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

Abstract

The application discloses a two-phase well testing analysis method considering Fick 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 Fick 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 Fick diffusion and channeling in the pseudo-three-pore medium.

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 Fick 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. However, 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 Fick 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 well testing analysis of shale gas wells with Fick diffusion and channeling flow in pseudo-three-pore media.

[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 Fick diffusion and channeling in pseudo-three-pore media comprises the following steps:

[0007] S1: building 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 Fick diffusion, and then moves to the fracture system in the form of channeling from the matrix 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 two-phase pseudo-pressure difference at the shale gas well bottom hole 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 measured data at the well bottom, 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 scheme can realize well test interpretation considering gas-liquid two-phase flow existing in the fracture system in the physical model considering Fick diffusion and channeling in pseudo-three-pore media, so as to improve the accuracy of well test interpretation result. 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 is shown in the flowchart;

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

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

[0019] Figure 5 The physical model schematic diagram of the circular shale gas reservoir fractured horizontal well of the present application is shown in the flowchart. 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 Figure 1 shown, the present application considers a two-phase well test analysis method of Fick diffusion and channeling in pseudo-triple-pore media, which includes 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 Fick diffusion, and then moves to the fracture system from the matrix macropores in the form of channeling;

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

[0029] Specifically, according to as Figure 2The physical model shown, in step S2, the gas-liquid two-phase flow in microcracks, respectively, the establishment of gas and liquid flow equation, at the same time, since the gas desorbed from the matrix surface, will first into the matrix macro-pore, and then from the macro-pore channeling to the microcrack system, therefore, the microcrack gas flow equation also contains shale gas channeling term q from the macro-pore to the microcrack m ; then, the fracture system two-phase flow equation is as follows:

[0030]

[0031] In the formula, ρ g is the density of 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 r-direction fracture gas phase relative permeability component, k frw is the r-direction fracture water phase relative permeability component, k frg and k frw are fractions; P f is the pressure of the fracture system, unit Pa; μ g is the viscosity of 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 , since the gas desorbed from the matrix surface, to Fick diffusion to the matrix macro-pore, at the same time, shale gas is in the form of channeling from the matrix to the microcrack; wherein, the form of channeling includes pseudo-steady channeling and non-steady channeling.

[0033] The diffusion amount q F is calculated by Fick's law: C m is the molar concentration of shale gas in the matrix under non-steady state, mol / m 3 ;

[0034] When the matrix diffuses to the fracture respectively, the change rate of concentration with time is:

[0035] D F is the Fick diffusion coefficient, m 2 / s;

[0036] When the matrix diffusion to the fracture is pseudo-steady state diffusion, the rate of change of concentration with time is:

[0037] C E (P f ) is the gas concentration at the interface between the matrix rock and the fracture system, mol / m 3 ; C m is the volume concentration of shale gas in the matrix under pseudo-steady state conditions, mol / m 3 ;

[0038] Thus, if the shale gas moves to the matrix macro-pore in the form of Fick steady state diffusion, the Fick diffusion amount is expressed as:

[0039]

[0040] If the shale gas moves to the matrix macro-pore in the form of Fick non-steady state diffusion, the Fick diffusion amount is expressed as:

[0041]

[0042] Where M g is the molar mass of the gas; r m is the radial coordinate of the matrix rock, R m is the radius of the spherical matrix body.

[0043] Specifically, when the migration of the matrix macro-pore to the micro-fracture system is in the form of non-steady state channeling, because there is only gas phase flow in the matrix, the single-phase flow equation of the matrix system and the channeling flow amount expression are as follows:

[0044]

[0045] Further, when the migration of the matrix macro-pore to the micro-fracture system is in the form of steady state channeling, because there is only gas phase flow in the matrix, the single-phase flow equation of the matrix system and the channeling flow amount expression are as follows:

[0046]

[0047] Where k m is the matrix permeability; p gm is the density of shale gas in the matrix, p gf is the density of shale gas in the fracture; k mrg is the r-direction matrix gas phase relative permeability component, k ap is the apparent permeability of the matrix in the pressure field and concentration field; P m is the fracture pressure; f m is the matrix porosity; and q difcalculated from the Langmuir isotherm adsorption equation; a is a shape factor, a = 15 / r for a sphere 2 , r is the radius of the sphere.

[0048] Specifically, the step S3 further comprises: introducing two-phase dimensionless variables 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 converted into a matrix system virtual two-phase flow equation;

[0049] The simplified two-phase flow equation of the fracture system and the matrix system virtual two-phase flow equation are Laplace transformed with respect to time t D ;

[0050] The two-phase dimensionless pseudo-pressure difference microscopic flow equation is obtained by simultaneously solving the Laplace-transformed two-phase flow equation of the fracture system and the matrix system virtual two-phase flow equation.

[0051] Further, if the macro-pore moves to the fracture system in a non-steady-state channeling manner, the simplified two-phase flow equation of the fracture system is:

[0052]

[0053] The specific simplification process is as follows:

[0054] 1. Introducing two-phase pseudo-pressure and two-phase compressibility coefficient, the fracture pressure in the two-phase flow equation of the fracture system 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 coefficient is: t = p g C g S g + p w C w S w , in the formula, C t is the two-phase compressibility coefficient, C g is the gas compressibility coefficient, the unit is Pa -1 .

