Two-phase well test analysis method considering knudsen diffusion and crossflow in pseudo-triple porosity media

Abstract

The application discloses a two-phase well testing analysis method considering Knudsen diffusion and channeling in a pseudo-three-pore medium, and the method further considers gas-liquid two-phase flow existing in a fracture system in a micro-physical model considering Knudsen diffusion and channeling in the pseudo-three-pore medium, establishes corresponding fracture system two-phase flow equations and matrix system single-phase flow equations, and solves shale gas well bottom two-phase pseudo-pressure difference and a double-logarithmic characteristic curve graph of a pseudo-pressure derivative and time by using the established fracture system two-phase flow equations and matrix system single-phase flow equations; finally, the characteristic curve graph and sensitivity parameters corresponding to fitting of the characteristic curve generated based on bottom measured data are taken as well testing interpretation results. Therefore, the application can realize well testing interpretation considering gas-liquid two-phase flow existing in the fracture system in the micro-physical model considering Knudsen diffusion and channeling in the pseudo-three-pore medium, and improve the accuracy of well testing interpretation results.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of shale gas well testing analysis, and particularly relates to a two-phase well testing analysis method considering Knudsen diffusion and channeling in pseudo-three-pore media. BACKGROUND

[0002] Due to the characteristics of tight and ultra-low permeability of shale gas reservoirs, in actual development, shale gas wells usually need to be fractured to form hydraulic fractures to obtain economic productivity. Shale gas reservoirs have the characteristics of low porosity and multi-scale, and the size of matrix pores ranges from several to several hundred nanometers, which makes shale gas exist in various migration modes such as Darcy flow, slippage effect and diffusion in the reservoir, and the percolation mechanism of shale gas well reservoir is very complex.

[0003] In actual development, shale gas wells are usually accompanied by the production of fracturing flowback fluid and shale gas. Due to the presence of fracturing fluid in the reservoir, the fluid flow in the reservoir is gas-liquid two-phase flow. However, most of the current research on shale gas well testing analysis only considers single-phase unstable flow in shale gas reservoirs, and does not consider the influence caused by the flow of fracturing fluid in the reservoir, which makes the accuracy of well testing interpretation insufficient. For example, the patent application with the application number 202110386841.0, "Shale reservoir well testing intelligent interpretation and analysis method and device based on deep learning", establishes a physical model for shale gas well testing. However, the model only considers single-phase gas flow and does not consider the influence caused by the flow of fracturing fluid in the reservoir. SUMMARY

[0004] In view of the above-mentioned deficiencies of the prior art, the purpose of the present application is to provide a two-phase well testing analysis method considering Knudsen diffusion and channeling in pseudo-three-pore media, which solves the problem of insufficient accuracy of well testing interpretation caused by not considering gas-liquid two-phase flow in the current shale gas well testing analysis of pseudo-three-pore media with Knudsen diffusion and channeling flow.

[0005] To achieve the above-mentioned purpose of the application, the present application provides the following technical solutions:

[0006] The two-phase well testing analysis method considering Knudsen diffusion and channeling in pseudo-three-pore media comprises the following steps:

[0007] S1: build a shale gas reservoir micro-physical model for well testing analysis, wherein the assumption conditions of the shale gas reservoir micro-physical model include: the shale gas reservoir is a pseudo-three-pore medium composed of matrix nanometer pores, matrix macro-pores and fracture systems, there is single-phase flow of shale gas in the matrix system, there is gas-liquid two-phase flow in the fracture system, and after shale gas desorbs from the matrix surface to the matrix nanometer pores, it moves to the matrix macro-pores in the form of Knudsen diffusion, and then moves to the fracture system in the form of channeling from the macro-pores;

[0008] S2: according to the shale gas reservoir micro-physical model, the fracture system two-phase flow equation and the matrix system single-phase flow equation are respectively established;

[0009] S3: the fracture system two-phase flow equation and the matrix system single-phase flow equation are combined to obtain a two-phase dimensionless pseudo-pressure difference micro-flow equation;

[0010] S4: according to the two-phase dimensionless pseudo-pressure difference micro-flow equation, and in combination with the outer boundary condition of the circular shale gas reservoir, a point source solution of two-phase dimensionless pseudo-pressure in the circular shale gas reservoir is obtained;

[0011] S5: according to the continuous point source solution of two-phase dimensionless pseudo-pressure in the circular shale gas reservoir, and in combination with the macro-physical model of the shale gas well, a shale gas well bottom hole pressure solution in the Laplace space is obtained;

[0012] S6: the shale gas well bottom hole pressure solution in the Laplace space is inverted into the real space, the shale gas well bottom hole two-phase pseudo-pressure difference and the double logarithmic characteristic curve of the pseudo-pressure derivative and time are calculated to obtain a characteristic curve chart;

[0013] S7: according to the characteristic curve generated based on the bottom hole measured data, the sensitivity parameters of the characteristic curve chart are adjusted, and the characteristic curve chart and the corresponding sensitivity parameters when fitting the characteristic curve are taken as the well test analysis result.

[0014] Therefore, the above-mentioned scheme can realize the well test interpretation of considering the gas-liquid two-phase flow of Knudsen diffusion and channeling flow in the fracture system in the micro-physical model considering Knudsen diffusion and non-steady channeling flow in the pseudo-three-pore medium, so that the accuracy of the well test interpretation result is improved. BRIEF DESCRIPTION OF DRAWINGS

[0015] Figure 1 The well test analysis method provided by the present application is shown in the flowchart;

[0016] Figure 2 The physical model schematic diagram for shale gas well test analysis of the present application;

[0017] Figure 3 The physical model schematic diagram of the circular shale gas reservoir of the present application;

[0018] Figure 4 The physical model schematic diagram of the circular shale gas reservoir fractured vertical well of the present application;

[0019] Figure 5 The physical model schematic diagram of the circular shale gas reservoir fractured horizontal well of the present application. DETAILED DESCRIPTION

[0020] Following, the embodiments of the present application are illustrated by specific examples, and other advantages and effects of the present application can be easily understood by those skilled in the art from the disclosure of the present specification. The present application can also be implemented or applied by other different specific embodiments, and various modifications or changes can be made to the details in the present specification based on different views and applications without departing from the spirit of the present application.

