CN112084726A - Shale gas adsorption desorption diffusion kinetic model and interpretation method - Google Patents

Shale gas adsorption desorption diffusion kinetic model and interpretation method Download PDF

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CN112084726A
CN112084726A CN202010973807.9A CN202010973807A CN112084726A CN 112084726 A CN112084726 A CN 112084726A CN 202010973807 A CN202010973807 A CN 202010973807A CN 112084726 A CN112084726 A CN 112084726A
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刘建仪
袁华
刘治彬
蒋橹
何汶亭
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Abstract

The invention discloses a shale gas adsorption desorption diffusion kinetic model and an explanation method, which comprises the following steps: constructing a shale gas adsorption desorption diffusion dynamic model; introducing an Arrhenius formula to obtain calculation formulas of parameters such as diffusion rate constant, diffusion activation energy, adsorption pre-exponential factor, adsorption activation energy and the like; performing Laplace transformation on the kinetic mathematical model to obtain a concentration expression of Laplace space, and performing Stehfest numerical inversion to obtain an adsorbed gas concentration expression of real space; calculating to obtain an adsorption and desorption mass fraction and a mass fraction derivative curve; and fitting experimental data to calculate to obtain kinetic parameters, and predicting adsorption desorption diffusion kinetic curves at different temperatures. The invention considers the fitting error of the adsorption mass fraction and the derivative thereof of the theoretical model and the experimental data, fits the experimental data to calculate and obtain the kinetic parameters such as activation energy, pre-exponential factor and the like, and simultaneously predicts the adsorption desorption diffusion kinetic curve under different temperature conditions.

Description

Shale gas adsorption desorption diffusion kinetic model and interpretation method
Technical Field
The invention relates to the field of shale gas development, in particular to a shale gas adsorption desorption diffusion kinetic model and an explanation method.
Background
Shale gas is an unconventional oil and gas resource and becomes a hot spot of global oil and gas exploration and development due to the characteristics of abundant reserves and clean energy. The adsorption, desorption and diffusion mechanism of shale gas is studied earlier in foreign countries, and relatively later in domestic countries, but with the increasing of the exploration and development force of shale gas in recent years in China, the energy consumption proportion of shale gas in China is gradually increased.
At present, the research on shale gas adsorption and desorption at home and abroad mainly focuses on a static process, and the research on the dynamic process of adsorption, desorption and diffusion is relatively less. The research on the adsorption and diffusion kinetic process of the shale gas is beneficial to deeply analyzing the adsorption and diffusion characteristics and the control mechanism in shale particles and improving the accuracy of dynamic evaluation of shale gas reservoir production, and meanwhile, the shale gas adsorption, desorption and diffusion is the key for researching shale gas reserve estimation, flow rule, capacity prediction, dynamic analysis, recovery ratio calculation and numerical simulation.
Numerous scholars at home and abroad research an adsorption-desorption diffusion kinetic model, the model is the most widely applied in an intraparticle diffusion model which is originally proposed by Boyd and Webber (1963), and the model is a diffusion model without considering adsorption and desorption; brusseau et al (1991) simplify the adsorption and desorption terms to linear relations, and obtain an approximate solution of a kinetic curve by combining a Fick diffusion equation; plum shiqiang et al (2015) propose a new dynamic diffusion coefficient model capable of accurately describing a full-time diffusion process, and a normal diffusion coefficient of a classical model is an average value of dynamic diffusion coefficients of the new model; yang Zehao et al (2016) combine Langmuir adsorption kinetics with classical Fick diffusion kinetics, propose a new dynamic adsorption diffusion model (DAD model), combine the experimental data of dynamic adsorption, fit and get diffusion coefficient, absorption rate coefficient and desorption rate coefficient; afshi Davarpanah and the like (2019) research adsorption diffusion kinetic experiments and mathematical models of methane, provide a corrected single-hole diffusion mathematical model, and fit experimental data to obtain model parameters such as diffusion coefficients and the like. However, the current shale gas adsorption desorption diffusion kinetic model has the following problems:
(1) the shale gas diffusion model does not comprehensively consider the influences of adsorption and desorption, rock density and porosity, and high-order terms are ignored in the model solving process, so that the calculation result is an approximate solution;
(2) an Arrhenius formula is not introduced into the diffusion model, and parameters such as a diffusion rate constant, diffusion activation energy, adsorption pre-exponential factor, adsorption activation energy and the like cannot be obtained by fitting kinetic experimental data;
(3) the adsorption rate coefficient and desorption rate coefficient are temperature dependent and cannot predict adsorption diffusion kinetic curves at different temperatures.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides a shale gas adsorption desorption diffusion kinetic model and an explanation method.
