Two-phase well test analysis method considering Fick diffusion in pseudo double-porosity media

Abstract

The present invention discloses a two-phase well test analysis method considering Fick diffusion in a pseudo double-porosity medium. The method further considers the gas-liquid two-phase flow existing in the fracture system in the microscopic physical model considering Fick diffusion in the pseudo double-porosity medium, and establishes the corresponding two-phase flow equation for the fracture system and the single-phase flow equation for the matrix system. By using the established two-phase flow equation for the fracture system and the single-phase flow equation for the matrix system, the double-logarithmic characteristic curve graph of the two-phase pseudo-pressure difference and the pseudo-pressure derivative at the bottom hole of the shale gas well with respect to time is solved. Finally, the sensitivity parameters corresponding to the fitting of the characteristic curve graph with the characteristic curve generated based on the measured bottom hole data are used as the well test interpretation results. Therefore, the present invention can achieve well test interpretation considering the gas-liquid two-phase flow existing in the fracture system in the microscopic physical model considering Fick diffusion in the pseudo double-porosity medium, and improves the accuracy of the well test interpretation results.

Description

Technical Field

[0001] The present invention relates to the field of well test analysis of shale gas wells, and particularly to a two-phase well test analysis method considering Fick diffusion in a pseudo dual-porosity medium. Background Art

[0002] Due to the characteristics of tight and extra-low permeability of shale gas reservoirs, in actual development, shale gas wells usually need to be subjected to reservoir fracturing to form hydraulic fractures to obtain economic production capacity. The shale gas reservoir has the characteristics of extra-low porosity and multi-scale, and the matrix pore size ranges from a few to several hundred nanometers, which makes various migration modes such as Darcy flow, slippage effect, and diffusion exist for shale gas in the reservoir, and the seepage mechanism of the shale gas well reservoir is very complex.

[0003] In actual development, most shale gas wells produce both fracturing flowback fluid and shale gas simultaneously. Due to the presence of fracturing fluid in the reservoir, the fluid flow in the reservoir is gas-liquid two-phase flow. However, most current studies on well test analysis of shale gas wells only consider the single-phase unstable flow in the shale gas well reservoir and do not consider the influence caused by the flow of fracturing fluid in the reservoir, resulting in insufficient accuracy of well test interpretation results. For example, in the patent application "Intelligent Interpretation Analysis Method and Device for Well Test of Shale Reservoir Based on Deep Learning" with the application number 202110386841.0, a physical model for well test of shale gas reservoir wells is established, but 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 of the Invention

[0004] In view of the above deficiencies of the prior art, the purpose of the present invention is to provide a two-phase well test analysis method considering Fick diffusion in a pseudo dual-porosity medium, so as to solve the problem that the accuracy of well test interpretation results is insufficient due to the lack of consideration of gas-liquid two-phase flow in the current well test analysis considering Fick diffusion of shale gas in a pseudo dual-porosity shale reservoir.

[0005] To achieve the above invention purpose, the present invention provides the following technical solutions:

[0006] A two-phase well test analysis method considering Fick diffusion in a pseudo dual-porosity medium, comprising the following steps:

[0007] S1: Establish a microscopic physical model of a shale gas reservoir for well test analysis. Among them, the assumption conditions of the microscopic physical model of the shale gas reservoir include: the shale gas reservoir is a pseudo dual-porosity medium composed of a matrix system and a fracture system. 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 the shale gas desorbs from the matrix surface, it migrates to the fracture system in the form of Fick diffusion;

[0008] S2: According to 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; among them, the two-phase flow equation for the fracture system is:

[0009]

[0010]

[0011] And the single-phase flow equation for the matrix system is:

[0012]

[0013] If the Fick diffusion is Fick steady-state diffusion, the rate of change of concentration with time is:

[0014] If the Fick diffusion is Fick non-steady-state diffusion, the rate of change of concentration with time is:

[0015] Where ρ g is the density of the gas; ρ w is the density of water; k f is the fracture permeability component, k frg is the relative permeability component of the gas phase in the fracture in the r direction, k frw is the relative permeability component of the water phase in the fracture in the r direction; P f is the fracture pressure; μ g is the viscosity of the gas, μ w is the viscosity of water; φ f is the porosity of the fracture system; S g is the gas saturation in the fracture, S w is the water saturation; C m is the gas concentration in the matrix, D F is the Fick diffusion coefficient, R m is the radius of the spherical matrix body, C E (p f ) is the gas concentration at the junction of the matrix rock block and the microfracture system, r m is the radial coordinate of the matrix rock block;

[0016] S3: Combine the two-phase flow equation for the fracture system and the single-phase flow equation for the matrix system to obtain the two-phase dimensionless pseudo-pressure difference micro-flow equation;