[0021] As shown in Figure 1 The present application considers a two-phase well test analysis method of Knudsen diffusion and channeling in pseudo-triple-pore media, which comprises the following steps:

[0022] S1: building a shale gas reservoir micro-physical model for well test analysis, wherein the assumption conditions of the shale gas reservoir micro-physical model include that the shale gas reservoir is a pseudo-triple-pore medium composed of matrix nanopores, matrix macropores and a fracture system, there is single-phase flow of shale gas in the matrix system, there is two-phase flow of gas and liquid in the fracture system, and after shale gas is desorbed from the matrix surface to the matrix nanopores, it moves to the matrix macropores in the form of Knudsen diffusion, and then moves to the fracture system in the form of channeling from the macropores;

[0023] S2: respectively establishing a fracture system two-phase flow equation and a matrix system single-phase flow equation according to the shale gas reservoir micro-physical model;

[0024] S3: combining the fracture system two-phase flow equation and the matrix system single-phase flow equation to obtain a two-phase dimensionless pseudo-pressure difference micro-flow equation;

[0025] S4: obtaining a point source solution of two-phase dimensionless pseudo-pressure in a circular shale gas reservoir according to the two-phase dimensionless pseudo-pressure difference micro-flow equation and in combination with the outer boundary condition of the circular shale gas reservoir;

[0026] S5: obtaining a shale gas well bottom hole pressure solution in Laplace space according to the continuous point source solution of two-phase dimensionless pseudo-pressure in the circular shale gas reservoir and in combination with a macro-physical model of the shale gas well;

[0027] S6: inverting the shale gas well bottom hole pressure solution in Laplace space to real space to calculate a two-phase pseudo-pressure difference at the bottom hole of the shale gas well and a double logarithmic characteristic curve of pseudo-pressure derivative and time to obtain a characteristic curve chart;

[0028] S7: adjusting the sensitivity parameters of the characteristic curve chart according to the generated characteristic curve based on the measured data at the bottom hole, and taking the sensitivity parameters corresponding to the fitting of the characteristic curve as the well test analysis results.

[0029] Specifically, in step S2, according to as shown in Figure 2The micro-physical model shows that gas-liquid two-phase flow exists in the micro-fracture, and the gas phase and liquid phase flow equations are established respectively. Meanwhile, since the gas desorbed from the matrix surface will first enter the macro-pore of the matrix, and then flow into the micro-fracture system from the macro-pore, the gas phase flow equation of the micro-fracture also contains the flow term q of the shale gas from the macro-pore to the micro-fracture m ; then, the two-phase flow equation of the fracture system is as follows:

[0030]

[0031] In the formula, ρ g is the density of the shale gas, ρ w is the density of water, unit: kg / m 3 ; k f is the fracture permeability, unit: m 2 ; k frg is the relative permeability component of the gas phase in the r direction of the fracture, k frw is the relative permeability component of the water phase in the r direction of the fracture, k frg and k frw are fractions; P f is the pressure of the fracture system, unit: Pa; μ g is the viscosity of the gas, μ w is the viscosity of water, unit: Pa·s; φ f is the porosity of the fracture system, φ f is a small number; S g is the gas saturation in the fracture, S w is the gas saturation, S g and S w are fractions.

[0032] According to the physical model as shown in Figure 2 , two kinds of flow exist in the nanometer matrix pore, one is the Darcy flow caused by the pressure difference, and the other is the Knudsen diffusion caused by the concentration difference. Among them,

[0033] The mass flux generated by the pressure difference is: In the formula, J mp represents the mass flux generated by the pressure difference, kg / (m·s);

[0034] The mass flux generated by the concentration difference is: In the formula, J K is the mass flux generated by the concentration difference, unit: kg / (m·s); D k is the Knudsen diffusion constant, unit: m 2 / s; M g is the molar mass of the gas, unit: kg / mol; C gmN is the number of shale gas moles in the unit matrix pore volume, unit: mol / m 3 ; Z is the deviation factor of shale gas under the temperature and pressure conditions of the matrix, dimensionless; T is the temperature in the matrix, unit: K.

[0035] Therefore, the mass flux generated by the pressure field and the concentration field together is: In the formula, J is the total shale gas mass flux in the pressure field and the concentration field, kg / (m·s); at the same time, the apparent permeability k ap is introduced by combining Darcy's law, which reflects the influence of the pressure field and the concentration field on the matrix permeability. ap m +D k μC gm ; In the formula, k ap is the apparent permeability of the matrix in the pressure field and the concentration field.

[0036] Specifically, when shale gas is transported from the matrix to the microcrack in the form of non-steady-state channeling, the flow equation of the gas phase in the matrix macro-pore and the channeling flow can be expressed as: That is, the single-phase flow equation of the matrix system can be obtained; in the formula, ρ gm is the density of shale gas in the matrix, ρ gf is the density of shale gas in the crack, unit: kg / m 3 ; k mrg is the relative permeability component of the matrix gas phase in the r direction, k mrg is a fraction; p m is the pressure in the matrix pore, unit: Pa; φ m is the matrix porosity, φ m is a fraction; the adsorption and desorption amount q dif is calculated by the Langmuir isothermal adsorption equation, that is,

[0037] Further, step S3 further includes:

[0038] 1. Introducing two-phase pseudo-pressure and two-phase compressibility coefficient to convert the crack pressure in the crack two-phase flow equation into crack two-phase pseudo-pressure; specifically, the two-phase pseudo-pressure is: In the formula, m(p) is the two-phase pseudo-pressure, unit: kg / (m 3 ·s); the two-phase compressibility coefficient is: C t = ρ g C g S g + ρ w C w S w , in the formula, C t is the two-phase compressibility coefficient, C g is the gas compressibility coefficient, unit: Pa​-1 .