The purpose of the invention is realized by the following technical scheme:
a shale gas adsorption desorption diffusion kinetic model and an explanation method comprise the following steps:
s1, substituting a free gas/adsorbed gas concentration equation, a shale porosity equation and an adsorption/desorption rate equation into each other according to a Langmuir theory to obtain an equation of change of adsorbed gas in shale particles along with time; establishing a total mass change equation of free gas/adsorbed gas within delta t time, and establishing an adsorption desorption diffusion equation according to a mass conservation law on the basis of the obtained initial condition, the obtained inner boundary condition and the obtained outer boundary condition, wherein the equation is used as a shale gas adsorption desorption diffusion kinetic model;
s2, introducing an Arrhenius formula into the adsorption-desorption diffusion kinetic model to obtain a relational expression among an adsorption rate coefficient, an adsorption-indicating pre-factor and adsorption activation energy, a relational expression among a desorption rate coefficient, a desorption-indicating pre-factor and desorption activation energy, and a relational expression among a diffusion coefficient, a diffusion rate constant and diffusion activation energy;
s3, performing Laplace transformation and solving on the dynamic model to obtain a concentration expression of Laplace space, and performing Stehfest numerical inversion to obtain an adsorbed gas concentration expression of real space;
s4; carrying out concentration integration on the shale spherical particles from the outer surface to the center of the shale spherical particles to obtain an adsorption and desorption gas quantity expression, taking the ratio of the accumulated adsorption and desorption gas quantity at any moment to the total adsorption and desorption gas quantity at the final balance as an adsorption and desorption mass fraction, and obtaining a mass fraction derivative curve through the adsorption and desorption mass fraction;
s5, obtaining adsorption and desorption mass fraction kinetic experiment data by using a gravimetric method experiment;
and S6, fitting experimental data through an optimization algorithm to calculate kinetic parameters such as activation energy, pre-exponential factors and the like, and predicting adsorption desorption diffusion kinetic curves under different temperature conditions.
Further, the step S2 includes the following sub-steps:
s101, obtaining the concentration of the free gas and the porosity of the shale according to the mass of the free gas, the total volume of pores of shale particles and the total volume of the shale particles, obtaining an adsorption rate according to a Langmuir adsorption kinetics theory, and calculating an adsorption rate coefficient;
s102, obtaining the concentration of the adsorbed gas according to the fact that the concentration of the adsorbed gas is equal to the shale gas adsorption amount measured in an experiment; according to the Langmuir adsorption kinetics theory, obtaining a desorption rate, and calculating a desorption rate coefficient;
s103, simplifying the equation to obtain an exchange rate equation between the free gas and the adsorbed gas in the shale particles:
Figure BDA0002685027840000021
where ρ isbIs the average density of shale particles in g/cm3；csThe concentration of the adsorbed gas is unit g/g; t is time, unit s; k is a radical ofaIs the adsorption rate coefficient, in units of s-1(ii) a Phi is the porosity of the shale and is dimensionless; c is the concentration of free gas in unitsg/cm3；kdIs a desorption rate coefficient, in units of s-1；
S104, obtaining a infinitesimal dv according to the constructed physical model of the shale spherical particles, and obtaining the volume of the concentric spherical ring;
s105, according to the law of conservation of mass, the sum of the dv shale gas amount of the inflow control body and the outflow control body is equal to the shale gas increment in the control body, so that the total mass change of free gas in the time delta t in the concentric sphere is calculated:
Figure BDA0002685027840000031
the total mass change of the free gas in the concentric sphere at time Δ t was calculated:
Figure BDA0002685027840000032
wherein c is the concentration of free gas and the unit is g/cm3(ii) a t is time, unit s; rhobIs the average density of shale particles in g/cm3；csThe concentration of the adsorption gas in the shale particles is unit g/g; phi is the porosity of the shale and is dimensionless;
s106, obtaining a diffusion coefficient based on an Arrhenius formula;
s107, acquiring an initial condition, an inner boundary condition and an outer boundary condition;
s108, within the time delta t in the concentric sphere, the shale gas increment caused by diffusion follows Fick second law, so that the gas adsorption-desorption diffusion equation in the spherical particles is calculated:
Figure BDA0002685027840000033
wherein c is the concentration of free gas and the unit is g/cm3(ii) a t is time, unit s; rhobIs the average density of shale particles in g/cm3；csIn shale granulesThe concentration of the adsorbed gas of (2), unit g/g; daIs a diffusion coefficient in cm2S; r is the radius corresponding to any spherical surface of the spherical particles, and the unit is cm; phi is the shale porosity and is dimensionless.
Further, an adsorption-desorption diffusion kinetic curve can be obtained through the adsorption gas concentration expression.