[0017] S4: According to the two-phase dimensionless pseudo-pressure difference micro-flow equation and combined with the outer boundary conditions of the circular shale gas reservoir, obtain the point source solution of the two-phase dimensionless pseudo-pressure in the circular shale gas reservoir;

[0018] 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 a shale gas well, the bottom-hole pressure solution of the shale gas well in Laplace space is obtained;

[0019] S6: Invert the bottom-hole pressure solution of the shale gas well in Laplace space into real space, calculate the double logarithm characteristic curve of the two-phase pseudo-pressure difference and pseudo-pressure derivative at the bottom of the shale gas well with respect to time, and obtain the characteristic curve graph;

[0020] S7: According to the characteristic curve generated based on the measured bottom-hole data, adjust the sensitivity parameters of the characteristic curve graph, and use the sensitivity parameters corresponding to the fitting of the characteristic curve graph and the characteristic curve as the well test analysis result.

[0021] Therefore, through the above scheme, the present invention can realize well test interpretation considering gas-liquid two-phase flow existing in the fracture system in the microscopic physical model considering Fick diffusion of shale gas in a pseudo dual-porosity shale reservoir, thereby improving the accuracy of well test interpretation results. Brief Description of the Drawings

[0022] Figure 1 is a schematic flow chart of the well test analysis method provided by the present invention;

[0023] Figure 2 is a schematic diagram of the physical model for well test analysis of a shale gas well of the present invention;

[0024] Figure 3 is a schematic diagram of the physical model of a circular shale gas reservoir of the present invention;

[0025] Figure 4 is a schematic diagram of the physical model of a fractured vertical well in a circular shale gas reservoir of the present invention;

[0026] Figure 5 is a schematic diagram of the physical model of a fractured horizontal well in a circular shale gas reservoir of the present invention. Detailed Embodiments

[0027] The following uses specific specific examples to illustrate the embodiments of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and various details in this specification can also be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention.

[0028] As Figure 1 shown, the two-phase well test analysis method of the present invention considering Fick diffusion in a pseudo dual-porosity medium includes the following steps:

[0029] S1: Establish a microscopic physical model of shale gas reservoir for well test analysis. The assumed conditions of the microscopic physical model of the shale gas reservoir include: the shale gas reservoir is a pseudo dual-porosity medium composed of a matrix system and a fracture system. There is single-phase flow of shale gas in the matrix system, two-phase flow of gas and liquid in the fracture system, and after desorbing from the matrix surface, the shale gas migrates to the fracture system in the way of Fick diffusion;

[0030] S2: According to the microscopic physical model of the shale gas reservoir, establish two-phase flow equations for the fracture system and single-phase flow equations for the matrix system respectively;

[0031] S3: Combine the two-phase flow equations of the fracture system and the single-phase flow equations of the matrix system to obtain a two-phase dimensionless pseudo-pressure difference microscopic flow equation;

[0032] S4: According to the two-phase dimensionless pseudo-pressure difference microscopic flow equation and in combination with the outer boundary conditions of a circular shale gas reservoir, obtain the point-source solution of the two-phase dimensionless pseudo-pressure in the circular shale gas reservoir;

[0033] S5: According to the continuous point-source solution of the two-phase dimensionless pseudo-pressure in the circular shale gas reservoir and in combination with the macroscopic physical model of the shale gas well, obtain the bottom-hole pressure solution of the shale gas well in Laplace space;

[0034] S6: Invert the bottom-hole pressure solution of the shale gas well in Laplace space into 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 versus time to obtain a characteristic curve graph;

[0035] S7: According to the characteristic curve generated based on the measured bottom-hole data, adjust the sensitivity parameters of the characteristic curve graph, and use the sensitivity parameters corresponding to the fitting of the characteristic curve graph and the characteristic curve as the well test analysis result.

[0036] Specifically, in step S2, according to the Figure 2 physical model shown, there is two-phase flow of gas and liquid in the microfractures. Establish gas-phase and liquid-phase flow equations respectively. At the same time, the gas-phase flow equation in the microfractures also includes a Fick diffusion term. Therefore, the gas-phase flow equation in the microfractures is as follows:

[0037]

[0038] In the formula, q F is the Fick diffusion term, kg / (m·s), ρ g is the density of the gas, with the unit of kg / m 3 ; k f is the fracture permeability component, with the unit of m 2 ; k frg is the r-direction fracture gas-phase relative permeability component, with the unit of decimal; Pf is the fracture pressure, with the unit of Pa; μ g is the viscosity of the gas, with the unit of Pa·s; φ f is the porosity of the fracture system, with the unit of decimal; S g is the gas saturation in the fracture, with the unit of decimal.