[0039] Thus, the two-phase flow equation of the fracture system in spherical coordinates can be expressed as:

[0040]

[0041] where k app represents the comprehensive permeability of the matrix considering both Knudsen diffusion and Darcy flow in the matrix.

[0042] 2. The single-phase flow equation of the matrix system is converted to the following form:

[0043]

[0044] Next, the matrix gas phase pseudo-pressure and the apparent two-phase pseudo-pressure of the matrix are defined as follows:

[0045]

[0046] where P m is the pressure in the matrix pores; m*(P m ) is the matrix gas phase pseudo-pressure; μ mg is the viscosity of the gas in the matrix, and μ mw is the viscosity of water in the matrix, with the unit of Pa·s.

[0047] The density and the compressibility factor of the gas are defined as follows:

[0048]

[0049] where ρ g is the density of the gas; p m is the pressure in the matrix pores; M g is the molar mass of the gas; Z is the real gas deviation factor; R is the universal gas constant, 0.008314 MPa·m 3 / (kmol·K); T is the absolute temperature; and C g is the gas compressibility factor, Pa -1 .

[0050] In order to connect the fracture equation and the matrix equation, the single-phase pseudo-pressure of the matrix system and the apparent two-phase pseudo-pressure of the matrix are linked, and the following substitution is made between the matrix single-phase pseudo-pressure and the apparent two-phase pseudo-pressure of the matrix through a mathematical relationship:

[0051]

[0052] Then, the matrix flow equation in the spherical coordinate system is simplified by introducing the matrix gas phase pseudo-pressure, the density of the gas, the compressibility factor, and the relationship between the matrix gas phase pseudo-pressure and the apparent two-phase pseudo-pressure of the matrix, and the following equation is obtained:

[0053]

[0054] The adsorption and desorption amount q dif Convert to two-phase pseudo-pressure form:

[0055]

[0056] Introduce The matrix system single-phase flow equation can be further simplified as:

[0057]

[0058] After the above equation simplification, the following two-phase dimensionless variables are introduced to further simplify the fracture system and matrix system flow equations;

[0059] Matrix single-phase dimensionless radius:

[0060] Fracture two-phase dimensionless radius:

[0061] Matrix single-phase dimensionless time:

[0062] Comprehensive mass-conductivity coefficient: Λ = φ f C t + φ m ρ gi C mgi ;

[0063] Matrix dimensionless mass-elastic storage ratio:

[0064] Fracture dimensionless mass-elastic storage ratio:

[0065] Matrix dimensionless mass channeling coefficient:

[0066] Fracture-to-matrix permeability ratio:

[0067] Matrix single-phase dimensionless pseudo-pressure difference: Δm * m pm = m * (p i )- m * (p m ) ;

[0068] Matrix apparent two-phase dimensionless pseudo-pressure difference: Δm pm = m(p i )- m(p m ) ;

[0069] Fracture two-phase dimensionless pseudo-pressure difference: Δm pf = m(p i ) - m(p f );

[0070] After simplification, the fracture system two-phase flow equation is: The apparent two-phase flow equation of the matrix system is:

[0071] Further, the above simplified fracture system two-phase flow equation and the apparent two-phase flow equation of the matrix system are Laplace transformed with respect to time t D , and the corresponding results are:

[0072]

[0073] The fracture system two-phase flow equation and the apparent two-phase flow equation of the matrix system are simultaneously Laplace transformed, and the two-phase dimensionless pseudo-pressure difference microscopic flow equation is obtained:

[0074]

[0075] wherein,

[0076] Specifically, when the shale gas is transported from the matrix to the micro-fracture in the form of steady-state channeling, the flow equation of the gas phase in the matrix macro-pore and the channeling flow can be expressed as: The matrix system single-phase flow equation can be obtained; in the formula, ρ gm is the density of the shale gas in the matrix, ρ gf is the density of the shale gas in the fracture, the unit is kg / m 3 ; k mrg is the r-direction matrix gas phase relative permeability component, k mrg is a fraction; p m is the pressure in the matrix pore, the unit is Pa; φ m is the matrix porosity, φ m is a fraction; the adsorption and desorption amount q dif is calculated by the Langmuir isothermal adsorption equation, that is,

[0077] Further, step S3 further includes:

[0078] 1. Introducing two-phase pseudo-pressure and two-phase compressibility, the fracture pressure in the fracture two-phase flow equation is converted into the fracture two-phase pseudo-pressure; specifically, the two-phase pseudo-pressure is: In the formula, m(p) is the two-phase pseudo-pressure, the unit is kg / (m 3 ·s); the two-phase compressibility is: C t = ρ gC g S g +ρ w C w S w , where C t is the two-phase compressibility, C g is the gas compressibility, in Pa -1 .

[0079] Thus, the two-phase flow equation in the fracture system in spherical coordinates can be expressed as:

[0080]

[0081] where k app represents the comprehensive permeability of the matrix considering both Knudsen diffusion and Darcy flow in the matrix.

[0082] 2. The single-phase flow equation of the matrix system is converted to the following form:

[0083]

[0084] Next, the matrix gas phase pseudo pressure and the apparent two-phase pseudo pressure in the matrix are defined as follows:

[0085]

[0086] where p m is the pressure in the matrix pore; m * (p m ) is the matrix gas phase pseudo pressure; μ mg is the viscosity of the gas in the matrix, in Pa·s; and μ mw is the viscosity of water in the matrix, in Pa·s.