Further, the step S4 includes the following sub-steps:
s401, performing concentration integration on the spherical particles of the shale from the outer surface to the center of the spherical particles, and eliminating a variable r to obtain an integral expression of the change of the total amount of the adsorption and desorption gas of the spherical particles at any moment along with time;
s402, carrying out numerical integration on an integral expression of the change of the total amount of the adsorption and desorption gas of the spherical particles at any time along with time to obtain an expression of the total adsorption/desorption gas amount and time of the spherical particles;
s403, comparing the accumulated adsorption and desorption amount at any moment in the spherical particles with the total adsorption and desorption amount in the final balance to obtain the adsorption and desorption mass fraction;
s404, on the basis of the mass fraction curve, obtaining an adsorption and desorption mass fraction derivative by using a numerical derivation method.
Further, for the shale adsorption-desorption diffusion kinetics experiment in the step S4, based on the assumed conditions of consistent temperature, pressure and sample pore structure in the sample cylinder, when the experiment reaches dynamic equilibrium, all spherical particles in the shale sample cylinder should satisfy the same mass fraction kinetics curve.
Further, the step S5 includes the following sub-steps:
s501, converting the pressure of the experimental gas into the concentration of the free gas, wherein the adsorption, desorption and diffusion model represents the change relation of the concentration along with time, and the gas pressure in the adsorption, desorption and diffusion experiment can be converted into the concentration of the free gas according to a gas state equation;
and S502, acquiring an adsorption and desorption mass fraction kinetic curve measured by an experiment by using a gravimetric method.
The invention has the beneficial effects that:
1. the method is based on the diffusion of gas in the shale spherical particles, considers the influence of adsorption and desorption, utilizes the Langmuir adsorption kinetics theory and combines the Arrhenius formula, and establishes an adsorption, desorption and diffusion kinetics model in the shale spherical particles according to the Fick diffusion law and the substance balance theory.
2. Obtaining shale gas adsorption and desorption mass fraction and mass fraction derivative curves through Laplace transformation, Stehfest numerical inversion and numerical integration calculation, obtaining adsorption and desorption mass fraction kinetic experimental data by combining a gravimetric method experiment, comprehensively considering the adsorption mass fraction and the fitting error of the derivative of the theoretical model and the experimental data, obtaining kinetic parameters such as activation energy and index factors by fitting the experimental data, and meanwhile predicting adsorption and desorption diffusion kinetic curves under different temperature conditions.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
Fig. 2 is a schematic diagram of adsorption desorption diffusion of shale spherical particles.
FIG. 3 is a graph comparing the experimental value and the fitting value of the adsorption mass fraction of the shale sample A under the conditions of 2MPa and 60 ℃.
FIG. 4 is a graph comparing the experimental value and the predicted value of the adsorption mass fraction of the shale sample A under the condition of 2 MPa.
FIG. 5 is a graph comparing the experimental mass fraction desorption values with the fitted mass fraction desorption values of shale sample B at 15MPa and 60 ℃.
FIG. 6 is a graph comparing the experimental value and the predicted value of the desorption mass fraction of the shale sample B under the condition of 15 MPa.
Detailed Description
In order to more clearly understand the technical features, objects, and effects of the present invention, embodiments of the present invention will now be described with reference to the accompanying drawings.
In this embodiment, as shown in fig. 1 and fig. 2, a shale gas adsorption desorption diffusion kinetic model and an explanation method thereof include the following steps:
s1, constructing a shale spherical particle adsorption desorption diffusion physical model;
s2, constructing a mathematical model of the concentration change of the shale gas adsorption phase and the free phase;
s3, constructing a shale gas adsorption desorption diffusion kinetic mathematical model;
s4, performing Laplace transformation and solving on the dynamic model to obtain a concentration expression of Laplace space, and performing Stehfest numerical inversion to obtain an adsorbed gas concentration expression of real space;
s5; carrying out concentration integration on the shale spherical particles from the outer surface to the center of the shale spherical particles to obtain an adsorption and desorption gas quantity expression, and comparing the accumulated adsorption and desorption gas quantity at any moment with the total adsorption and desorption gas quantity at the final balance to obtain an adsorption and desorption mass fraction and mass fraction derivative curve;
s6, obtaining adsorption and desorption mass fraction kinetic experiment data by using a gravimetric method experiment;
and S7, fitting experimental data through an optimization algorithm to calculate kinetic parameters such as activation energy, pre-exponential factors and the like, and predicting adsorption desorption diffusion kinetic curves under different temperature conditions.
In this embodiment, the establishment of the physical model in step S1 includes the following assumed conditions:
a, supposing that a shale sample used for an experiment consists of countless spherical particles, the particles are taken from a matrix, and the pore structures in the particles are uniform;
b, the adsorbed methane molecules have no interaction, and the adsorption and desorption are in dynamic balance;
c, the migration of methane in the shale particles meets the mass conservation law and the continuity equation and follows the Fick second law;
and d, for the adsorption, desorption and diffusion experiment of the shale, keeping the temperature constant, and taking the pressure as the average pressure in the experiment process.