[0055] That is:

[0056]

[0057] Therefore, the flow equation of the fracture system in the spherical coordinates can be expressed as:

[0058]

[0059] 2. The matrix gas phase pseudo-pressure is defined as follows:

[0060]

[0061] 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, Pa.s.

[0062] The density of the gas and the compressibility factor are defined as follows:

[0063]

[0064] 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; C g is the rock compressibility, Pa -1 .

[0065] Then, using the above defined matrix gas phase pseudo pressure, the density of the gas and the compressibility factor, the matrix system flow equation is simplified, and the rock compressibility is taken as the value at the initial state; and the following is obtained:

[0066]

[0067] In order to connect the fracture equation and the matrix equation, the single phase pseudo pressure of the matrix system needs to be 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:

[0068]

[0069] Then, the relationship between the matrix gas phase pseudo pressure and the matrix apparent two phase pseudo pressure is combined, and the following two phase dimensionless variables are introduced to further simplify the two phase flow equation of the fracture system and the single phase flow equation of the matrix system;

[0070] Fracture two phase dimensionless radius: Dimensionless time: Steady state diffusion comprehensive mass transfer coefficient: Matrix dimensionless mass elastic storage ratio: Fracture dimensionless mass elastic storage ratio: Matrix-fracture mass cross flow coefficient: Steady state diffusion matrix mass cross flow coefficient: Matrix single phase dimensionless pseudo pressure difference: Δ* m pm =m * (p i )-m * (p m );

[0071] Matrix two-phase dimensionless pseudo-pressure difference: Δm pm =m(p i )-m(p m );

[0072] Fracture two-phase dimensionless pseudo-pressure difference: Δm pf =m(p i )-m(p f );

[0073] Matrix diffusion concentration difference: ΔC mD =C m (p m )-C m (p i );

[0074] Matrix equilibrium diffusion concentration difference: ΔC ED =C E (p m )-C m (p i );

[0075] After simplification, the fracture system two-phase flow equation is obtained as follows: If the Fick diffusion is non-steady-state diffusion, the matrix system two-phase flow equation is as follows:

[0076]

[0077] If the Fick diffusion is pseudo-steady-state diffusion, the matrix system two-phase flow equation is as follows:

[0078]

[0079] Further, if the macro-pore moves to the fracture system in a pseudo-steady-state channeling manner, the simplified fracture system two-phase flow equation is as follows:

[0080]

[0081] The specific simplification process is as follows:

[0082] 1. Introducing two-phase pseudo-pressure and two-phase compressibility coefficient, the fracture pressure in the fracture system two-phase flow equation is converted into the fracture two-phase pseudo-pressure; specifically, the two-phase pseudo-pressure is as follows: 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 = p g C g S g + p w C w S w , wherein C t is the two-phase compressibility, C g is the gas compressibility, with the unit of Pa -1 .

[0083] Thus, the following is obtained:

[0084]

[0085] Thus, the fracture system flow equation in spherical coordinates can be expressed as:

[0086]

[0087] 2、The matrix gas phase pseudo-pressure is defined as follows:

[0088]

[0089] wherein p m is the pressure in the matrix pore; m * (p m ) is the matrix gas phase pseudo-pressure; and μ mg is the viscosity of the gas in the matrix, Pa·s.

[0090] The density and compressibility of the gas are defined as follows:

[0091]

[0092] wherein 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 rock compressibility, Pa -1 .

[0093] The matrix system single-phase flow equation is simplified using the above-defined matrix gas phase pseudo-pressure, the density and compressibility of the gas, and the rock compressibility is taken as the value in the initial state; that is, the following is obtained:

[0094]

[0095] In order to connect the fracture equation and the matrix equation, it is necessary to link the single-phase pseudo-pressure of the matrix system with the apparent two-phase pseudo-pressure. Through mathematical relations, the matrix single-phase pseudo-pressure and the matrix apparent two-phase pseudo-pressure are replaced as follows:

[0096]

[0097] Then, combining the relationship between the matrix gas phase pseudo-pressure and the matrix apparent two-phase pseudo-pressure, the following two-phase dimensionless variables are introduced to further simplify the fracture system two-phase flow equation and the matrix system single-phase flow equation;

[0098] Dimensionless radius of two-phase crack: Dimensionless time: Steady-state diffusion comprehensive mass transfer coefficient: Dimensionless mass elastic storage ratio of matrix: Dimensionless mass elastic storage ratio of fracture: Matrix fracture mass crossflow coefficient: Steady-state diffusion matrix mass cross-flow coefficient: Matrix single-phase dimensionless pseudo-pressure difference: Δ * m pm =m * (p i )-m * (p m );

[0099] Dimensionless pseudo-pressure difference between the two phases as viewed by the matrix: Δm pm =m(p i )-m(p m );

[0100] Dimensionless pseudo-pressure difference between two phases in the fracture: Δm pf =m(p i )-m(p f );

[0101] Matrix diffusion concentration difference: ΔC mD =C m (p m )-C m (p i );