[0039] The microfracture liquid-phase flow equation is as follows:

[0040]

[0041] In the formula, ρ w is the density of water, with the unit of kg / m 3 ; k frw is the r-direction relative permeability component of the fracture water phase, with the unit of decimal; μ w is the viscosity of water, with the unit of Pa·s; S w is the gas saturation, with the unit of decimal.

[0042] When shale gas migrates from the matrix to the microfracture in the form of Fick diffusion, the diffusion amount q is calculated by Fick's law F : C m is the molar concentration of shale gas in the matrix under unsteady conditions, mol / m 3 ;

[0043] When the diffusion from the matrix to the fracture is unsteady diffusion respectively, the rate of change of concentration with time is:

[0044] D F is the Fick diffusion coefficient, m 2 / s;

[0045] And when the diffusion from the matrix to the fracture is pseudo-steady diffusion respectively, the rate of change of concentration with time is:

[0046] C E (P f ) is the gas concentration at the junction of the matrix rock block and the fracture system, mol / m 3 ; C m is the volume concentration of shale gas in the matrix under pseudo-steady conditions, mol / m 3 ;

[0047] Therefore, if shale gas migrates to the macro-pores of the matrix in the form of Fick steady-state diffusion, the Fick diffusion amount is expressed as:

[0048]

[0049] If shale gas migrates into the matrix macro-pores in the way of Fick unsteady diffusion, the Fick diffusion amount is expressed as:

[0050]

[0051] Wherein, M g is the molar mass of the gas; r m is the radial coordinate of the matrix rock block, and R m is the radius of the spherical matrix body.

[0052] Therefore, in step S3, by combining the two-phase flow equation of the fracture system and the single-phase flow equation of the matrix system, the following equation is obtained:

[0053]

[0054] Further, step S3 specifically includes the following steps:

[0055] First, by combining the two-phase flow equation of the fracture system and the single-phase flow equation of the matrix system, the following equation is obtained:

[0056]

[0057] And by introducing the two-phase pseudo-pressure and the two-phase compressibility, the above equation is transformed into the following equation:

[0058]

[0059] Wherein, the two-phase pseudo-pressure is: The unit is kg / (m 3 ·s); the two-phase compressibility is: C t = ρ g C g S g + ρ w C w S w , the unit is Pa -1 , wherein, C g is the gas compressibility, and the unit is Pa -1 ; C w is the formation water compressibility, and the unit is Pa -1 ;

[0060] Then, the simplified two-phase flow equation of the fracture system is transformed into the cylindrical coordinate system, and the following equation is obtained:

[0061]

[0062] Finally, by introducing a plurality of two-phase dimensionless variables, the two-phase flow equation of the fracture system transformed into the cylindrical coordinate system is made dimensionless; specifically, the following two-phase dimensionless variables are introduced:

[0063] Fracture two-phase dimensionless radius

[0064] Matrix dimensionless radius

[0065] Dimensionless time

[0066] Pseudo-steady-state mass transfer coefficient Fracture dimensionless mass elastic storage ratio Pseudo-steady-state diffusion matrix dimensionless mass crossflow coefficient Matrix single-phase dimensionless pseudo-pressure difference Δ * m pm = m * (p i ) - m * (p m );

[0067] Matrix apparent two-phase dimensionless pseudo-pressure difference Δm pm = m(p i ) - m(p m );

[0068] Fracture two-phase dimensionless pseudo-pressure difference Δm pf = m(p i ) - m(p f );

[0069] Dimensionless concentration ΔC mD = C(p m ) - C(p i );

[0070] Dimensionless equilibrium concentration ΔC ED = C(p f ) - C(p i );

[0071] Based on the above-introduced two-phase dimensionless variables, the two-phase dimensionless pseudo-pressure difference microscopic flow equation can be obtained as follows:

[0072]

[0073] If the migration mode of gas from the matrix surface to the fracture is pseudo-steady-state diffusion, then according to Fick's first law, we know that:

[0074]

[0075] Nondimensionalize the above equation:

[0076]

[0077] Take the Laplace transform of the above equation with respect to the dimensionless time t D :

[0078]

[0079] Then we get:

[0080]

[0081] Furthermore, we can obtain:

[0082]

[0083] According to the Langmuir isothermal adsorption equation and the relationship between volume and concentration:

[0084]

[0085] And let We can get:

[0086]

[0087] Combined with We can get:

[0088]

[0089] Substitute it into the two-phase dimensionless pseudo-pressure difference microscopic flow equation:

[0090]

[0091] Let Then we obtain the following two-phase dimensionless pseudo-pressure difference microscopic flow equation:

[0092]

[0093] where r D is the dimensionless radius of the fracture, dimensionless; is the two-phase dimensionless pseudo-pressure difference in the fracture after Laplace transform, dimensionless; S is the Laplace variable, dimensionless; ω is the dimensionless mass elastic storage ratio, dimensionless; λ is the dimensionless mass crossflow coefficient, dimensionless; σ is the adsorption-desorption mass coefficient, dimensionless.