In this embodiment, the step S2 includes the following sub-steps:
s201, since the free gas diffuses inside the pores of the shale particles, the concentration of the free gas is:
Figure BDA0002685027840000051
wherein: c represents the concentration of free gas in g/cm3；mfRepresents the mass of free gas in g; v. ofpRepresents the total pore volume of the shale particles in cm3；
The porosity of the shale is:
Figure BDA0002685027840000052
wherein: phi represents the porosity of the shale and is dimensionless; v. ofbRepresenting the total volume of shale particles in cm3；
According to the Langmuir adsorption kinetics theory, the adsorption rate is proportional to the concentration of diffused free gas, resulting in an adsorption rate:
Rads＝kac
wherein: radsDenotes the adsorption rate in g/(cm)3s)；kaDenotes the adsorption rate coefficient in units of s-1(ii) a c represents the concentration of free gas in the shale particles in g/cm3；
Based on the Arrhenius formula, the adsorption rate coefficient can be derived:
Figure BDA0002685027840000053
wherein: k is a radical ofa，∞Denotes the adsorption pre-factor, unit s-1(ii) a Ea represents adsorption activation energy, unit J/mol; r represents the molar gas constant in units of J/(mol K); t represents temperature, in K;
s202, for the concentration of the adsorbed gas, the gas is equal to the shale adsorbed gas amount measured by the experiment, namely the concentration of the adsorbed gas is as follows:
Figure BDA0002685027840000061
wherein: c. CsRepresents the adsorbed gas concentration in shale particles in g/g；maRepresents the mass of adsorbed gas in g; v. ofbRepresents the total volume of shale particles in cm3，ρbRepresenting the average density of the shale particles in g/cm3；
According to Langmuir kinetic theory, the desorption rate is in direct proportion to the concentration of the adsorption phase, and the desorption rate R is obtaineddes：
Rdes＝kdcs
Wherein: rdesRepresents the desorption rate in g/(g s); k is a radical ofdDenotes the desorption rate coefficient in units of s-1；
According to the Arrhenius formula, obtaining a desorption rate coefficient:
Figure BDA0002685027840000062
wherein: k is a radical ofd，∞Denotes the desorption finger-front factor, unit s-1；EdRepresents desorption activation energy, unit J/mol;
s203, therefore, the change of the adsorbed gas inside the shale particles with time t can be expressed as:
Figure BDA0002685027840000063
s204, based on the shale spherical particle physical model, taking a infinitesimal body dv, wherein the volume of the concentric spherical ring can be expressed as:
Figure BDA0002685027840000064
wherein: r is0Represents the corresponding radius of the outer surface of the spherical particle, in cm; r represents the radius corresponding to any spherical surface of the spherical particles in cm;
s205, according to the mass conservation, the sum of the dv shale gas amount of the inflow and outflow control body is equal to the shale gas increment inside the control body, so as to establish an adsorption desorption diffusion equation, namely:
during time Δ t in the concentric sphere, the total mass of the free gas changes:
Figure BDA0002685027840000065
during time Δ t in the concentric sphere, the total mass of adsorbed gas changed:
Figure BDA0002685027840000066
within Δ t time in concentric sphere, the shale gas increment due to diffusion follows Fick's second law, expressed as:
Figure BDA0002685027840000071
the following can be obtained:
Figure BDA0002685027840000072
further simplifying to obtain a shale gas adsorption desorption diffusion equation:
Figure BDA0002685027840000073
in this embodiment, the step S3 includes the following sub-steps:
s301, obtaining a diffusion coefficient according to an Arrhenius formula:
Figure BDA0002685027840000074
wherein: d0Denotes the diffusion rate constant in cm2S; t represents temperature, in K; eDRepresents the absorption and diffusion activation energy, and the unit is J/mol; r represents the molar gas constant in units of J/(mol K);
s302, the outer boundary of the physical model of the shale granules is the outer surface of the spherical granules, and the inner boundary is the center of the spherical granules. It is assumed that during the adsorption-desorption diffusion experimental test, the initial free gas concentration inside the spherical particles is ciThe gas concentration at the outer boundary of the particle is c0And is kept constant, the initial adsorbed gas concentration on the particle surface is csi。
Initial conditions:
c＝ci,0＜r＜r0,t＝0
cs＝csi,0＜r＜r0,t＝0
inner boundary conditions:
Figure BDA0002685027840000075
outer boundary conditions:
c＝c0，r＝r0，t＞0
s303, the exchange rate equation between the free gas and the adsorbed gas is as follows:
Figure BDA0002685027840000076
