Non-coal measure stratum natural gas displacement mathematical model construction method

Abstract

The invention discloses a non-coal measure stratum natural gas displacement mathematical model construction method. The method comprises the following steps that gas migration equations are established, wherein the gas migration equations comprise the step of establishing a rock mass pore gas diffusion motion control equation and the step of establishing a stratum fracture gas seepage equation; establishing a stratum rock mass deformation control equation; establishing a dynamic porosity and permeability change control equation: establishing a rock mass porosity dynamic change equation, establishing a stratum fracture rate dynamic change equation and establishing a permeability dynamic change equation; and establishing a non-coal measure stratum natural gas displacement mathematical model based on the above steps. The non-coal measure stratum natural gas displacement mathematical model is established by combining the non-coal measure stratum natural gas displacement characteristics, and a scheme which is good in treatment effect, low in cost and high in efficiency is provided for non-coal measure stratum natural gas emission.

Description

Technical Field

[0001] The present invention relates to the technical field of tunnel construction engineering, and in particular to a method for constructing a mathematical model of natural gas displacement in non-coal formations. Background Art

[0002] In tunnels constructed using the shield method, natural gas in the stratum enters the tunnel through weak sealing points such as the cutterhead cutting joints, slag conveyors, segment splicing, and lining, posing a huge threat to tunnel construction safety. Active natural gas emission measures refer to the emission of harmful gases in the stratum in front of the shield tunneling face to ensure the safe passage of the shield machine through strata containing harmful gases. Active natural gas emission measures are mainly used in shallow buried shield tunnels of subways. For example, Chengdu Metro Line 19 and Hangzhou Metro Line 1 have adopted natural gas extraction measures, and Chengdu Metro Line 30 has drilled holes for natural emission of stratum natural gas. Currently, the active natural gas emission measures commonly used in tunnel construction are mainly divided into two types: natural emission through drilling and natural gas extraction. However, these two measures usually have disadvantages such as long treatment time and poor treatment effect for high-concentration natural gas strata and stable natural gas supply strata.

[0003] Given that passive protection measures for gas tunnels have disadvantages such as high cost and "treating the symptoms but not the root cause", while the two active control measures of natural gas drainage through drilling and natural gas extraction have problems such as unsatisfactory control effects and high time cost, there is an urgent need for a method to construct a mathematical model of natural gas displacement in non-coal formations. Summary of the Invention

[0004] The purpose of the present invention is to overcome the deficiencies of the prior art and provide a method for constructing a mathematical model for natural gas displacement in non-coal-bearing formations.

[0005] The object of the present invention is achieved through the following technical solutions:

[0006] A method for constructing a mathematical model for natural gas displacement in non-coal-bearing formations comprises the following steps:

[0007] S1. Establishing gas migration equations: This includes establishing the governing equations for gas diffusion in rock pores and establishing the gas seepage equations in formation fractures;

[0008] S2. Establish the governing equation for formation rock deformation;

[0009] S3. Establishment of dynamic porosity and permeability change control equations: including establishment of dynamic change equations for rock mass porosity, establishment of dynamic change equations for formation fracture rate, and establishment of dynamic change equations for permeability;

[0010] S4. Based on steps S1-S3, a mathematical model for natural gas displacement in non-coal-bearing formations is established.

[0011] Furthermore, in step S1, establishing the governing equation for gas diffusion motion in rock pores includes the following steps:

[0012] S11. Establish a gas mass exchange equation: methane, the main component of natural gas, is used as the displaced gas and nitrogen is used as the injected gas. During the displacement process, the free natural gas in the pores of the rock matrix diffuses outward into the formation fractures as a mass source, allowing the diffusion and seepage process to continue. At the same time, the injected nitrogen diffuses into the rock pores with low nitrogen concentration as a mass source. The flux formula for the mass exchange between the rock matrix pores and the formation fracture system is expressed as:

[0013]

[0014] Where C m1 、C m2 —respectively, the concentrations of methane and nitrogen in the pores of the matrix rock, kg / m 3 ;