[0094] If the gas migration mode from the matrix surface to the fracture is unsteady diffusion, then according to Fick's second law, we know:

[0095]

[0096] Transformed into the spherical coordinate case, the unsteady diffusion equation in the matrix rock block is:

[0097]

[0098] The initial conditions are:

[0099] C m (r m , t)| t＝0 = C m (p i )

[0100] The inner boundary conditions are:

[0101]

[0102] The outer boundary conditions are:

[0103]

[0104] The dimensionless unsteady-state diffusion equation of the matrix:

[0105]

[0106] Taking the Laplace transform of the above equation with respect to time t D :

[0107]

[0108] The inner boundary conditions are:

[0109]

[0110] The outer boundary conditions are:

[0111]

[0112] Substituting into and solving the equation, we get:

[0113]

[0114] From the inner boundary conditions, when r mD → 0, the limit of the dimensionless variable of concentration exists and is finite, that is the limit exists and is finite, so b = 0. Also, according to the outer boundary conditions, Substituting a and b into the general solution of , we get:

[0115]

[0116] According to the Langmuir isothermal adsorption equation:

[0117]

[0118] Transform the above equation:

[0119]

[0120] Substitute the above equation into to obtain:

[0121]

[0122] Take the derivative of both sides of the above equation with respect to r mD to get:

[0123]

[0124] Substitute into the two-phase dimensionless pseudo-pressure difference microscopic flow equation:

[0125]

[0126] Let Then the following two-phase dimensionless pseudo-pressure difference microscopic flow equation is obtained:

[0127]

[0128] where r D is the dimensionless radius of the fracture, dimensionless; is the two-phase dimensionless pseudo-pressure difference of the fracture after Laplace transform, dimensionless; S is the Laplace variable, dimensionless; ω is the dimensionless mass elastic storage ratio, dimensionless; λ is the dimensionless mass crossflow coefficient, dimensionless; σ is the adsorption-desorption mass coefficient, dimensionless.

[0129] Specifically, in step S4, first, according to the two-phase dimensionless pseudo-pressure difference microscopic flow equation, establish the two-phase dimensionless pseudo-pressure difference microscopic flow equation corresponding to any cylindrical microelement continuous point source in the circular shale reservoir;

[0130] Introduce the Fourier finite cosine transform into the two-phase dimensionless pseudo-pressure difference microscopic flow equation corresponding to any cylindrical microelement continuous point source in the circular shale reservoir, and combine with the outer boundary condition of the circular shale gas reservoir to obtain the point source solution of the two-phase dimensionless pseudo-pressure corresponding to any cylindrical microelement continuous point source in the circular shale gas reservoir; As in the physical model shown in Figure 3 , 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:

[0131] (1) Both the top and bottom surfaces of the shale reservoir are closed, the outer boundary in the horizontal direction is infinite, the reservoir thickness is h, and the outer boundary radius of the reservoir is r e ;

[0132] (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 passing through the origin is the z-axis; when converted to the cylindrical coordinate system, only the r and z directions are considered.

[0133] (3) The horizontal permeability of the shale reservoir is k fh , and the vertical permeability is k fz ;

[0134] (4) The center of a cylindrical microelement is located at (0, 0, z w ) in the rectangular coordinate system, that is, at (0, z w ) in the cylindrical coordinate system; its radius is r and its height is z w ;

[0135] (5) The gas-liquid two-phase flows into the cylinder from the side surface of the cylindrical microelement, and the flow rate of shale gas flowing into the cylindrical microelement under standard surface conditions is q gscin , and the flow rate of the fracturing flowback fluid flowing into the cylindrical microelement under standard surface conditions is q wscin , ignoring the effects of capillary force and gravity.