[0102] Matrix equilibrium diffusion concentration difference: ΔC ED =C E (p m )-C m (p i );

[0103] After simplification, the two-phase flow equation of the fracture system is obtained as follows: If Fick diffusion is non-steady state diffusion, the matrix system is considered as two-phase flow equations as follows:

[0104]

[0105] If Fick diffusion is quasi-steady state diffusion, the matrix system is considered as two-phase flow equations as follows:

[0106]

[0107] Then, Laplace transform is taken for the simplified fracture system two-phase flow equations and matrix system two-phase flow equations with respect to time t, i.e.: D

[0108] If macro-pore moves into fracture system in the way of non-steady state channeling, Fick diffusion is non-steady state diffusion, then the corresponding result is obtained as follows:

[0109]

[0110] According to Fick's second law, we have:

[0111]

[0112] In the case of spherical coordinates, the non-steady state diffusion equation in matrix rock is transformed as follows:

[0113]

[0114] Combining the aforementioned definition of dimensionless variable, the non-steady state diffusion equation in matrix system is:

[0115]

[0116] Laplace transform is taken for the above equation with respect to time t: D

[0117]

[0118] Further, we have:

[0119]

[0120] According to Langmuir isothermal adsorption equation, we have:

[0121]

[0122] Transform the above equation:

[0123]

[0124] Substitute the above equation into , we have: ​​

[0125]

[0126] Taking the derivative of the above equation with respect to r mD at both ends, we get:

[0127]

[0128] Substituting the L-transformed matrix system into the two-phase flow equation, we get:

[0129]

[0130] Substituting the L-transformed matrix system into the two-phase flow equation, we get:

[0131]

[0132] If the macro-pore moves to the fracture system in the form of non-steady-state channeling, Fick diffusion is steady-state diffusion, and the corresponding result is obtained:

[0133]

[0134]

[0135] Taking the L-transform of with respect to the dimensionless time t D , we get

[0136] According to the Langmuir isothermal adsorption equation, we get:

[0137]

[0138] Substituting the L-transformed matrix system into the two-phase flow equation, we get:

[0139]

[0140] Further, Substituting the L-transformed matrix system into the two-phase flow equation, we get:

[0141]

[0142] Let , then:

[0143]

[0144] If the macro-pore moves to the fracture system in the form of steady-state channeling, Fick diffusion is non-steady-state diffusion, and the corresponding result is obtained:

[0145]

[0146] From Fick's second law:

[0147]

[0148] In spherical coordinates, the unsteady diffusion equation in the matrix block is:

[0149]

[0150] Introducing the aforementioned dimensionless definitions, the unsteady diffusion equation in the matrix system is:

[0151]

[0152] Taking the Laplace transform of the above equation with respect to time t: tD

[0153]

[0154] Further, we can obtain:

[0155]

[0156] According to the Langmuir isothermal adsorption equation, we can obtain:

[0157]

[0158] Transforming the above equation:

[0159]

[0160] Substituting the above equation into the equation: we can obtain:

[0161]

[0162] Taking the derivative of both ends of the above equation with respect to x: rmD

[0163]

[0164] Substituting into the matrix system two-phase flow equation, we obtain:

[0165]

[0166] Substituting into the fracture system two-phase flow equation, we obtain the two-phase dimensionless pseudo-pressure difference microscopic flow equation:

[0167]

[0168] ​​If the macro-pore moves to the fracture system in the form of steady-state channeling, Fick diffusion is steady-state diffusion, and the following is obtained:

[0169] According to the Langmuir isothermal adsorption equation, the following is obtained:

[0170]

[0171] Substituting the Laplace transform into the matrix system two-phase flow equation, the following is obtained:

[0172]

[0173] The relationship between the matrix dimensionless two-phase pseudo-pressure after Laplace transform and the fracture dimensionless two-phase pseudo-pressure is obtained:

[0174] Substituting the Laplace transform into the fracture system two-phase flow equation, the following is obtained:

[0175]

[0176] The fracture dimensionless pseudo-pressure difference equation is further obtained:

[0177]

[0178] Further, in step S4, first, according to the two-phase dimensionless pseudo-pressure difference microscopic flow equation, the two-phase dimensionless pseudo-pressure difference microscopic flow equation corresponding to the continuous point source of any cylindrical microelement in the circular shale reservoir is established;

[0179] The Fourier finite cosine transform is introduced into the two-phase dimensionless pseudo-pressure difference microscopic flow equation of the continuous point source of any cylindrical microelement in the circular shale reservoir, and the point source solution equation of the two-phase pseudo-pressure of the continuous point source of any cylindrical microelement in the circular shale gas reservoir is obtained in combination with the outer boundary condition of the shale gas reservoir; as shown in the physical model Figure 3 In the physical model shown, the shale gas reservoir is a circular shale gas reservoir, and the basic assumptions for the continuous point source of any cylindrical microelement in the circular shale reservoir are as follows:

[0180] (1) The top and bottom surfaces of the shale reservoir are closed, the horizontal direction outer boundary is infinite, the reservoir thickness is h, and the outer boundary radius of the reservoir is r e ;