[0087] The density of the gas and the compressibility factor are defined as follows:

[0088]

[0089] where ρ g is the density of the gas; p m is the pressure in the matrix pore; M g is the molar mass of the gas; Z is the real gas deviation factor; R is the universal gas constant, 0.008314 MPa·m 3 / (kmol·K); T is the absolute temperature; and C g is the gas compressibility, in Pa -1 .

[0090] In order to connect the fracture equation and the matrix equation, the single-phase pseudo-pressure of the matrix system is linked with the apparent two-phase pseudo-pressure, and the following substitution is made between the matrix single-phase pseudo-pressure and the matrix apparent two-phase pseudo-pressure through a mathematical relationship:

[0091]

[0092] Then, the matrix flow equation in the spherical coordinate system is simplified by introducing the matrix gas-phase pseudo-pressure, the density of the gas, the compressibility coefficient, and the relationship between the matrix gas-phase pseudo-pressure and the matrix apparent two-phase pseudo-pressure, and the following equation is obtained:

[0093]

[0094] The adsorption and desorption amount q dif is converted into the apparent two-phase pseudo-pressure form:

[0095]

[0096] The following equation is introduced: and the apparent two-phase flow equation of the matrix system is obtained:

[0097]

[0098] After the above equation is simplified, the following two-phase dimensionless variables are introduced to further simplify the two-phase flow equation of the fracture system and the apparent two-phase flow equation of the matrix system:

[0099] The matrix single-phase dimensionless radius is:

[0100] The fracture two-phase dimensionless radius is:

[0101] The matrix single-phase dimensionless time is:

[0102] The comprehensive mass-conductivity coefficient is Λ = φ f C t + φ m ρ gi C mgi ;

[0103] The matrix dimensionless mass-elastic storage ratio is:

[0104] The fracture dimensionless mass-elastic storage ratio is:

[0105] The matrix dimensionless mass-crossflow coefficient is:

[0106] The ratio of the apparent permeability of the fracture to the permeability of the matrix is:

[0107] Matrix single-phase dimensionless pseudo-pressure difference: Δ * m pm =m * (p i )-m * (p m );

[0108] Matrix apparent dimensionless pseudo-pressure difference between two phases: Δm pm =m(p i )-m(p m );

[0109] Dimensionless pseudo-pressure difference between two phases in the crack: Δm pf =m(p i )-m(p f );

[0110] After simplification, the two-phase flow equation for the fracture system is: The apparent two-phase flow equation for the matrix system is:

[0111] Furthermore, the simplified two-phase flow equations for the fracture system and the apparent two-phase flow equations for the matrix system are then recalculated with respect to time t. D The Laplace transform yields:

[0112]

[0113] Combining the Laplace transform equations for the two-phase flow in the fracture system and the apparent two-phase flow equations for the matrix system, we obtain:

[0114]

[0115] Substituting the above equation into... From this, we can obtain:

[0116]

[0117] make The two-phase dimensionless pseudo-pressure difference microflow equation is then:

[0118] Specifically, in step S4, firstly, based on the two-phase dimensionless pseudo-pressure difference micro-flow equation, the corresponding two-phase dimensionless pseudo-pressure difference micro-flow equation for any cylindrical continuous point source in a circular shale reservoir is established.

[0119] By introducing Fourier finite cosine transform into the microscopic flow equation of a two-phase dimensionless pseudo-pressure difference for a continuous cylindrical point source in a circular shale reservoir, and combining this with the outer boundary case of the shale gas reservoir, the point source solution equation of the two-phase pseudo-pressure for a continuous cylindrical point source in a circular shale gas reservoir is obtained; for example... Figure 3In the physical model shown, the shale gas reservoir is a circular shale gas reservoir, and the basic assumptions for any cylindrical continuous point source in the circular shale reservoir are as follows:

[0120] (1) The top and bottom surfaces of the shale reservoir are closed, the horizontal outer boundary is infinite, the reservoir thickness is h, and the outer boundary radius of the reservoir is r e ;

[0121] (2) For the established rectangular coordinate system, a point on the bottom surface of the shale reservoir is taken as the coordinate origin, the bottom surface of the shale reservoir is the x0y plane, and the axis perpendicular to the bottom surface of the reservoir through the origin is the z axis; converted to cylindrical coordinates, only the r and z directions are considered;

[0122] (3) The horizontal permeability of the shale reservoir is k fh , and the vertical permeability is k fz ;

[0123] (4) A cylindrical element with its center at (0, 0, z w ) in the rectangular coordinate system, i.e. (0, z w ) in the cylindrical coordinate system; its radius is r, and its height is z w ;

[0124] (5) Gas and liquid phases flow into the cylindrical element from the side, and the flow rate of shale gas into the cylindrical element at the surface standard condition is q gscin , the flow rate of fracturing flowback fluid into the cylindrical element at the surface standard condition is q wscin , and the effects of capillary force and gravity are ignored.