[0136] For the convenience of research, the cylindrical coordinate will be used to interpret the model. Therefore, the two-phase flow equation in the fracture can be expressed as follows in the cylindrical coordinate:

[0137]

[0138] At the same time, since the assumed cylindrical microelement is an infinitesimal quantity, when the gas-water two-phase fluid flows in from its side surface, this microelement can be regarded as a continuous point sink, and the corresponding expression of the inner boundary condition can be obtained as:

[0139]

[0140] In the formula, ρ gsc is the density of shale gas under standard surface conditions, and ρ wsc is the density of the fracturing flowback fluid under standard surface conditions; the upper and lower boundary conditions are: The outer boundary condition is:

[0141] Moreover, the following two-phase dimensionless variables are introduced to simplify the above equations and boundary conditions to obtain:

[0142] The dimensionless two-phase flow equation in the fracture under the cylindrical coordinate is:

[0143] The inner boundary condition is:

[0144] The upper and lower boundary conditions are:

[0145] The outer boundary conditions are as follows:

[0146] After taking the Laplace transform with respect to time t D the dimensionless flow equations for the two phases in the fracture can be transformed into:

[0147]

[0148] Relating the above equation to the dimensionless fracture gas-liquid two-phase flow equation obtained in step S3 of the present invention, the Laplace variable s in the above equation is replaced as follows:

[0149]

[0150] The inner boundary conditions are as follows:

[0151] The upper and lower boundary conditions are as follows: The outer boundary conditions are as follows:

[0152] For solving the initial boundary value problem, first let:

[0153]

[0154] And substitute the above equation into Combined with the top and bottom closed boundary conditions, using the Sturm-Liouville eigenvalue theory, the eigenvalue equation for z D can be obtained:

[0155]

[0156] At the same time, according to the Sturm-Liouville eigenvalue theory, the eigenvalues of are α n =(nπ) 2 Then the corresponding eigenfunction system can be expressed as:

[0157] Z n (z D )=cosnπz D , n = 0, 1, 2···

[0158] Therefore, taking cosnπz D as the kernel function, introducing the Fourier finite cosine transform into and defining as the function after the Fourier finite cosine transform of with respect to z D the following can be obtained:

[0159]

[0160] Inner boundary expression:

[0161]

[0162] Outer boundary expression:

[0163]

[0164] Let Then the two-phase flow equation of the fracture can be written in the following form:

[0165]

[0166] Since is the Bessel equation with an imaginary argument, its general solution can be known as:

[0167]

[0168] Where: a n , b n are undetermined constants.

[0169] Substitute n = 0, 1, 2, ··· into the inner boundary expression, we can get:

[0170]

[0171] According to the properties of Bessel functions, when x → 0, I1(x) = 0, Therefore, from the above inner boundary expression, we can get:

[0172]

[0173] Therefore:

[0174]

[0175] Substitute it into the outer boundary expression:

[0176] Let Then:

[0177]

[0178] Substitute into we can get:

[0179]

[0180] Performing the inverse Fourier cosine transform on the above equation, we can obtain the point source solution of the two-phase pseudo-pressure of the dimensionless fracture gas-liquid two-phase flow equation, that is:

[0181]

[0182] In the formula:

[0183] If the outer boundary of the shale gas reservoir is closed at the top and bottom and the lateral outer boundary is closed, the mass flow rate of the fluid passing through the outer boundary is 0 at this time. After conversion to the dimensionless two-phase pseudo-pressure difference form after Laplace transform, it is:

[0184]

[0185] At this time, C is a constant, then δ = 0. Substitute it into the point source solution of the two-phase pseudo-pressure of the dimensionless fracture gas-liquid two-phase flow equation, that is

[0186]

[0187] Furthermore, when the outer boundary of the shale gas reservoir is closed at the top and bottom and the lateral outer boundary is closed, the point source solution of the two-phase pseudo-pressure of the corresponding dimensionless fracture gas-liquid two-phase flow equation is obtained as:

[0188]

[0189] In the formula, q gscins and q wscins are usually constants. Therefore:

[0190]

[0191] If the outer boundary of the shale gas reservoir is closed at the top and bottom and the lateral outer boundary is infinite, the expression of the outer boundary at this time is:

[0192]

[0193] Also according to the properties of the Bessel function, I0(∞) = ∞. Substitute it into to obtain: a n = c n = 0; then substitute a n = c n = 0 into to obtain that when the outer boundary of the shale gas reservoir is closed at the top and bottom and the lateral outer boundary is infinite, the point source solution of the two-phase pseudo-pressure of the corresponding dimensionless fracture gas-liquid two-phase flow equation is:

[0194]

[0195] Among them, ρ gsc is the density of the gas under standard surface conditions, with the unit of g / cm 3 ; ρ wsc is the density of water under standard surface conditions, with the unit of g / cm 3 ; qgscins is the flow rate of shale gas flowing into the cylindrical microelement under surface standard conditions, with the unit of m / s; q wscins is the flow rate of fracturing flowback fluid flowing into the cylindrical microelement under surface standard conditions, with the unit of m / s; h is the thickness of the shale reservoir; 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 outer boundary radius of the shale reservoir; z wD is the dimensionless distance, dimensionless; z D is the dimensionless height in the z direction; I0 and I1 are the Bessel functions of the first kind of imaginary argument, dimensionless; K0 and K1 are the Bessel functions of the second kind of imaginary argument, dimensionless.