[0015] C f1 、C f2 —respectively, the concentrations of methane and nitrogen in formation fractures, kg / m 3 ;

[0016] Q1, Q2—gas mass exchange rates between matrix rock pores and formation fractures of methane and nitrogen, kg / (m 3 s);

[0017] D1, D2—methane and nitrogen gas diffusion constants, respectively, m 2 / s;

[0018] σ—matrix shape factor, σ=3π 2 / L 2 , m -2 ;

[0019] L—spacing between formation fractures, m;

[0020] The gas concentration is obtained from the gas state equation:

[0021]

[0022] Where, M1 and M2 are the molar masses of methane and nitrogen, kg / mol respectively;

[0023] P m1 、P m2 —pressures of methane and nitrogen in the matrix rock pore system, Pa;

[0024] P f1 、P f2 —pressures of methane and nitrogen in the formation fracture system, Pa, respectively;

[0025] D1, D2—methane and nitrogen gas diffusion constants, respectively, m 2 / s;

[0026] R—gas state constant, J / (mol·K);

[0027] T—temperature of the entire displacement system, K;

[0028] Substituting the gas concentration formula (4-3) into (4-1) yields the mass exchange flux between the rock mass pores and the formation fractures:

[0029]

[0030] S12. Establish the matrix gas diffusion equation: Use Fick's law to describe the diffusion movement of gas in the pores of the rock matrix:

[0031]

[0032] Right now:

[0033]

[0034] Where J— is the diffusion flux, kg / (m 2 s);

[0035] α, β, γ—represent the unit vectors in the x, y, and z directions, respectively, and are dimensionless;

[0036] c—volume concentration of diffusing substances, kg / m 3 ;

[0037] D—is the gas diffusion constant, m 2 / s;

[0038] Combining Fick's diffusion theorem, the mass conservation theorem, and the gas state equation, the continuity equation for the diffusion motion of methane and nitrogen in the rock pore system is obtained based on the fact that the change in gas concentration in the rock pores is the sum of the gas diffusion flux and the mass exchange flux of the rock pore-stratum fracture system:

[0039]

[0040] Substituting equation (4-3) into equation (4-7), we obtain the governing equation for gas diffusion motion in the matrix:

[0041]

[0042] Where, —Matrix porosity.

[0043] Furthermore, in step S1, establishing the control equation of gas seepage movement in formation fractures includes the following steps:

[0044] S13. Establish the seepage equation of gas in formation fractures:

[0045]

[0046] Where, v1—seepage rate, m / s;

[0047] k—permeability of porous media, m 2 ;

[0048] μ—fluid dynamic viscosity coefficient Pa·s;

[0049] Within Δt, the change in gas mass in any control volume of a porous media formation fracture system is equal to the difference in mass between the fluid flowing into and out of the volume plus the mass generated or absorbed by the control volume itself. The conservation of mass in the formation fracture system is:

[0050]

[0051] Combining the law of conservation of mass and Darcy's seepage theorem, the fracture system in the formation drives seepage under the combined pressure of methane and nitrogen:

[0052]

[0053] Where, — is the formation fracture ratio;

[0054] P f —Total gas pressure in the formation fracture system, Pa;

[0055] μ1, μ2—dynamic viscosity coefficients of methane and nitrogen, Pa·s, respectively.

[0056] Furthermore, in step S2, establishing the control equation for formation rock deformation includes the following steps: S21, establishing the effective stress equation for rock deformation:

[0057] σ′ ij =σ ij -P (4-12)

[0058] Where σ′ ij —effective normal stress on the plane, Pa;

[0059] σ ij —total normal stress on the plane, Pa;

[0060] P—pore pressure of formation rock mass, Pa;

[0061] S22. Based on formula (4-12), the correction is introduced. The effective stress equation after the correction is:

[0062] σ′ ij =σ ij -(β f P f +β m P m )δ ij (4-13)

[0063]

[0064] Where, β f —Correction coefficient of effective stress in formation fractures, dimensionless;

[0065] β m —Correction coefficient of effective stress in rock mass pores, dimensionless;

