative total volume on pressure, but only in the pressure range at which the sample is saturated with liquid. As a liquid in this case, it is desirable to use a liquid that does not wet the sample and for which the contact angle θ is known. As an example, the pore saturation curve of the material (Fig. 7) shows a section of the pressure dependence of the relative total volume in the pressure range at which pore filling occurs. With increasing pressure, more and more pores are filled, and with increasing pressure in accordance with formula (24), pores of ever smaller diameter are filled. The saturation curve of the rock sample (Fig. 7) and formula (24) under the assumption that the pores have a cylindrical shape and a contact angle θ is known allows calculating the pore size distribution.

На основе полученных значений ρ(r_i) , рассчитанных по формуле (25), строится гистограмма распределения пор по размерам в виде дифференциальных кривых распределения пор по размерам.The determination of pore size distribution is as follows. The indentation curve (Fig. 7) determines the minimum P _min and maximum P _max pressure, in the range of which the pores are filled, as well as the value V $\genfrac{}{}{0ex}{}{\text{*}}{\text{Σ}}$ / V _L corresponding to pressure P _max . By the formula (24), the maximum r _max and minimum r _min radius of the capillaries are determined, which correspond to the pressures P _min and P _max, respectively. The range of radii between r _max and r _{min is} divided into uniform intervals (the number of sections is selected for reasons of the required number of points in the histogram of pore size distribution). After that, for each value of the resulting pore radius r _i according to formula (24), the pressures P _{i are} determined at which the pores of a given radius are filled. According to the dependence of the relative total volume on pressure (Fig. 7) for two adjacent pressure values P _i and P _{i + 1, the} difference in the relative volume Δ (V / V _L ) _{i is found} . The volume fraction of pores of a given radius g _{i is} calculated as the ratio Δ (V _Σ / V _L ) _i . to a complete change in the relative volume of 1-V $\genfrac{}{}{0ex}{}{\text{*}}{\text{Σ}}$ / V _L. Thus, for each value of the pore radius r _{i, the} volume fraction of such pores ρ (r _i ) is calculated by the following formula:

Based on the obtained ρ (r _i ) values calculated by formula (25), a histogram of the pore size distribution is constructed in the form of differential curves of pore size distribution.

The procedure for measuring pore size distribution includes the following steps:
(a) measuring the volume V _{s of a} sample of material with high accuracy and placing it in chamber 1;
(b) filling the chamber 1 through a hermetically sealed cover 2 with a liquid 3 (Fig. 1) of volume V _i and obtaining the dependence of the relative total volume on pressure (Fig. 7) when the liquid is pressed into the pore space of the material;
(c) - from the obtained dependence, the determination of the minimum P _min and maximum P _max pressure, in the range of which the pores of the sample material are filled, as well as V $\genfrac{}{}{0ex}{}{\text{*}}{\text{Σ}}$ / V _L corresponding to pressure P ^* ;
(d) - according to formula (24), the maximum r _max and minimum r _min radius of the capillaries are determined, which correspond to pressures P _min and P _max, respectively;
(e) - the range of radii between r _max and r _{min is} divided, for example, into uniform intervals (the number of sections is selected from considerations of the required number of points in the histogram of pore size distribution);
(e) - for each of the obtained values of the radius of the pores r _i according to the formula (24), the pressures P _i are determined at which the pores of a certain radius are filled;
(g) - according to the dependence of the relative volume on the pressure constructed in paragraph (b), the dependence of the indented volume V _in liquid on the pressure P _{i is} determined (Fig. 8);
(h) - according to the dependence of the relative volume on the pressure constructed in paragraph (g), the dependence of the depressed volume V _in liquid on the pore size r _{i is} determined (Fig. 9);
(i) - based on the data obtained in paragraph (h), using the differential formula: f (r) = dV / dr, a differential pore size distribution curve is constructed for this material (Fig. 10).

(k) - the normalized distribution function g (r) is determined by dividing the differential distribution function f (r) by the total pore volume V _in obtained by either integrating the differential distribution function f (r) (Fig. 10), or from Fig. 9 (Fig. 11).