[0181] (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 reservoir is the x0y plane, and the axis perpendicular to the bottom surface of the reservoir is the z axis; converted into cylindrical coordinates, only r and z directions are considered;

[0182] (3) The horizontal permeability of the shale reservoir is k fh , and the vertical permeability is k fz ;

[0183] (4) A cylindrical microelement has its center located at (0, 0, z w ) in a rectangular coordinate system, i.e., (0, z w ) in a cylindrical coordinate system; the radius of the cylindrical microelement is r, and the height is z w ;

[0184] (5) The gas and liquid phases flow into the cylindrical microelement from the side of the cylindrical microelement, and the flow rate of the shale gas flowing into the cylindrical microelement at the surface standard condition is q gscin , the flow rate of the fracturing flowback fluid flowing into the cylindrical microelement at the surface standard condition is q wscin , and the effects of capillary force and gravity are ignored.

[0185] For the convenience of research, the cylindrical coordinate system is used to interpret the model. Thus, the fracture two-phase flow equation can be expressed in the cylindrical coordinate system as follows:

[0186]

[0187] At the same time, since the assumed cylindrical microelement is an infinitesimal amount, when the gas and water two-phase fluid flows into the cylindrical microelement from the side, the microelement can be regarded as a continuous point sink, and thus the corresponding inner boundary condition expression is as follows:

[0188]

[0189] In the formula, p gsc is the density of the shale gas at the surface standard condition, p wsc is the density of the fracturing flowback fluid at the surface standard condition; the upper and lower boundary conditions are as follows: The outer boundary condition is as follows:

[0190] Further, the following two-phase dimensionless variables are introduced to simplify the above equation and boundary conditions:

[0191] The fracture two-phase dimensionless flow equation in the cylindrical coordinate system is as follows:

[0192] The inner boundary condition is as follows:

[0193] The upper and lower boundary conditions are as follows:

[0194] The outer boundary condition is as follows:

[0195] After the time t DAfter the Laplace transformation, the two-phase dimensionless fracture flow equation can be transformed into:

[0196]

[0197] The above formula is associated with the dimensionless fracture gas-liquid two-phase flow equation obtained in step S3 in the present application, and the Laplace variable s in the above formula is replaced as follows:

[0198]

[0199] The inner boundary condition is:

[0200] The upper and lower boundary conditions are: The outer boundary condition is:

[0201] For solving the initial boundary value problem, first:

[0202]

[0203] Substitute the above formula into , and combine the top and bottom sealing boundary conditions, use the Sturm-Liouville eigenvalue theory, and the eigenvalue equation about z D can be obtained:

[0204]

[0205] At the same time, according to the Sturm-Liouville eigenvalue theory, the eigenvalue of is n = (nπ) 2 , so the corresponding eigenfunction system can be expressed as:

[0206] Z n (z D ) = cosnπz D , n = 0, 1, 2,...

[0207] Therefore, taking cosnπz D as the kernel function, the Fourier finite cosine transformation is introduced into , and is defined as The function after the Fourier finite cosine transformation about z D can be obtained:

[0208]

[0209] The inner boundary expression is:

[0210]

[0211] Outer boundary expression:

[0212]

[0213] Let The fracture two-phase flow equation can be written as follows:

[0214]

[0215] Since The Bessel equation of the dummy variable is known to have the general solution:

[0216]

[0217] where a n and b n are undetermined constants.

[0218] Substituting n = 0, 1, 2, ··· into the inner boundary expression, we get:

[0219]

[0220] According to the properties of the Bessel function, when x→0, I1(x) = 0, Thus, from the above inner boundary expression, we have:

[0221]

[0222] Thus:

[0223]

[0224] Substituting it into the outer boundary expression:

[0225] Let Then:

[0226]

[0227] Substituting n = 0, 1, 2, ··· into the inner boundary expression, we get:

[0228]

[0229] Performing the inverse Fourier cosine transform on the above equation, we get the point source solution of the dimensionless two-phase pseudo-pressure of the fracture gas-liquid two-phase flow equation, i.e.,

[0230]

[0231] In the equation:

[0232] If the outer boundary of the shale gas reservoir is top and bottom face closed, and the side face outer boundary is closed, the mass flow rate of the fluid passing through the outer boundary is 0, which is converted into the dimensionless two-phase pseudo-pressure difference form after the Laplace transform, that is:

[0233]

[0234] At this time, C is a constant, then δ = 0, which is substituted into the point source solution of the two-phase pseudo-pressure of the dimensionless fracture gas-liquid two-phase flow equation, that is

[0235]

[0236] Further, the point source solution of the two-phase pseudo-pressure of the dimensionless fracture gas-liquid two-phase flow equation corresponding to the outer boundary condition of the shale gas reservoir that the top and bottom face are closed and the side face outer boundary is closed is obtained as follows:

[0237]

[0238] In the formula, q gscins and q wscins are usually constants, so:

[0239]

[0240] If the outer boundary of the shale gas reservoir is top and bottom face closed, and the side face outer boundary is infinite, at this time the expression of the outer boundary is:

[0241]