s304, the gas adsorption desorption diffusion equation in the spherical particles is as follows:
Figure BDA0002685027840000081
in this embodiment, the step S4 includes the following sub-steps:
s401, for convenient solution, defining the following relation:
Δc＝c-ci
Δcs＝cs-csi
Δcv＝c0-ci
the function can be expressed according to the Laplace transform as follows:
Figure BDA0002685027840000082
Figure BDA0002685027840000083
on the adsorption and desorption diffusion equation
Figure BDA0002685027840000084
The Laplace space of the Laplace transformation is expressed as follows:
Figure BDA0002685027840000085
the exchange rate equation between free gas and adsorbed gas in Laplace space can be expressed as:
Figure BDA0002685027840000086
the outer boundary condition in Laplace space can be expressed as:
Figure BDA0002685027840000087
the inner boundary condition can be expressed in Laplace space as:
Figure BDA0002685027840000088
substituting the expression in sub-step S404 into the expression in sub-step S403, the adsorption desorption diffusion equation can be expressed as:
Figure BDA0002685027840000089
substituting the expression of the outer boundary condition in the Laplace space in the substep S405 and the expression of the inner boundary condition in the Laplace space in the substep S406 into the adsorption, desorption and diffusion equation in the substep S407 can solve the concentration expression of the Laplace space:
Figure BDA0002685027840000091
Figure BDA0002685027840000092
in the formula:
Figure BDA0002685027840000093
Figure BDA0002685027840000094
Figure BDA0002685027840000095
Figure BDA0002685027840000096
s402, obtaining an adsorption gas concentration expression in an actual space by adopting a Stehfest numerical inversion method, wherein the adsorption gas concentration expression in the actual space is as follows:
Figure BDA0002685027840000097
Figure BDA0002685027840000098
Figure BDA0002685027840000099
in the present embodiment, the adsorbed gas concentration expressed by the adsorbed gas concentration expression is a function of the radius r and the time t; the adsorption-desorption diffusion kinetic curve can be obtained by the expression of the concentration of the adsorbed gas, if c0>ciThen the adsorption gas concentration expression represents the adsorption diffusion kinetic process; if c is0<ciThe adsorbed gas concentration expression then represents the desorption diffusion kinetics.
In this embodiment, the step S5 includes the following sub-steps:
s501, performing concentration integration on the spherical particles of the shale from the outer surface to the center of the spherical particles, eliminating a variable r, and obtaining the change relation of the total amount Mtcs of the adsorption and desorption gas of the spherical particles at any moment along with time t, wherein the integral expression can be written as:
Figure BDA00026850278400000910
s502, due to the above csThe method is obtained through Stehfest numerical inversion calculation, and an analytical expression of integral cannot be obtained. Therefore, the expression of the total adsorption and desorption gas quantity Mtcs of the spherical particles and the time t can be obtained by numerically integrating the formula;
defining the f-function:
f＝4πr2ρb(cs-csi)
the numerical integral expression can therefore be written as:
Figure BDA0002685027840000101
if the time t tends to infinity, the adsorption and desorption are in a stable dynamic equilibrium state, and M is∞csCan be expressed as:
Figure BDA0002685027840000102
s503, adsorbing and desorbing mass fraction FmodIs any time in the spherical particlesThe ratio of the cumulative amount of adsorption/desorption to the total amount of adsorption/desorption at the final equilibrium, FmodCan be expressed as:
Figure BDA0002685027840000103
mass fraction of adsorption/desorption FmodCan be further expressed as:
Figure BDA0002685027840000104
for shale adsorption desorption diffusion kinetic experiments, based on the assumed conditions that the temperature, the pressure and the sample pore structure in a sample cylinder are consistent, when the experiment reaches dynamic balance, all spherical particles in the shale sample cylinder should meet the same mass fraction kinetic curve;
and S504, calculating the mass fraction derivative of adsorption and desorption by using a numerical derivation method on the basis of the mass fraction curve. The mass fraction derivative is calculated using the difference instead of the differential:
Figure BDA0002685027840000105
the mass fraction derivative is further calculated by weighted averaging:
Figure BDA0002685027840000106
wherein: fm'odThe mass fraction derivative of adsorption and desorption at any moment obtained by model calculation is represented and dimensionless; t is tjRepresenting an arbitrary point in time, in units of s.