 The most successfully claimed methods for measuring porosity and pore size distribution can be industrially used in geology, geophysics, soil science when studying the process of swelling of clays, soils, in the mining and oil and gas industries in determining porosity and saturation of rocks, in construction, in medicine and pharmacology in the production of tablets and determination of porosity and mechanical properties of bones, cartilage, joints, in the production of granular mineral raw materials, polymer films, in the production of ceramic or reflective surfaces, filters, membranes, the determination of porosity and pore size distribution of adsorbents and catalysts, as well as in other areas in which capillary-porous media are used.

Sources of information:
1. US patent N 2874565, n. 73-38, publ. 1959

 2. Polyakov E.N. "Methodology for the study of the physical properties of oil and gas reservoirs" M, 1981, Nedra, p. 181.

 3. US patent N 3882714, G 01 N 15/08, publ. 1975

Claims (25)

 1. The method of measuring porosity, including measuring the initial volume of the material, placing the material in the chamber with the liquid, increasing the pressure in the chamber to press the liquid into the material, measuring the volume of the liquid pressed into the material, taking into account the correction in the measured volume of the liquid related to the presence of gas in the material , determining the porosity of the material in relation to the measured volume of liquid pressed into the material, as amended, to the original volume of the material, characterized in that before placing the material in the chamber with the liquid is placed in this liquid, the pressure in the chamber is increased and the compressibility of the liquid is measured, while measuring the volume of liquid pressed into the material, the temporal characteristic of the pressure in the chamber and / or volume of liquid is measured to find the inflection point in it, by which the moment of full filling is determined the pores of the material with liquid, after the pores of the material are completely filled with liquid, the pressure in the chamber is additionally increased and the deformability of the material is measured; correction, associated with its compressibility, and take into account in the measured initial volume of the material an additional correction related to the deformability of the material, determine the porosity of the material, taking into account additional amendments.  2. The method according to claim 1, characterized in that after the complete indentation of the liquid into the material, the pressure in the chamber is further increased to a value exceeding the value of the pressure of full filling of the pores by at least 5 times.  3. The method according to claim 2, characterized in that the additional pressure increase in the chamber is produced up to a pressure of 300 MPa.  4. The method according to claim 1, characterized in that they increase the pressure in the chamber and / or additionally increase the pressure in the chamber at a constant speed.  5. The method according to claim 1, characterized in that the pressure in the chamber is increased and / or the pressure in the chamber is increased stepwise.  6. The method according to claim 1, characterized in that the pressure in the chamber is increased and / or the pressure in the chamber is further increased at a speed that provides an isothermal condition for the liquid is pressed into the material.  7. The method according to claim 1, characterized in that the moment t * of the complete filling of the pores of the material with liquid is determined by the inflection point on the baric dependence of the derivative of the total volume ∂V / ∂t when the liquid is pressed into the material and by the time dependence of the pressure change.  8. The method according to claim 1, characterized in that the moment t * of the complete filling of the pores of the material with liquid is determined by the inflection point on the time dependence of the derivative of the total volume ∂V / ∂t when the liquid is pressed into the material.  9. The method according to claim 1, characterized in that the moment t * of the complete filling of the pores of the material with liquid is determined by the inflection point on the time dependence of the pressure when the liquid is pressed into the material.  10. The method according to claim 1, characterized in that the time t * of the complete filling of the pores of the material with liquid is determined by the inflection point on the baric dependence of the total volume of the liquid when it is pressed into the material and by the time dependence of the pressure change. 11. The method according to claim 1, characterized in that the compressibility β _W liquid is determined by the formula

where V _L is the volume of fluid at atmospheric pressure P _at ;
V * _L is the volume of liquid at a pressure value of P * at the inflection point of the pressure dependence of the relative total volume;
ΔP = P * -P _at - pressure increment. 12. The method according to claim 11, characterized in that the deformability β _{s of the} material is determined by the formula

where V _s is the volume of the material at atmospheric pressure P _at ;
V _Σ1 , V _Σ2 - total volumes, including the volume of liquid and the volume of the material placed in it, at a pressure of P ₁ and P ₂ , respectively, with a small pressure difference from those in the pressure range of large P *;
V _l ¹ is the volume of liquid at a pressure of P ₁ for the case when liquid and material are placed in the chamber;
V _L1 , V _L2 - fluid volumes for pressure P ₁ and P ₂ , respectively. 13. The method according to p. 12, characterized in that the porosity m * of the material is determined by the formula
m ^* = m • [1 + β _s • (P ^* -P _at )],
Where