[0196] Specifically, in step S5, since there are two types of fracturing vertical wells and fracturing horizontal wells in shale gas wells, and they correspond to different physical models, it is necessary to study the bottom-hole pressure responses of fracturing vertical wells and fracturing horizontal wells separately.

[0197] As Figure 4 shown in the physical model of the shale gas well, the assumptions are as follows:

[0198] Assumption 1: The shale gas well is a fully opened infinite conductivity fracturing vertical well in a circular reservoir;

[0199] Assumption 2: The fracturing fracture is a symmetric two-wing fracture with respect to the wellbore, and its half-length is x f ;

[0200] Assumption 3: The shale gas well produces at a constant surface production rate q sc or a constant bottom-hole pressure p wf for production;

[0201] 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 fracture has infinite conductivity;

[0202] Assumption 5: 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 perpendicular to the fracture surface. When calculating the bottom-hole pressure, the rectangular coordinate system is converted into a cylindrical coordinate system;

[0203] Then, combining 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 reservoir thickness direction and then along the fracture direction to solve the bottom-hole pressure response of the shale gas well in the Laplace space.

[0204] Specifically, if the outer boundary of the shale gas reservoir is closed at the top and bottom, and the outer boundary of the side is closed, the pressure response expression at the bottom of the circular reservoir fractured vertical well is:

[0205]

[0206] Where:

[0207] Since the trigonometric function cos(nπz wD ) has the property of integrating to zero over a symmetric domain, and note that Moreover, the flow distribution of the fractured vertical well is uniform, so the mass flow rate of gas produced by the gas well can be converted by multiplying the density of shale gas, the point source intensity and the area of the fracture. Similarly, the relationship between the mass flow rate of water produced by the gas well and the density of the fracturing return fluid, the 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 ); Combined with the above relationship, the pressure response expression of the bottom hole of the circular reservoir fracture vertical well is simplified to:

[0208]

[0209] Introduce the following two-phase dimensionless variables By non-dimensionalizing the above expression, we get:

[0210]

[0211] In the Substituting into the above formula, we can get:

[0212]

[0213] When studying the pressure response on the fracture surface, y w =0, then y wD =0; at the same time, in order to use the bottom hole pressure dynamic distribution of uniform flow fracture to characterize the pressure dynamic distribution of infinite conductivity fracture, it is necessary to first select a suitable equivalent observation point, whose coordinates are as follows: In order to follow the properties of the Bessel function, when α→x D When |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 numerical integration results, it is necessary to perform an integral transformation, let but: Therefore, the bottom-hole pressure response equation of a shale gas well in the Laplace space is as follows:

[0214]

[0215] The above equation is the dimensionless bottom-hole pressure dynamic distribution of a uniformly flowing fracture. When calculating the pressure distribution of an infinitely conductive fracture, the abscissa value of the observation point is x D = 0.732.

[0216] If the outer boundary of the shale gas reservoir is closed at the top and bottom and infinitely large at the side boundaries, the pressure response expression at the bottom of a vertically fractured well in a circular reservoir is:

[0217]

[0218] After simplification, we get: Then substitute Further simplify to: Then introduce After dimensionlessization, we can get:

[0219]

[0220] Next, substitute into the above equation, we can get:

[0221]

[0222] When studying the pressure response on the fracture surface, in order to use the bottom-hole pressure dynamic distribution of a uniformly flowing fracture to characterize the pressure dynamic distribution of an infinitely conductive fracture, we can get:

[0223]

[0224] Among them, α is the shape factor, dimensionless.

[0225] In addition, for the physical model of a shale gas well as shown in Figure 5 , the following assumptions are made:

[0226] Assumption 1: The shale gas well is a fully opened infinitely conductive fractured horizontal well in a circular reservoir;

[0227] Assumption 2: The total number of fractured cracks is M, the fracture surface is perpendicular to the central axis of the horizontal section of the wellbore, and is distributed along the central axis of the wellbore;

[0228] Assumption 3: The fractured cracks are symmetric double-wing cracks or asymmetric double-wing cracks with respect to the wellbore, infinitely conductive and with a length of x f ;

[0229] Assumption 4: Each fracture is divided into 2N units, and the flow distribution inside each fracture unit i is uniform;

[0230] Assumption 5: Shale gas wells produce at a constant surface production rate q sc Or constant bottom hole pressure p wf To carry out production;

[0231] Assumption 6: The gas is slightly compressible and has a constant viscosity and compressibility coefficient; the effects of capillary force and gravity are ignored;

[0232] 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, and 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 direction and is positive upward, and the x-axis is perpendicular to the above two directions. When calculating the bottom hole pressure, the rectangular coordinate system is converted into a cylindrical coordinate system;

[0233] 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 the M×2N fracture units, and 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 bottom hole pressure response of the shale gas well in the Laplace space.