[0242] According to the properties of the Bessel function, I0(∞) = ∞, which is substituted into , and the following can be obtained: a n = c n = 0; again, a n = c n = 0 is substituted into , and the point source solution of the two-phase pseudo-pressure of the dimensionless fracture gas-liquid two-phase flow equation corresponding to the outer boundary condition of the shale gas reservoir that the top and bottom face are closed and the side face outer boundary is infinite is obtained as follows:

[0243]

[0244] In the formula, ρ gsc is the density of the gas at the ground standard condition, with the unit of g / cm 3 ; ρ wsc is the density of the water at the ground standard condition, with the unit of g / cm 3 ; q gscins is the flow rate of the shale gas flowing into the cylindrical element at the ground standard condition, with the unit of m / s; q wscinsis the flow rate of the fracturing flowback fluid flowing into the microelement of the cylindrical body under the ground standard condition, in m / s; h is the thickness of the shale reservoir; n = 0, 1, 2, …; k fh is the horizontal permeability of the shale reservoir; r e is the outer boundary radius of the shale reservoir; r eD is the dimensionless radius of the outer boundary of the shale reservoir; z wD is the dimensionless distance, dimensionless; z D is the dimensionless height in the z direction; I0, I1 are the first kind of virtual quantity Bessel functions, dimensionless; K0, K1 are the second kind of virtual quantity Bessel functions, dimensionless.

[0245] Further, in step S5, since there are two types of fracturing vertical wells and fracturing horizontal wells in the shale gas well, and different physical models are corresponded, the bottom hole pressure responses of the fracturing vertical wells and the fracturing horizontal wells need to be respectively researched.

[0246] As shown in the physical model of the shale gas well, Figure 4 the assumption conditions are as follows:

[0247] Assumption condition one: the shale gas well is a completely fractured infinite conductivity fracturing vertical well in a circular reservoir;

[0248] Assumption condition two: the fracturing fracture is a symmetric double-wing fracture about the wellbore, and the half length thereof is x f .

[0249] Assumption condition three: the shale gas well is produced at a constant surface production q sc or a constant bottom hole pressure p wf .

[0250] Assumption condition four: the gas is slightly compressible, and has a constant viscosity and a constant compressibility coefficient; the effects of capillary force and gravity are ignored; the fracture is infinite conductivity;

[0251] Assumption condition five: a rectangular coordinate system is established 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 being perpendicular to the fracture surface, and the rectangular coordinate system is converted into a cylindrical coordinate system when the bottom hole pressure is calculated.

[0252] Then, in combination with the assumption conditions of the physical model of the shale gas well, the point source solution of the dimensionless fracture gas-liquid two-phase flow equation is integrated along the reservoir thickness direction, and then integrated along the fracture direction, so as to solve the bottom hole pressure response of the shale gas well in the Laplace space.

[0253] Specifically, if the outer boundary condition of the shale gas reservoir is that the top and bottom surfaces are closed and the side surface outer boundary is closed, the pressure response expression of the bottom hole of the circular reservoir fracturing vertical well is:

[0254]

[0255] where:

[0256] Since the trigonometric function cos(nπz wD ) has the property of integrating to 0 over the symmetric domain, and noting that Moreover, the flow distribution of the fractured vertical well is uniform, the mass flow rate of gas production of the gas well can be converted by the product of the density of shale gas, point source intensity and the area of the fracture, and the relationship between the mass flow rate of water production of the gas well and the density of fracturing flowback fluid, point source intensity and the area of the fracture can be obtained: gsc q gsc +ρ wsc q wsc =2hx f (ρ gsc q gscins +ρ wsc q wscins ); combining the above relationship, the expression of the pressure response at the bottom of the fractured vertical well in the circular reservoir is simplified as:

[0257]

[0258] Two dimensionless variables of two phases are introduced as follows The above expression is dimensionless, and the following is obtained:

[0259]

[0260] Substituting into the above expression, the following is obtained:

[0261]

[0262] Since the pressure response on the fracture surface is studied, y w =0, so y wD =0; at the same time, in order to use the dynamic pressure distribution of the bottom of the uniform flow fracture to represent the dynamic pressure distribution of the infinite conductivity fracture, a suitable equivalent observation point needs to be selected first, and its coordinates are as follows: In order to 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 wrong result of numerical integration, the integral transformation is needed, let Then: Therefore, the shale gas well bottom pressure response equation in the Laplace space is:

[0263]

[0264] The above formula is the dimensionless bottom hole pressure dynamic distribution of uniform flow fracture. When calculating the pressure distribution of infinite conductivity fracture, the value of the horizontal coordinate of the observation point is x D =0.732.

[0265] If the outer boundary of the shale gas reservoir is closed at the top and bottom, and the outer boundary of the side is infinite, the pressure response expression of the bottom hole of the circular reservoir fractured vertical well is:

[0266]

[0267] Simplifying, we get: Substitute again Further simplified to: reintroduction By non-dimensionalization, we can get:

[0268]

[0269] Next, Substituting into the above formula, we can get:

[0270]

[0271] When studying the pressure response on the fracture surface, in order to use the dynamic distribution of bottom hole pressure of uniform flow fracture to characterize the dynamic distribution of pressure in infinite conductivity fracture, it can be obtained:

[0272]

[0273] in, α is the shape factor, dimensionless.