In this embodiment, the step S6 includes the following sub-steps:
s601, the adsorption, desorption and diffusion model represents the change relation of concentration along with time, the gas pressure in the adsorption, desorption and diffusion experiment can be converted into the concentration of free gas according to a gas state equation, and the conversion formula can be written as follows:
Figure BDA0002685027840000107
wherein: c represents the gas concentration in g/cm 3; n represents the amount of gaseous species in mol; v represents the gas volume in cm 3; p represents the experimental pressure in MPa; z represents a deviation factor; r represents a gas constant in units of J/(mol k); t represents the experimental temperature in K;
s602, acquiring shale gas adsorption desorption diffusion dynamics experiment data through a gravimetric method experiment, mainly comprising 3 main steps of a blank experiment, a buoyancy experiment and an adsorption experiment, and specifically referring to NB/T10117-; as can be known from gravimetric experiments, the adsorption isotherm only describes the change relationship of the adsorption quantity with the pressure in the equilibrium state, and cannot represent the dynamic process of gas adsorption and diffusion; the adsorption isotherm shows the corresponding adsorption amount under different pressures, the adsorption kinetics curve shows the dynamic change curve of the adsorption amount along with time, the change curve of the total mass change quantity delta MP of the sample barrel and the sample along with time t is directly measured by a gravimetric method experiment, and the sample barrel volume V obtained by combining a blank experiment and a buoyancy experiment in the gravimetric methodscWith the sample volume VsThe conversion expression can be used for converting the delta MP into a kinetic curve of the change of the adsorption quantity at any time along with time, and is as follows:
mex＝ΔMP+(Vsc+Vs)×ρg
wherein: m isexRepresents the amount of adsorption at any time in units of g; vscRepresenting the volume of the sample barrel in cm3；VsRepresents the sample volume; unit cm3；
The adsorption capacity to reach equilibrium under the experimental pressure is me∞Then the mass fraction F of adsorption and desorption measured experimentallyexpCan be expressed as:
Figure BDA0002685027840000111
wherein: fexpRepresents the mass fraction of adsorption and desorption at any moment, and is dimensionless; m ise∞Represents the amount of adsorption in g that reaches equilibrium under the experimental pressure.
In this example, the experimentally measured derivative of mass fraction of adsorption/desorption Fe'xpCalculation Process and Fm'odSimilarly.
In this embodiment, adsorption/desorption kinetic experimental data can be obtained by a gravimetric experiment, and an adsorption/desorption diffusion kinetic parameter interpretation method is established based on the experimental data. In order to increase the parameter interpretation precision, the mass fraction derivative of adsorption and desorption is added into error calculation, and a genetic algorithm is adopted to fit kinetic experiment data, so that the kinetic parameters of adsorption and desorption diffusion can be obtained. The error between the experimental data and the model calculation result is expressed by an objective function Evi, and the expression can be written as:
Figure BDA0002685027840000112
wherein: fmodRepresenting the mass fraction of adsorption and desorption obtained by model calculation; fexpExpressing the mass fraction of adsorption and desorption calculated by experimental data; fm'odRepresenting the adsorption and desorption mass fraction derivative obtained by model calculation; fe'xpRepresenting the mass fraction derivative of adsorption and desorption calculated by experimental data; n is the number of experimental data.
Example 1, adsorption diffusion kinetics example analysis of shale sample a;
and (3) processing the total mass variation experimental data of the sample barrel and the sample obtained by the shale sample A under the conditions of 2MPa and 60 ℃ by a gravimetric method to obtain an adsorption mass fraction kinetic curve. The shale sample A had a porosity of 0.046 and a rock density of 2.62g/cm3The radius of the shale spherical particles is 0.02 cm.
The adsorption mass fraction kinetic experimental data were fitted by a genetic algorithm, and the fitting results are shown in fig. 3. The relevant kinetic parameters of shale sample a obtained by fitting the experimental data are shown in table 1.
Kinetic parameters Unit of Value of
Constant of diffusion rate cm2/s 2.15×10-3
Activation energy of diffusion J/mol 1.10×104
Adsorption of prepro-factor s-1 503
Activation energy of adsorption J/mol 1.23×104
Desorption of finger pre-factor s-1 959
Activation energy of desorption J/mol 1.59×104
TABLE 1 shale sample A at 2MPa and 60 deg.C adsorption and diffusion kinetic parameter interpretation results
As can be seen from the Arrhenius law, the activation energy, the prespecial factor and the diffusion rate constant are independent of temperature. Therefore, the shale sample A can be used for explaining the obtained adsorption diffusion kinetic parameters under the conditions of 2MPa and 60 ℃ to predict the kinetic curve at 90 ℃ and 110 ℃, and the prediction result is shown in FIG. 4.
Example 2, desorption diffusion kinetics example analysis of shale sample B;
and processing the total mass variation experimental data of the sample barrel and the sample obtained by the shale sample B under the conditions of 15MPa and 60 ℃ to obtain a desorption mass fraction kinetic curve. The porosity of shale sample A was 0.038, the rock density was 2.65g/cm3, and the spherical particle radius of the shale was 0.02 cm. The desorption mass fraction kinetics experimental data were fitted by genetic algorithm, and the fitting results are shown in fig. 5.