where v $\genfrac{}{}{0ex}{}{\text{*}}{\text{Σ}}$ - total volume at pressure P *;
V ₁ = V _L - V _S.  14. A method of measuring the pore size distribution, including measuring the initial volume of the material, placing the material in the chamber with the liquid, increasing the pressure in the chamber to press the liquid into the material, measuring the volume of the liquid pressed into the material, taking into account the correction in the measured volume of the liquid due to the presence of in a gas material, determining the size distribution of pores in relation to the measured volume of liquid pressed into the material, as amended, to the initial volume of the material, while measuring the volume of liquid of the cavity pressed into the material, measure the change in the relative volume of the liquid being pressed into the material from pressure and divide it into discrete intervals, which determine the pore size distribution, characterized in that before placing the material in the chamber with the liquid, this liquid is placed in it, increase the pressure in the chamber and measure the compressibility of the liquid, while measuring the volume of liquid pressed into the material, simultaneously measure the temporal response of the pressure in the chamber and / or the volume of liquid to find the point of inflection at which the moment of complete filling of the pores of the material with liquid is determined, after the pores of the material are completely filled with liquid, the pressure in the chamber is additionally increased and the deformability of the material is measured, the additional correction associated with its compressibility is taken into account in the measured volume of the liquid, and the measured the initial volume of the material an additional amendment associated with the deformability of the material, determine the pore size distribution taking into account the mentioned additional amendments.  15. The method according to 14, characterized in that after the complete indentation of the liquid into the material, the pressure in the chamber is additionally increased to a value that exceeds the value of the pressure of full filling of the pores by at least 5 times.  16. The method according to p. 15, characterized in that the additional increase in pressure in the chamber is produced up to a pressure of 300 MPa.  17. The method according to 14, characterized in that they increase the pressure in the chamber and / or additionally increase the pressure in the chamber at a constant speed.  18. The method according to 14, characterized in that the pressure in the chamber is increased and / or the pressure in the chamber is increased stepwise.  19. The method according to 14, characterized in that they increase the pressure in the chamber and / or additionally increase the pressure in the chamber at a speed that provides an isothermal condition for the liquid is pressed into the material.  20. The method according to 14, characterized in that the moment t * of the complete filling of the pores of the material with liquid is determined by the inflection point on the pressure dependence of the derivative of the total volume ∂V / ∂t when the liquid is pressed into the material and by the time dependence of the pressure change.  21. The method according to 14, characterized in that the moment t * of the complete filling of the pores of the material with liquid is determined by the inflection point on the time dependence of the derivative of the total volume ∂V / ∂t when the liquid is pressed into the material.  22. The method according to 14, characterized in that the moment t * of the complete filling of the pores of the material with liquid is determined by the inflection point on the time dependence of the pressure when the liquid is pressed into the material.  23. The method according to 14, characterized in that the moment t * of the complete filling of the pores of the material with liquid is determined by the inflection point on the baric dependence of the total volume of the liquid when it is pressed into the material and by the time dependence of the pressure change. 24. The method according to claim 14, characterized in that the fluid compressibility β _w is determined by the formula

where V _L is the volume of fluid at atmospheric pressure P _at ;
V * _L is the volume of liquid at a pressure value of P * at the inflection point of the pressure dependence of the relative total volume;
ΔP = P * -P _at - pressure increment. 25. The method according to paragraph 24, wherein the deformability β _{s of the} material is determined by the formula

where V _S is the volume of material at atmospheric pressure P _at ;
V _Σ1 , V _Σ2 - total volumes, including the volume of liquid and the volume of the material placed in it, at a pressure of P ₁ and P ₂ , respectively, with a small pressure difference from those in the pressure range of large P *;
V _l ¹ is the volume of liquid at a pressure of P ₁ for the case when liquid and material are placed in the chamber;
V _L1 , V _L2 - fluid volumes for pressure P ₁ and P ₂ , respectively.

Images