[0274] In addition, if Figure 5 The physical model of the shale gas well shown is based on the following assumptions:

[0275] Assumption 1: The shale gas well is a fully fractured horizontal well with infinite conductivity in a circular reservoir;

[0276] Assumption 2: The total number of hydraulic fractures is M, and the fracture planes are perpendicular to the central axis of the horizontal section of the wellbore and distributed along the central axis of the wellbore;

[0277] Assumption 3: The fracture is a symmetrical double-wing fracture or an asymmetrical double-wing fracture with infinite conductivity and a length of x. f ;

[0278] Assumption 4: Each fracture is divided into 2N units, and the flow distribution within each fracture unit i is uniform;

[0279] Assumption 5: Shale gas wells produce at a constant surface rate qsc Or constant bottom hole pressure p wf to carry out production;

[0280] Assumption 6: The gas is slightly compressible and has a constant viscosity and compressibility; the effects of capillary forces and gravity are ignored;

[0281] Assumption 7: Take the center of the intersection of the first fracture near the horizontal section of the wellbore and the wellbore as the origin of the rectangular coordinate system. Establish a coordinate system in which the y-axis coincides with and extends along the axis of the horizontal section of the wellbore, the z-axis is along the reservoir thickness and is positive upward, and the x-axis is perpendicular to the above two directions. When calculating the bottomhole pressure, the rectangular coordinate system is converted to a cylindrical coordinate system.

[0282] Combined with the assumptions 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 first integrated along the z direction to calculate the line source solution of the two-phase dimensionless pseudo-pressure of M×2N fracture units. Then, the line source solution of the two-phase dimensionless pseudo-pressure of the M×2N fracture units of the fractured horizontal well and the production accumulation equation are combined to solve the bottomhole pressure response of the shale gas well in Laplace space.

[0283] Specifically, the observation point is selected at the node of a crack unit i When (dimensionless coordinates are According to the principle of potential superposition, M×2N discrete crack units on M cracks are at point The total pressure response generated at The resulting superposition of pressure responses:

[0284]

[0285] Since the seepage resistance experienced by the fluid in infinite conductivity fractures and wellbores is much smaller than the resistance experienced at other locations in the reservoir, it is assumed that the fluid flowing in infinite conductivity fractures and wellbores does not produce a pressure drop. Therefore, it can be seen that the pseudo-pressure at any discrete node of the fracture is equal and equal to the bottomhole flowing pressure, that is:

[0286]

[0287] Since the fractured horizontal well has a constant mass production q msc To produce, its value should be equal to the sum of the two-phase mass flow rates of all fracture discrete units, that is: thus,

[0288] Based on M×2N dimensionless two-phase pseudo-pressure equations and a production accumulation equation, the expression is obtained as follows:

[0289]

[0290] wherein, is a coefficient matrix; q Di is the dimensionless mass production of two phases on the crack unit i, dimensionless, i = 1, 2, ···, 2N*M;

[0291] wherein, if the outer boundary condition of the shale gas reservoir is that the top and bottom surfaces are closed, and the lateral outer boundary is closed, then the element in the coefficient matrix is:

[0292] if the outer boundary condition of the shale gas reservoir is that the top and bottom surfaces are closed, and the lateral outer boundary is infinite, then the element in the coefficient matrix is:

[0293] wherein, for the sphere α = 15 / r 2 is a shape factor, with the unit of m -2 ; ΔL fDi is the dimensionless half length of the crack unit i, dimensionless; x Dj is the dimensionless longitudinal coordinate, dimensionless; x Di is the dimensionless transverse coordinate, dimensionless; y Dj is the dimensionless longitudinal coordinate of the cylindrical continuous point source, dimensionless; y wDi is the dimensionless transverse coordinate of the cylindrical continuous point source, dimensionless.

[0294] Specifically, in step S6, the shale gas well bottom hole pressure response in the Laplace space is inverted into the real space by using the stehfest numerical inversion method.

[0295] Specifically, in step S7, the well bottom measured data include: the depth of the production layer, the shale reservoir thickness, the reservoir temperature, the water saturation, the density of the gas, the viscosity of the gas, the gas compressibility coefficient, and the Langmuir pressure and the Langmuir volume measured according to the indoor experiment. The sensitivity parameters of the characteristic curve chart include: the formation pressure, the matrix permeability, the matrix crack mass channeling coefficient, the mass diffusion coefficient, the dimensionless crack half length, the dimensionless mass elastic storage ratio of the crack, and the dimensionless mass elastic storage ratio of the matrix.

[0296] The above only describes preferred embodiments of the present application and is not intended to limit the present application, and any modification, equivalent replacement and improvement made within the spirit and principle of the present application shall be included in the protection scope of the present application.