The relevant kinetic parameters of shale sample B obtained by fitting the experimental data are shown in table 2.
Kinetic parameters Unit of Value of
Constant of diffusion rate cm2/s 3.93×10-3
Activation energy of diffusion J/mol 1.00×104
Adsorption of prepro-factor s-1 865
Activation energy of adsorption J/mol 1.59×104
Desorption of finger pre-factor s-1 195
Activation energy of desorption J/mol 1.99×104
TABLE 2 shale sample B Desorption diffusion kinetic parameter interpretation results under 15MPa and 60 deg.C conditions
As can be seen from the Arrhenius law, the activation energy, the prespecial factor and the diffusion rate constant are independent of temperature. Therefore, the shale sample B can be used for explaining the obtained desorption diffusion kinetic parameters under the conditions of 15MPa and 60 ℃ to predict the kinetic curve at 90 ℃ and 110 ℃, and the prediction result is shown in FIG. 6.
Therefore, the new shale gas adsorption, desorption and diffusion model can be used for calculating the kinetic parameters such as the activation energy, the pre-exponential factor and the like under different experimental conditions, and the adsorption, desorption and diffusion kinetic parameters obtained by fitting the kinetic experimental data under one temperature can be used for predicting the adsorption, desorption and diffusion kinetic curves under different temperature conditions. The adsorption diffusion kinetic curve of the shale sample A and the desorption diffusion kinetic curve of the shale sample B are analyzed to obtain respective adsorption diffusion or desorption diffusion kinetic parameters, the kinetic parameter result obtained by fitting 60 ℃ experimental data is used for predicting the adsorption kinetic curves at 90 ℃ and 110 ℃, the predicted value is basically coincident with the experimental value, and the prediction effect is good.
The method comprises the steps of establishing an adsorption-desorption diffusion kinetic model in the shale spherical particles according to Fick diffusion law and substance balance theory by utilizing a Langmuir adsorption kinetic theory and combining an Arrhenius formula based on the diffusion of gas in the shale spherical particles and considering the influence of adsorption-desorption; obtaining shale gas adsorption and desorption mass fraction and mass fraction derivative curves through Laplace transformation, Stehfest numerical inversion and numerical integration calculation, obtaining adsorption and desorption mass fraction kinetic experimental data by combining a gravimetric method experiment, comprehensively considering the adsorption mass fraction and the fitting error of the derivative of the theoretical model and the experimental data, obtaining kinetic parameters such as activation energy and index factors by fitting the experimental data, and meanwhile predicting adsorption and desorption diffusion kinetic curves under different temperature conditions.
The foregoing shows and describes the general principles and broad features of the present invention and advantages thereof. It will be understood by those skilled in the art that the present invention is not limited to the embodiments described above, which are described in the specification and illustrated only to illustrate the principle of the present invention, but that various changes and modifications may be made therein without departing from the spirit and scope of the present invention, which fall within the scope of the invention as claimed. The scope of the invention is defined by the appended claims and equivalents thereof.

Claims (6)

1. A shale gas adsorption desorption diffusion kinetic model and an explanation method are characterized by comprising the following steps:
s1, substituting a free gas/adsorbed gas concentration equation, a shale porosity equation and an adsorption/desorption rate equation into each other according to a Langmuir theory to obtain an equation of change of adsorbed gas in shale particles along with time; establishing a total mass change equation of free gas/adsorbed gas within delta t time, and establishing an adsorption desorption diffusion equation according to a mass conservation law on the basis of the obtained initial condition, the obtained inner boundary condition and the obtained outer boundary condition, wherein the equation is used as a shale gas adsorption desorption diffusion kinetic model;
s2, introducing an Arrhenius formula into the adsorption-desorption diffusion kinetic model to obtain a relational expression among an adsorption rate coefficient, an adsorption-indicating pre-factor and adsorption activation energy, a relational expression among a desorption rate coefficient, a desorption-indicating pre-factor and desorption activation energy, and a relational expression among a diffusion coefficient, a diffusion rate constant and diffusion activation energy;
s3, performing Laplace transformation and solving on the dynamics mathematical model to obtain a concentration expression of Laplace space, and performing Stehfest numerical inversion to obtain an adsorbed gas concentration expression of real space;
s4; carrying out concentration integration on the shale spherical particles from the outer surface to the center of the shale spherical particles to obtain an adsorption and desorption gas quantity expression, taking the ratio of the accumulated adsorption and desorption gas quantity at any moment to the total adsorption and desorption gas quantity at the final balance as an adsorption and desorption mass fraction, and obtaining a mass fraction derivative curve according to the adsorption and desorption mass fraction;
s5, obtaining adsorption and desorption mass fraction kinetic experiment data by using a gravimetric method experiment;
and S6, fitting the experimental data through an optimization algorithm to obtain adsorption/desorption diffusion kinetic parameters, and predicting adsorption/desorption diffusion kinetic curves under different temperature conditions.