[0234] Specifically, the observation point is selected at the node of a crack unit i When (the dimensionless coordinate is According to the principle of potential superposition, M×2N discrete fracture units on M fractures are located at the point The total pressure response generated at The resulting superposition of pressure responses:

[0235]

[0236] Since the seepage resistance of the fluid in the infinite conductivity fracture and wellbore is much smaller than the resistance in other locations of the reservoir, it is considered that the fluid flows in the infinite conductivity fracture and wellbore without pressure drop. Therefore, it can be known that the pseudo-pressure at any discrete node of the fracture is equal and equal to the bottom hole flow pressure, that is:

[0237]

[0238] Since the fractured horizontal well has a fixed 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,

[0239] Based on M×2N dimensionless two-phase pseudo-pressure equations and a cumulative production equation, the expression is obtained as follows:

[0240]

[0241] Wherein, is the coefficient matrix; q Di is the dimensionless two-phase mass production on fracture element i, dimensionless, i = 1, 2, ···, 2N*M;

[0242] Wherein, if the outer boundary condition of the shale gas reservoir is closed at the top and bottom and closed at the lateral outer boundary, the elements in the coefficient matrix are:

[0243] If the outer boundary condition of the shale gas reservoir is closed at the top and bottom and infinitely large at the lateral outer boundary, the elements in the coefficient matrix are:

[0244] Wherein, for a sphere, α = 15 / r 2 , is the shape factor, with the unit of m -2 ; ΔL fDi is the dimensionless half-length of fracture element i, dimensionless; x Dj is the dimensionless ordinate, dimensionless; x Di is the dimensionless abscissa, dimensionless; y Dj is the dimensionless ordinate of the continuous point source of the cylinder, dimensionless; y wDi is the dimensionless abscissa of the continuous point source of the cylinder, dimensionless.

[0245] Specifically, in step S6, the Stehfest numerical inversion method is used to invert the bottom-hole pressure response of the shale gas well in the Laplace space into the real space.

[0246] Specifically, in step S7, the measured bottom-hole data includes: the mid-depth of the pay zone, the thickness of the shale reservoir, the reservoir temperature, the water saturation, the density of the gas, the viscosity of the gas, the gas compressibility, and the Langmuir pressure and Langmuir volume measured according to laboratory experiments. The sensitivity parameters of the characteristic curve chart include: the formation pressure, the permeability of the matrix, the mass cross-flow coefficient between the matrix and the fracture, the mass diffusion coefficient, the dimensionless fracture half-length, the dimensionless mass elastic storage ratio of the fracture, and the dimensionless mass elastic storage ratio of the matrix.