Claims

1. A two-phase well test analysis method considering Fick diffusion and crossflow in pseudo-three-porosity media, characterized by: The following steps are involved: S1: Establish a shale gas reservoir microphysical model for well test analysis, wherein the assumptions of the shale gas reservoir microphysical model include: the shale gas reservoir is a pseudo-three-porosity medium consisting of matrix nanopores, matrix macropores, and a fracture system; shale gas single-phase flow occurs in the matrix system, and gas-liquid two-phase flow occurs in the fracture system; and shale gas desorbs from the matrix surface into the matrix nanopores and then migrates into the matrix macropores by Fick diffusion, and then migrates from the matrix macropores into the fracture system by cross-flow; S2: Based on the shale gas reservoir microphysical model, a two-phase flow equation for the fracture system and a single-phase flow equation for the matrix system are established respectively; S3: 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 microscopic flow equation is obtained; S4: Based on the two-phase dimensionless pseudo-pressure difference microscopic flow equation and combined with the outer boundary of the circular shale gas reservoir, a 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 the circular shale gas reservoir and combined with the macroscopic physical model of the shale gas well, the bottomhole pressure solution of the shale gas well in Lagrangian space is obtained; S6: Invert the bottom hole pressure solution of the shale gas well in Lagrangian space into real space, calculate the two-phase pseudo-pressure difference at the bottom hole of the shale gas well and the double logarithmic characteristic curve of the pseudo-pressure derivative and time to obtain a characteristic curve chart; S7: adjusting the sensitivity parameters of the characteristic curve plate according to the characteristic curve generated based on the measured data at the bottom of the well, and using the sensitivity parameters corresponding to the fitting of the characteristic curve plate to the characteristic curve as the well test analysis result; Wherein, in step S2, the two-phase flow equation of the fracture system is established as: ； Moreover, if shale gas migrates into the matrix macropores in the manner of Fick steady-state diffusion, the Fick diffusion amount can be expressed as: ； If shale gas migrates into the matrix macropores in the manner of Fick non-steady-state diffusion, the Fick diffusion amount can be expressed as: ； Among them, ρ g is the density of shale gas; ρ w is the density of water; k f is the fracture permeability, k frg is the gas phase relative permeability component of the fracture in the r direction, k frw is the relative permeability component of water phase in the fracture in the r direction; p f is the crack pressure, q m is the cross-flow term; μ g is the viscosity of the gas, μ w is the viscosity of water; ϕ f is the fracture porosity, ϕ m is the matrix porosity; S g is the gas saturation, S w is gas saturation; C E (P f ) is the gas concentration at the interface between the matrix rock and the fracture system, D F is the Fick diffusion coefficient; C m is the gas concentration in the matrix; M g is the molar mass of the gas; r m is the radial coordinate of the matrix rock, R m is the radius of the spherical matrix; If shale gas migrates into the fracture system through the matrix macropores in a steady-state channeling manner, the established single-phase flow equation for the matrix system is: ; and the cross-flow rate is expressed as: ; If shale gas migrates into the fracture system through the matrix macropores in an unsteady-state channeling manner, the established single-phase flow equation for the matrix system is: ; and the cross-flow rate is expressed as: ; Among them, k m is the matrix permeability; ρ gm is the density of shale gas in the matrix, ρ gf is the density of shale gas in the fracture; k mrg is the relative permeability component of the matrix gas phase in the r direction, k ap is the apparent permeability of the matrix in the pressure field and concentration field; p m is the pressure in the matrix pores; ϕ m is the matrix porosity; adsorption and desorption amount q dif Calculated by the Langmuir isotherm adsorption equation; α is the shape factor, for a sphere α = 15 / r 2 , r is the radius of the sphere; Wherein, step S3 further includes the following steps: Introducing two-phase dimensionless variables 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 converted into a matrix system apparent two-phase flow equation; The simplified two-phase flow equation of the fracture system and the two-phase flow equation of the matrix system are analyzed with respect to time t D Laplace transform of The two-phase dimensionless pseudo-pressure difference microscopic flow equation is obtained by combining the two-phase flow equation of the fracture system after Laplace transformation and the two-phase flow equation of the matrix system.

2. The two-phase well testing analysis method considering Fick diffusion and crossflow in pseudo-three-porosity media according to claim 1, characterized in that: If the macropores migrate into the fracture system in an unsteady-state channeling manner, the simplified two-phase flow equation of the fracture system is: ； If Fick diffusion is non-steady-state diffusion, the matrix system is considered as a two-phase flow equation as follows: ； If Fick diffusion is a pseudo-steady-state diffusion, the matrix system is considered as a two-phase flow equation: ； If the macropores migrate into the fracture system in a pseudo-steady-state channeling manner, the simplified two-phase flow equation of the fracture system is: ； If Fick diffusion is non-steady-state diffusion, the matrix system is considered as a two-phase flow equation as follows: ； If Fick diffusion is a pseudo-steady-state diffusion, the matrix system is considered as a two-phase flow equation: ； Among them, r D is the dimensionless radius of the two phases of the crack, r mD is the dimensionless radius of the matrix single phase; Δm pf is the dimensionless pseudo-pressure difference between the two phases of the fracture, Δm pm is the dimensionless pseudo-pressure difference between the two phases as viewed from the matrix; ω f is the dimensionless elastic storage ratio of the fracture mass, ω m is the dimensionless mass elastic storage ratio of the matrix; t D is dimensionless time; M g is the molar mass of the gas; ρ sc is the density of shale gas under standard surface conditions; q sc is the shale gas production under standard surface conditions; λ is the mass crossflow coefficient of the unsteady diffusion matrix; m is the matrix mass crossflow coefficient; h is the shale reservoir thickness; μ gi is the viscosity of the gas under original conditions; ρ gi is the density of the gas under original conditions; k f is the fracture permeability; C mD is the matrix diffusion concentration; β=(1-ω m -ω f )(1-ϕ f -ϕ m ).