[0125] For convenience of study, cylindrical coordinates will be used to interpret the model. Thus, the fracture two-phase flow equation can be expressed in cylindrical coordinates as follows:

[0126]

[0127] At the same time, since the assumed cylindrical element is an infinitesimal quantity, when gas and water two-phase fluid flows into it from the side, the element can be regarded as a continuous point sink, and the corresponding inner boundary condition expression is obtained as follows:

[0128]

[0129] In the formula, ρ gsc is the density of shale gas at the surface standard condition, and ρ wsc is the density of fracturing flowback fluid at the surface standard condition; the upper and lower boundary conditions are: The outer boundary condition is:

[0130] Moreover, the following two-phase dimensionless variables are introduced to simplify the above equations and boundary conditions:

[0131] The equation for the two-phase dimensionless flow in the crack in cylindrical coordinates is:

[0132] The inner boundary conditions are:

[0133] The upper and lower boundary conditions are:

[0134] The outer boundary conditions are:

[0135] After time t D After performing the Laplace transform, the two-phase dimensionless flow equation of the crack can be transformed into:

[0136]

[0137] Relating the above equation to the dimensionless fracture gas-liquid two-phase flow equation obtained in step S3 of this invention, the Lagrange variable s in the above equation is replaced as follows:

[0138]

[0139] The inner boundary conditions are:

[0140] The upper and lower boundary conditions are: The outer boundary conditions are:

[0141] For solving the initial-boundary value problem, first let:

[0142] Δm pf =R(r) D )Z(z D )

[0143] Substitute the above formula into In the middle, and combined with the closed boundary conditions at the top and bottom, using the Sturm-Liouville eigenvalue theory, we can obtain the following about z. D Eigenvalue equations:

[0144]

[0145] At the same time, according to the Sturm-Liouville eigenvalue theory, The eigenvalue is α n =(nπ) 2 Then the corresponding characteristic function system can be expressed as:

[0146] Z n (z D )=cosnπz D n = 0, 1, 2, ...

[0147] Thus, cosnπz D As the kernel function, we introduce the Fourier finite cosine transform into and define as Regarding z D The function after the Fourier finite cosine transform is given by

[0148]

[0149] The inner boundary expression is:

[0150]

[0151] The outer boundary expression is:

[0152]

[0153] Let The fracture two-phase flow equation can be written as follows:

[0154]

[0155] Since is the Bessel equation of the imaginary argument, its general solution is:

[0156]

[0157] where: a n , b n are undetermined constants.

[0158] Substituting into the inner boundary expression, we get:

[0159]

[0160] According to the properties of the Bessel function, when x→0, I1(x)=0, Thus, from the above inner boundary expression, we get:

[0161]

[0162] Thus:

[0163]

[0164] Substituting it into the outer boundary expression:

[0165] Let Then:

[0166]

[0167] Will Substitution From this, we can obtain:

[0168]

[0169] Performing an inverse Fourier cosine transform on the above equation yields the point source solution of the two-phase pseudo-pressure in the dimensionless fracture gas-liquid two-phase flow equation, namely:

[0170]

[0171] In the formula:

[0172] If the outer boundary of a shale gas reservoir is closed at the top and bottom surfaces and closed at the side surfaces, then the mass flow rate of the fluid passing through the outer boundary is 0. This can be converted into a dimensionless two-phase pseudo-pressure difference form after Laplace transform:

[0173]

[0174] at this time, If C is a constant, then δ = 0. Substituting this into the point source solution of the two-phase pseudo-pressure in the dimensionless fracture gas-liquid two-phase flow equation, we get...

[0175]

[0176] Furthermore, when the outer boundary of the shale gas reservoir is closed at the top and bottom surfaces and the outer boundary of the side surfaces, the point source solution of the two-phase pseudo-pressure of the corresponding dimensionless fracture gas-liquid two-phase flow equation is:

[0177]

[0178] In the formula, q gscins and q wscins It is usually a constant, therefore:

[0179]

[0180] If the outer boundary of a shale gas reservoir is closed at the top and bottom, and the lateral outer boundary is infinitely large, then the expression for the outer boundary is:

[0181]

[0182] Furthermore, according to the properties of Bessel functions, I0(∞)=∞, substituting... From this, we can obtain: a n =c n =0; then set a n =c n =0 Substitute In this context, when the outer boundary of a shale gas reservoir is closed at the top and bottom surfaces and infinitely large at the lateral outer boundaries, the point source solution of the two-phase pseudo-pressure in the dimensionless fracture gas-liquid two-phase flow equation is:

[0183]

[0184] Where, ρ gsc The density of gases under standard ground conditions is expressed in g / cm³. 3 ;ρ wsc The density of water at standard surface conditions, expressed in g / cm³. 3 ;q gscins q represents the flow rate of shale gas flowing into a cylindrical micro-element under standard surface conditions, expressed in m³ / s. wscins The flow rate of fracturing flowback fluid into the cylindrical micro-element under standard surface conditions is expressed in m / s; h is the thickness of the shale reservoir. k fh r represents the horizontal permeability of the shale reservoir. e r is the outer boundary radius of the shale reservoir. eD z is the dimensionless radius of the outer boundary of the shale reservoir; wD z is a dimensionless distance, which is dimensionless. D Let I0 be the dimensionless height in the z-direction; I1 and I0 are dimensionless imaginary argument Bessel functions of the first kind; K0 and K1 are dimensionless imaginary argument Bessel functions of the second kind.

[0185] Specifically, in step S5, since there are two types of shale gas wells, namely fractured vertical wells and fractured horizontal wells, and they correspond to different physical models, it is necessary to study the bottom hole pressure response of fractured vertical wells and fractured horizontal wells respectively.

[0186] like Figure 4 The physical model of the shale gas well shown is based on the following assumptions:

[0187] Assumption 1: The shale gas well is a fully fractured, infinitely conductive vertical well in a circular reservoir;

[0188] Assumption 2: The fracturing fracture is a symmetrical double-wing fracture about the wellbore, and its half-length is x. f ;

[0189] Assumption 3: Shale gas wells produce a constant surface yield q sc Or a constant bottom hole pressure p wf Production begins;

[0190] Assumption 4: The gas is slightly compressible and has a constant viscosity and compressibility coefficient; the effects of capillary force and gravity are ignored; the crack is an infinite flow channel;

[0191] Assumption condition five: taking the length of the fracture surface as the x-axis, the height of the fracture surface as the z-axis, and the y-axis perpendicular to the fracture surface to establish a rectangular coordinate system, and converting the rectangular coordinate system into a cylindrical coordinate system when calculating the bottom hole pressure;

[0192] Then, combining the assumption conditions of the physical model of the shale gas well, the point source solution of the dimensionless two-phase fracture gas-liquid flow equation is integrated along the reservoir thickness direction and then along the fracture direction to solve the shale gas well bottom hole pressure response in the Laplace space.