[0247] The above are only the preferred embodiments of the present invention, and are not intended to limit the present invention. Any modifications, equivalent replacements, and improvements made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A two-phase well test analysis method considering Fick diffusion in a pseudo double-porosity medium, characterized in that, It includes the following steps: S1: Establish a microscopic physical model of shale gas reservoir for well test analysis. Among them, the assumption conditions of the microscopic physical model of the shale gas reservoir include: the shale gas reservoir is a pseudo-dual-porosity medium composed of a matrix system and a fracture system. There is single-phase flow of shale gas in the matrix system, two-phase flow of gas and liquid in the fracture system, and after the shale gas desorbs from the matrix surface, it migrates to the fracture system in the way of Fick diffusion; S2: According to the microscopic physical model of the shale gas reservoir, establish two-phase flow equations for the fracture system and single-phase flow equations for the matrix system respectively. Among them, the two-phase flow equations for the fracture system are: And the single-phase flow equations for the matrix system are: If Fick diffusion is Fick steady-state diffusion, the rate of change of concentration with time is: If Fick diffusion is Fick unsteady-state diffusion, the rate of change of concentration with time is: where ρ g is the density of the gas; ρ w is the density of water; k f is the fracture permeability component, k frg is the relative gas permeability component in the r - direction of the fracture, k frw is the relative water permeability component in the r - direction of the fracture; P f is the fracture pressure; μ g is the viscosity of the gas, μ w is the viscosity of water; φ f is the fracture system porosity; S g is the gas saturation in the fracture, S w is the water saturation; C m is the gas concentration in the matrix, D F is the Fick diffusion coefficient, R m is the radius of the spherical matrix body, C E (p f ) is the gas concentration at the interface between the matrix rock block and the micro - fracture system, r m is the radial coordinate of the matrix rock block; M g is the molar mass of the gas; S3: Combine the two-phase flow equations for the fracture system and the single-phase flow equations for the matrix system to obtain a two-phase dimensionless pseudo-pressure difference microscopic flow equation; S4: According to the two-phase dimensionless pseudo-pressure difference microscopic flow equation and combined with the outer boundary conditions of the circular shale gas reservoir, obtain the point-source solution of the two-phase dimensionless pseudo-pressure in the circular shale gas reservoir; S5: According to 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, obtain the bottom-hole pressure solution of the shale gas well in the Laplace space; S6: Invert the bottom-hole pressure solution of the shale gas well in the Laplace space into the 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 to obtain a characteristic curve graph; S7: According to the characteristic curve generated based on the bottom-hole measured data, adjust the sensitivity parameters of the characteristic curve graph, and take the sensitivity parameters corresponding to the fitting of the characteristic curve graph and the characteristic curve as the well test analysis results; Among them, step S3 also includes the following steps: After combining the two-phase flow equations for the fracture system and the single-phase flow equations for the matrix system, introduce two-phase pseudo-pressure and two-phase compressibility to simplify the two-phase flow equations for the fracture system, and transform the simplified two-phase flow equations for the fracture system to the cylindrical coordinate system; Then introduce two-phase dimensionless variables to dimensionlessize the two-phase flow equations for the fracture system transformed to the cylindrical coordinate system to obtain the two-phase dimensionless pseudo-pressure difference microscopic flow equation. Among them, the two-phase dimensionless pseudo-pressure difference microscopic flow equation is: If Fick diffusion is Fick steady-state diffusion, If Fick diffusion is Fick unsteady-state diffusion, where r D is the dimensionless radius of the fracture; is the dimensionless pseudo-pressure difference of the two phases in the fracture after Laplace transform; s is the Laplace variable; ω is the dimensionless mass elastic storage ratio; λ is the dimensionless mass crossflow coefficient; σ is the adsorption-desorption mass coefficient.

2. The two-phase well test analysis method considering Fick diffusion in a pseudo double-porosity medium as described in claim 1, wherein Step S4 also includes the following steps: According to the two-phase dimensionless pseudo-pressure difference microscopic flow equation, establish a two-phase dimensionless pseudo-pressure difference microscopic flow equation corresponding to any cylindrical microelement continuous point source in the circular shale reservoir; Introduce Fourier finite cosine transform into the two-phase dimensionless pseudo-pressure difference microscopic flow equation corresponding to any cylindrical microelement continuous point source in the circular shale reservoir, and combined with the outer boundary conditions of the circular shale gas reservoir, obtain the point-source solution of the two-phase dimensionless pseudo-pressure corresponding to any cylindrical microelement continuous point source in the circular shale gas reservoir.

3. The two-phase well test analysis method considering Fick diffusion in a pseudo double-porosity medium according to claim 2, wherein Step S5 also includes: If the assumption conditions of the macroscopic physical model of the shale gas well are as follows: Assumption 1: The shale gas well is an infinitely conductive fractured vertical well fully opened in a circular shale gas reservoir; Hypothesis 2: The fracturing cracks are symmetric double-wing cracks with respect to the wellbore, and their half-length is x f ; Assumption 3: The shale gas well produces at a constant surface production rate q sc or a constant bottom-hole pressure p wf for production; 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 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, and when calculating the bottom hole pressure, the rectangular coordinate system is converted into 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 Laplace space.

4. The two-phase well test analysis method considering Fick diffusion in a pseudo-dual-porosity medium according to claim 2, characterized in that Step S5 also 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 surface is perpendicular to the central axis of the horizontal section of the wellbore and distributed along the central axis of the wellbore; Hypothesis condition three: The fracture is a symmetric bi-wing fracture or an asymmetric bi-wing fracture with infinite conductivity and length x with respect to the wellbore f ; Assumption 4: Each fracture is divided into 2N units, and the flow distribution inside each fracture unit i is uniform; Assumption Five: The shale gas well produces at a constant surface production rate q sc or a constant bottom-hole pressure p wf for production; 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 seven: 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, and 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 direction and is positive upward, and the x-axis is perpendicular to the above y-axis and z-axis directions. When calculating the bottom hole pressure, the rectangular coordinate system is converted into 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, and 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 bottom hole pressure solution of the shale gas well in the Laplace space.

5. The two-phase well test analysis method considering Fick diffusion in a pseudo double-porosity medium as described in claim 1, wherein, 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 into real space.

6. The two-phase well test analysis method considering Fick diffusion in a pseudo double-porosity medium according to any one of claims 1 to 5, 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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