2. The shale gas adsorption desorption diffusion kinetic model and interpretation method according to claim 1, wherein the adsorption desorption diffusion equation is constructed by the following steps:
s101, obtaining the concentration of the free gas and the porosity of the shale according to the mass of the free gas, the total volume of pores of shale particles and the total volume of the shale particles, and obtaining an adsorption rate solving equation according to a Langmuir adsorption kinetics theory;
s102, the concentration of the adsorbed gas is equal to the shale gas adsorption amount measured in the experiment, and a desorption rate solving equation is obtained according to the Langmuir adsorption kinetics theory;
s103, simplifying the equation to obtain an exchange rate equation between the free gas and the adsorbed gas in the shale particles:
Figure FDA0002685027830000011
where ρ isbIs the average density of shale particles in g/cm3；csThe concentration of the adsorbed gas is unit g/g; t is time, unit s; k is a radical ofaIs the adsorption rate coefficient, in units of s-1(ii) a Phi is the porosity of the shale and is dimensionless; c is free gas concentration in g/cm3；kdIs a desorption rate coefficient, in units of s-1；
S104, taking the dv of the concentric spherical ring infinitesimal body to obtain the volume of the concentric spherical ring;
s105, calculating the total mass change of the free gas in the time delta t in the concentric sphere according to the mass conservation law:
Figure FDA0002685027830000021
the total mass change of the adsorbed gas within the time Δ t in the concentric sphere was calculated:
Figure FDA0002685027830000022
wherein c is the concentration of free gas and the unit is g/cm3(ii) a t is time, unit s; rhobIs the average density of shale particles in g/cm3；csThe concentration of the adsorption gas in the shale particles is unit g/g; phi is the porosity of the shale and is dimensionless;
s106, in the time delta t in the concentric sphere, according to the shale gas increment caused by diffusion and following the Fick second law, obtaining a gas adsorption desorption diffusion equation in the spherical particles:
Figure FDA0002685027830000023
wherein c is the concentration of free gas and the unit is g/cm3(ii) a t is time, unit s; rhobIs the average density of shale particles in g/cm3；csThe concentration of the adsorption gas in the shale particles is unit g/g; daIs a diffusion coefficient in cm2S; r is the radius corresponding to any spherical surface of the spherical particles, and the unit is cm; phi is the shale porosity and is dimensionless.
3. The shale gas adsorption desorption diffusion kinetic model and the interpretation method thereof as claimed in claim 1, wherein an adsorption desorption diffusion kinetic curve can be obtained through the adsorption gas concentration expression.
4. The shale gas adsorption desorption diffusion kinetic model and interpretation method of claim 1, wherein the step S4 comprises the following sub-steps:
s401, performing concentration integration on the spherical particles of the shale from the outer surface to the center of the spherical particles, and eliminating a variable r to obtain an integral expression of the change of the total amount of the adsorption and desorption gas of the spherical particles at any moment along with time;
s402, carrying out numerical integration on an integral expression of the change of the total amount of the adsorption and desorption gas of the spherical particles at any time along with time to obtain an expression of the total amount of the adsorption and desorption gas of the spherical particles and the time;
s403, the mass fraction of adsorption and desorption is the ratio of the accumulated adsorption and desorption amount at any moment in the spherical particles to the total adsorption and desorption amount at the final balance;
s404, on the basis of the mass fraction curve, obtaining an adsorption and desorption mass fraction derivative by using a numerical derivation method.
5. The shale gas adsorption desorption diffusion kinetic model and interpretation method of claim 1, wherein for the shale adsorption desorption diffusion kinetic experiment in the step S4, based on the assumed conditions of consistent temperature, pressure and sample pore structure in the sample cylinder, when the experiment reaches dynamic equilibrium, all spherical particles in the shale sample cylinder should satisfy the same mass fraction kinetic curve.
6. The shale gas adsorption desorption diffusion kinetic model and interpretation method of claim 1, wherein the step S5 comprises the following sub-steps:
s501, converting the experimental gas pressure into the concentration of the free gas, wherein the adsorption, desorption and diffusion model represents the change relation of the concentration along with time, and converting the gas pressure in the adsorption, desorption and diffusion experiment into the concentration of the free gas according to a gas state equation;
and S502, acquiring an adsorption and desorption mass fraction kinetic curve measured by an experiment by using a gravimetric method.
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