[0193] Specifically, if the outer boundary condition of the shale gas reservoir is that the top and bottom surfaces are closed and the side outer boundary is closed, the pressure response expression of the bottom hole of the circular reservoir fractured vertical well is:

[0194]

[0195] In the formula:

[0196] Since the trigonometric function cos(nπz wD ) has the property of integrating to 0 in the symmetric domain, and noting that Moreover, the flow distribution of the fractured vertical well is uniform, the gas well gas mass flow rate can be converted by the product of the density of shale gas, the point source intensity and the area of the fracture, and the relationship between the gas well water mass flow rate and the fracturing flowback fluid density, the point source intensity and the area of the fracture is obtained: ρ gsc q gsc +ρ wsc q wsc =2hx f (ρ gsc q gscins +ρ wsc q wscins ); combining the above relationship, the pressure response expression of the bottom hole of the circular reservoir fractured vertical well is simplified as:

[0197]

[0198] Introducing the following two-phase dimensionless variables The above expression is dimensionless, and the following is obtained:

[0199]

[0200] Substituting into the above formula, the following is obtained:

[0201]

[0202] Since the pressure response on the fracture surface is studied, y w = 0, so y wD= 0; meanwhile, in order to use the wellbore pressure dynamic distribution of the uniform flux fracture to characterize the pressure dynamic distribution of the infinite conductivity fracture, a suitable equivalent observation point needs to be selected, and its coordinates are as follows: In order to further according to the properties of the Bessel function, when α→x D | x D - α |→ 0, Therefore, at this integral interpolation point, the function value is infinite, but in fact it is bounded. In order to avoid the generation of false numerical integral results, integral transformation is needed, let Then: Therefore, the shale gas wellbore pressure response equation in the Laplace space is:

[0203]

[0204] The above formula is the dimensionless wellbore pressure dynamic distribution of the uniform flux fracture. When calculating the pressure distribution of the infinite conductivity fracture, the value of the observation point horizontal coordinate is x D = 0.732.

[0205] And if the outer boundary of the shale gas reservoir is the top and bottom surface closed and the side surface outer boundary infinite, the pressure response expression of the fractured vertical well of the circular reservoir is:

[0206]

[0207] Simplify to get: Again, substitute Further simplify to: Again, introduce Non-dimensionalization can be obtained:

[0208]

[0209] Next, substitute Into the above formula, we can get:

[0210]

[0211] Since the study of the pressure response on the fracture surface, in order to use the wellbore pressure dynamic distribution of the uniform flux fracture to characterize the pressure dynamic distribution of the infinite conductivity fracture, we can get:

[0212]

[0213] Where, α is the shape factor, dimensionless.

[0214] In addition, as shown in the physical model of the shale gas well, the assumption conditions are as follows: Figure 5

[0215] ​Assumption 1: The shale gas well is a fully open infinite conductivity fractured horizontal well in a circular reservoir;

[0216] Assumption 2: The total number of the fractured fractures is M, the fracture surface is perpendicular to the central axis of the wellbore horizontal section, and is distributed along the central axis of the wellbore;

[0217] Assumption 3: The fractured fracture is a symmetrical double-wing fracture or an asymmetrical double-wing fracture about the wellbore, has infinite conductivity, and its length is x f ;

[0218] Assumption 4: Each fracture is divided into 2N units, and the flow distribution inside each fracture unit i is uniform;

[0219] Assumption 5: The shale gas well is produced at a constant surface production q sc Or a constant bottom hole pressure p wf ;

[0220] Assumption 6: The gas is slightly compressible, and has constant viscosity and compressibility coefficient; the effects of capillary force and gravity are ignored;

[0221] Assumption 7: The origin of the coordinate system is established with the center of the intersection surface of the first fracture close to the wellbore horizontal section and the wellbore as the origin, the y-axis coincides with and extends along the wellbore horizontal section axis, the z-axis is along the reservoir thickness direction and positive upward, and the x-axis is perpendicular to the above two directions, and when calculating the bottom hole pressure, the rectangular coordinate system is converted into a cylindrical coordinate system;

[0222] Then, combined with the assumption conditions of the physical model of the shale gas well, the point source solution of the two-phase pseudo-pressure of the dimensionless fracture gas-liquid two-phase flow equation is integrated along the z direction, the line source solution of the two-phase dimensionless pseudo-pressure of Mx2N fracture units is calculated, and the line source solution of the two-phase dimensionless pseudo-pressure of the Mx2N fracture units of the fractured horizontal well and the production accumulation equation are solved to solve the shale gas well bottom hole pressure response in the Laplace space.

[0223] Specifically, the observation point is selected at the node of a certain fracture unit i The dimensionless coordinate is According to the principle of superposition of potential, the total pressure response of Mx2N discrete fracture units on M fractures at point Is equal to the superposition of the pressure response of each discrete unit at ;

[0224]

[0225] Because the seepage resistance of fluid in the infinite conductivity fracture and wellbore is far less than that in other locations of the reservoir, it is considered that the fluid flowing in the infinite conductivity fracture and wellbore does not produce pressure drop. Thus, it is known that the pseudo-pressure at any fracture discrete node is equal and equal to the bottom-hole flowing pressure, i.e.,

[0226]

[0227] Because the fractured horizontal well produces at a constant mass rate q msc , the value should be equal to the sum of the two-phase mass flow rates of all fracture discrete units, i.e., Thus,

[0228] According to the M*2N dimensionless two-phase pseudo-pressure equations and one production accumulation equation, the expression is obtained as:

[0229]

[0230] wherein, is a coefficient matrix; q Di is the two-phase dimensionless mass production on the fracture unit i, dimensionless, i = 1, 2, ···, 2N*M;

[0231] If the outer boundary condition of the shale gas reservoir is that the top and bottom surfaces are closed and the side outer boundary is closed, the elements in the coefficient matrix are:

[0232] If the outer boundary condition of the shale gas reservoir is that the top and bottom surfaces are closed and the side outer boundary is infinite, the elements in the coefficient matrix are:

[0233] wherein, for a sphere α = 15 / r 2 is a shape factor, with the unit of m -2 ; ΔL fDi is the dimensionless half length of the fracture unit i; x Dj is the dimensionless longitudinal coordinate; x Di is the dimensionless transverse coordinate; y Dj is the dimensionless longitudinal coordinate of the cylindrical continuous point source; y wDi is the dimensionless transverse coordinate of the cylindrical continuous point source.

[0234] Specifically, in step S6, the Laplace space shale gas well bottom-hole pressure response is inverted to the real space by using the stehfest numerical inversion method.

[0235] Specifically, in step S7, the bottom-hole measured data include: mid-depth of the producing formation, shale reservoir thickness, reservoir temperature, water saturation, gas density, gas viscosity, gas compressibility, and Langmuir pressure and volume measured based on laboratory experiments. The sensitivity parameters of the characteristic curve chart include: formation pressure, matrix permeability, matrix fracture mass channeling coefficient, mass diffusion coefficient, dimensionless fracture half-length, fracture dimensionless mass elastic reservoir ratio, and matrix dimensionless mass elastic reservoir ratio.

[0236] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A two-phase well test analysis method considering Knudsen diffusion and crossflow in pseudo-three-pore media, characterized in that, Includes the following steps: S1: Establish a microphysical model of the shale gas reservoir for well test analysis. The assumptions of the microphysical model of the shale gas reservoir include: the shale gas reservoir is a pseudo-triporous medium composed of matrix nanopores, matrix macropores and fracture system. There is single-phase flow of shale gas in the matrix system and two-phase flow of gas and liquid in the fracture system. After the shale gas is desorbed from the matrix surface to the matrix nanopores, it migrates to the matrix macropores by Knudsen diffusion and then migrates from the macropores to the fracture system by channeling. S2: Based on the microphysical model of the shale gas reservoir, establish the two-phase flow equation for the fracture system and the single-phase flow equation for the matrix system, respectively; S3: By combining the two-phase flow equation of the fracture system and the single-phase flow equation of the matrix system, a two-phase dimensionless pseudo-pressure difference micro-flow equation is obtained. S4: Based on the two-phase dimensionless pseudo-pressure difference micro-flow equation and combined with the outer boundary case of the circular shale gas reservoir, the point source solution of the two-phase dimensionless pseudo-pressure in the circular shale gas reservoir is obtained. S5: Based on the continuous point source solution of the two-phase dimensionless pseudo-pressure in a circular shale gas reservoir, and combined with the macroscopic physical model of the shale gas well, the bottom hole pressure solution of the shale gas well in Laplace space is obtained. S6: Invert the bottom pressure solution of shale gas well in Laplace space to real space, calculate the two-phase pseudo-pressure difference at the bottom of the shale gas well and the double logarithmic characteristic curve of the pseudo-pressure derivative with time, and obtain the characteristic curve chart. S7: Based on the characteristic curve generated from the bottom hole measured data, adjust the sensitivity parameters of the characteristic curve chart, and use the sensitivity parameters corresponding to the fitting of the characteristic curve chart with the characteristic curve as the well test analysis result.

2. The two-phase well test analysis method considering Knudsen diffusion and crossflow in pseudo-tripore media as described in claim 1, characterized in that, In step S2, the established two-phase flow equation for the fracture system is: ； Where, ρ g ρ is the density of shale gas. w The density of water; k f k represents the fracture permeability. frg Let k be the relative permeability component of the gas phase in the fracture along the r direction. frw P represents the relative permeability component of the water phase in the fracture along the r-direction. f For the crack pressure, q m For cross-current term; μ g Let μ be the viscosity of the gas. w ϕ is the viscosity of water. f S represents the porosity of the fracture system. g S represents the gas saturation level. w This represents the gas saturation level.

3. The two-phase well test analysis method considering Knudsen diffusion and crossflow in pseudo-tripore media as described in claim 2, characterized in that, In step S2, if the macropores migrate into the fracture system via unsteady flow, the established single-phase flow equation for the matrix system is: ； Where, k ap k represents the apparent permeability of the matrix under pressure and concentration fields. mrg P represents the relative permeability component of the matrix gas phase in the r-direction; m ϕ is the matrix pore pressure; m q represents matrix porosity; adsorption / desorption capacity. dif Calculated from the Langmuir isotherm adsorption equation; k m For matrix permeability; If macropores migrate into the fracture system via steady-state channeling, then the single-phase flow equation for the matrix system is: ； Where, ρ gm ρ represents the density of shale gas in the matrix. gf Let be the density of shale gas in the fracture; α is the shape factor, for a sphere α = 15 / r 2 , where r is the radius of the sphere.

4. The two-phase well test analysis method considering Knudsen diffusion and crossflow in pseudo-tripore media as described in claim 3, characterized in that, Step S3 also includes the following steps: Two-phase dimensionless variables are introduced to simplify the two-phase flow equation of the fracture system and the single-phase flow equation of the matrix system; wherein, the single-phase flow equation of the matrix system is transformed into the apparent two-phase flow equation of the matrix system. The simplified two-phase flow equations of the fracture system and the apparent two-phase flow equations of the matrix system are analyzed with respect to time t. D Laplace transform; By combining the two-phase flow equations of the fracture system after Laplace transform with the apparent two-phase flow equations of the matrix system, the dimensionless pseudo-pressure difference micro-flow equations of the two phases are obtained.