3. The two-phase well testing analysis method considering Fick diffusion and crossflow in pseudo-three-porosity media according to claim 2, characterized in that: If the macropores migrate into the fracture system in an unsteady-state channeling manner, the two-phase dimensionless pseudo-pressure difference microscopic flow equation is: ； If Fick diffusion is a pseudo-steady-state diffusion, then ; If Fick diffusion is non-steady-state diffusion, then ; If the macropores migrate into the fracture system in a steady-state channeling manner and the Fick diffusion is a pseudo-steady-state diffusion, then the two-phase dimensionless pseudo-pressure difference microscopic flow equation is: ; where g(s)= ; If the macropores migrate into the fracture system in a steady-state channeling manner and Fick diffusion is a non-steady-state diffusion, then the two-phase dimensionless pseudo-pressure difference microscopic flow equation is: ;in, ; Where σ is the adsorption-desorption quality coefficient; K + The fracture matrix permeability is extremely poor.

4. The two-phase well testing analysis method considering Fick diffusion and crossflow in pseudo-three-porosity media according to claim 3, characterized in that: Step S4 further includes the following steps: According to the two-phase dimensionless pseudo-pressure difference microscopic flow equation, a two-phase dimensionless pseudo-pressure difference microscopic flow equation corresponding to any cylindrical infinitesimal continuous point source in the circular shale reservoir is established; The Fourier finite cosine transform is introduced into the two-phase dimensionless pseudo-pressure difference microscopic flow equation corresponding to any cylindrical infinitesimal continuous point source in the circular shale gas reservoir. Combined with the outer boundary case of the circular shale gas reservoir, the point source solution of the two-phase dimensionless pseudo-pressure corresponding to any cylindrical infinitesimal continuous point source in the circular shale gas reservoir is obtained.

5. The two-phase well testing analysis method considering Fick diffusion and crossflow in pseudo-three-porosity media according to claim 4, characterized in that: Step S5 further includes: The assumptions of the macroscopic physical model of shale gas wells are as follows: Assumption 1: The shale gas well is a fully fractured vertical well with infinite conductivity in a circular shale gas reservoir; Assumption 2: The hydraulic 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 production rate q sc Or constant bottom hole pressure p wf to carry out production; Assumption 4: The gas is slightly compressible and has a constant viscosity and compressibility; the effects of capillary forces and gravity are neglected; the fracture has infinite conductivity; Assumption 5: With the length of the fracture surface as the x-axis and the height of the fracture surface as the z-axis, a rectangular coordinate system is established perpendicular to the y-axis of the fracture surface. When calculating the bottomhole 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 reservoir thickness direction and then integrated along the fracture direction to obtain the bottom hole pressure solution of the shale gas well in Lagrangian space.

6. The two-phase well testing analysis method considering Fick diffusion and cross-flow in pseudo-three-porosity media according to claim 4, characterized in that: Step S5 further includes: The assumptions of the macroscopic physical model of shale gas wells are as follows: Assumption 1: The shale gas well is a fully fractured horizontal well with infinite conductivity in a circular shale gas reservoir; Assumption 2: The total number of hydraulic fractures is M, and the fracture planes are perpendicular to the central axis of the horizontal section of the wellbore and distributed along the central axis of the wellbore; Assumption 3: The fracture is a symmetrical double-wing fracture or an asymmetrical double-wing fracture with infinite conductivity and a length of x. f ; Assumption 4: Each fracture is divided into 2N units, and the flow distribution within each fracture unit i is uniform; Assumption 5: Shale gas wells produce at a constant surface rate q sc Or constant bottom hole pressure p wf to carry out production; Assumption 6: The gas is slightly compressible and has a constant viscosity and compressibility; the effects of capillary forces and gravity are ignored; Assumption 7: Take the center of the intersection of the first fracture near the horizontal section of the wellbore and the wellbore as the origin of the rectangular coordinate system. Establish a coordinate system in which the y-axis coincides with and extends along the axis of the horizontal section of the wellbore, the z-axis is along the reservoir thickness and is positive upward, and the x-axis is perpendicular to the above two directions. When calculating the bottomhole 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 units. Then, the line source solution of the two-phase dimensionless pseudo-pressure of the M×2N fracture units of the fractured horizontal well and the production accumulation equation are combined to solve the bottomhole pressure solution of the shale gas well in Laplace space.

7. The two-phase well testing analysis method considering Fick diffusion and crossflow in pseudo-three-porosity media according to any one of claims 1 to 6, characterized in that: The sensitivity parameters include: formation pressure, matrix permeability, matrix fracture mass crossflow coefficient, mass diffusion coefficient, dimensionless fracture half-length, fracture dimensionless mass elastic storage ratio and matrix dimensionless mass elastic storage ratio.

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