5. The two-phase well test analysis method considering Knudsen diffusion and crossflow in pseudo-tripore media as described in claim 4, characterized in that, The method to transform the single-phase flow equation of the matrix system into a seemingly two-phase flow equation of the matrix system is as follows: If macropores migrate into the fracture system via unsteady channeling, then the matrix system is considered to have a two-phase flow equation: ； Where, r mD The matrix is ​​a single-phase dimensionless radius; Δm pm The matrix appears to be a dimensionless pseudo-pressure difference between the two phases; ω f For the dimensionless mass elastic storage capacity ratio of the crack, ω d The dimensionless mass elastic storage capacity ratio of the matrix; t D λ represents the single-phase dimensionless time of the matrix; λ is the dimensionless mass transfer coefficient of the matrix. k app k represents the combined matrix permeability considering both Knudsen diffusion and Darcy flow within the matrix. m Indicates matrix permeability; If macropores migrate into the fracture system via steady-state channeling, then the matrix system is considered to have a two-phase flow equation: ； Where, Δm pf The pressure difference between the two phases of the crack is dimensionless.

6. The two-phase well test analysis method considering Knudsen diffusion and crossflow in pseudo-tripore media as described in claim 5, characterized in that, Step S4 also includes the following steps: Based on the two-phase dimensionless pseudo-pressure difference micro-flow equation, a two-phase dimensionless pseudo-pressure difference micro-flow equation corresponding to a continuous point source of any cylindrical micro-element in a circular shale reservoir is established. By introducing Fourier finite cosine transform into the micro-flow equation of the two-phase dimensionless pseudo-pressure difference corresponding to a continuous point source of any cylindrical micro-element in a circular shale reservoir, and combining it with the outer boundary case of the circular shale gas reservoir, the point source solution of the two-phase dimensionless pseudo-pressure corresponding to a continuous point source of any cylindrical micro-element in the circular shale gas reservoir is obtained.

7. The two-phase well test analysis method considering Knudsen diffusion and crossflow in pseudo-tripore media as described in claim 6, characterized in that, Step S5 also includes: If the macroscopic physical model of a shale gas well makes the following assumptions: Assumption 1: The shale gas well is a fully fractured, infinitely conductive vertical well in a circular shale gas reservoir; Assumption 2: The fracturing fracture is a symmetrical double-wing fracture about the wellbore, and its half-length is x. f ; Assumption 3: Shale gas wells produce a constant surface yield q sc Or a constant bottom hole pressure p wf Production begins; Assumption 4: The gas is slightly compressible and has a constant viscosity and compressibility coefficient; the effects of capillary force and gravity are ignored; the crack is an infinite flow channel; Assumption 5: Establish a rectangular coordinate system with the length of the fracture surface as the x-axis, the height of the fracture surface as the z-axis, and the y-axis perpendicular to the fracture surface. When calculating the bottom hole pressure, convert the rectangular coordinate system to a cylindrical coordinate system. The point source solution of the two-phase dimensionless pseudo-pressure is then integrated along the reservoir thickness direction and then along the fracture direction to obtain the bottom hole pressure solution of the shale gas well in the Laplace space.

8. The two-phase well test analysis method considering Knudsen diffusion and crossflow in pseudo-tripore media as described in claim 6, characterized in that, Step S5 also includes: If the macroscopic physical model of a shale gas well makes the following assumptions: Assumption 1: The shale gas well is a fully fractured, infinitely conductive horizontal well in a circular shale gas reservoir; Assumption 2: The total number of fracturing fractures is M, the fracture surfaces are perpendicular to the central axis of the horizontal section of the wellbore, and they are distributed along the central axis of the wellbore; Assumption 3: The fracturing fracture is a symmetrical or asymmetrical double-wing fracture about the wellbore, with infinite flow and a length of x. f ; Assumption 4: Each crack is divided into 2N elements, and the flow rate is uniformly distributed within each crack element i. Assumption 5: Shale gas wells produce a constant surface yield q. sc Or a constant bottom hole pressure p wf Production begins; Assumption 6: The gas is slightly compressible and has a constant viscosity and compressibility coefficient; the effects of capillary force and gravity are ignored; Assumption 7: Establish a coordinate system with the center of the intersection of the first fracture near the horizontal section of the wellbore as the origin of the rectangular coordinate system. The y-axis coincides with and extends along the axis of the horizontal section of the wellbore, the z-axis is positive upward along the reservoir thickness direction, and the x-axis is perpendicular to the above two directions. When calculating the bottom hole pressure, the rectangular coordinate system is converted to a cylindrical coordinate system. The point source solution of the two-phase dimensionless pseudo-pressure is first integrated along the z-direction to calculate the line source solution of the two-phase dimensionless pseudo-pressure of M×2N fracture elements. Then, the line source solution of the two-phase dimensionless pseudo-pressure of the M×2N fracture elements of the fractured horizontal well and the production accumulation equation are combined to solve the bottom hole pressure solution of the shale gas well in Laplace space.

9. The two-phase well test analysis method considering Knudsen diffusion and crossflow in pseudo-tripore media as described in claim 1, characterized in that, In step S6, the Stehfest numerical inversion method is used to invert the bottom hole pressure solution of the shale gas well in Laplace space to the real space.

10. The two-phase well test analysis method considering Knudsen diffusion and crossflow in pseudo-triporous media as described in any one of claims 1 to 9, characterized in that, The sensitive parameters include: formation pressure, matrix permeability, matrix fracture mass channeling coefficient, mass diffusion coefficient, dimensionless fracture half-length, fracture dimensionless mass elastic reservoir ratio, and matrix dimensionless mass elastic reservoir ratio.

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