[0066] P f —Gas pressure in formation fractures, Pa;

[0067] P m —Gas pressure in rock pores, Pa;

[0068] K—rock mass bulk modulus, Pa;

[0069] K m —rock matrix bulk modulus, Pa;

[0070] K s —rock skeleton bulk modulus, Pa;

[0071] E—elastic modulus of rock mass, Pa;

[0072] E m —rock matrix elastic modulus, Pa;

[0073] ν—Poisson's ratio of rock mass, dimensionless;

[0074] φ m —rock porosity, dimensionless;

[0075] δi j—Kirsten constant, expressed as dimensionless;

[0076] S23. Establish an equilibrium equation for rock mass deformation: Ignore the inertial forces generated by natural gas flow and rock mass skeleton deformation, and neglect the gravity of the natural gas itself. Select a small parallelepiped unit from the natural gas-bearing rock mass, and define its three edges as dx, dy, and dz. Assume that the stresses acting on each face of the unit are uniformly distributed. The stresses on this unit are in equilibrium, satisfying the static equilibrium equation:

[0077]

[0078] Express formula (4-17) in the form of a tensor:

[0079] σ ij，j +F i =0(i,j=1,2,3) (4-18)

[0080] Where, F i —body force component, Pa;

[0081] σ ij，j —total stress component, Pa;

[0082] Express formula (4-17) in the form of effective stress:

[0083] σ′ ij，j +[(β f P f +β m P m )δ ij ] ，j +F i =0 (i, j=1,2,3)(4-19);

[0084] S24. Establish the geometric equation of rock mass deformation: Rock mass deformation satisfies the Cauchy equation:

[0085]

[0086] Its tensor form is:

[0087]

[0088] Where, ε ij — volumetric strain tensor;

[0089] u—deformation displacement;

[0090] S25. Establish constitutive relations: Assume that the rock mass is a homogeneous isotropic material. During displacement, the rock mass deformation is in the elastic deformation stage. The rock mass deformation in three dimensions follows the generalized Hooke's law:

[0091] σ′ ij =λδ ij ε v +2Gε ij (i, j = 1, 2, 3) (4-22)

[0092] Where, λ is the Mellar constant of the rock mass, λ = Ev / (1+v)(1-2v), Pa;

[0093] ε v — rock mass volume strain;

[0094] G—rock shear modulus, G=E / 2(1+v);

[0095] Considering the effects of rock pore gas pressure and formation fracture gas pressure in dual-porosity media, and substituting formula (4-22) into effective stress formula (4-13), we obtain:

[0096] σ ij -(β f P f +β m P m )δ ij -λδ ij ε v +2Gε ij =0 (4-23)

[0097] Substituting formula (4-22) into formula (4-19), we can obtain the governing equation for rock deformation during natural gas displacement:

[0098]

[0099] (4-24).

[0100] Furthermore, in step S3, establishing a dynamic change equation of rock mass porosity includes the following steps:

[0101] S31. The rock matrix is ​​considered as a porous medium, and its porosity is expressed as follows:

[0102]

[0103] V=V P +V S (4-26)

[0104] Where, φ m — rock matrix porosity, dimensionless;

[0105] V—rock matrix volume, m 3 ;

[0106] V P —pore volume, m 3 ;

[0107] V S —Solid particle volume, m 3 ;

[0108] The following expression is obtained based on the relationship between rock mass porosity and the components of rock mass matrix volume:

[0109]

[0110] Using the rock mass porosity model under strain conditions, the porosity in the rock matrix is ​​expressed as:

[0111]

[0112] Where, α is the effective stress coefficient, dimensionless;

[0113] β—effective stress coefficient, dimensionless;

[0114] K—rock mass bulk modulus, Pa;

[0115] K p —pore bulk modulus, Pa;

[0116] K s —bulk modulus of solid particles, Pa;

[0117] ε s —desorption adsorption strain;

[0118] The natural gas displacement process in non-coal-bearing strata does not consider gas desorption and adsorption, and the expression for the porosity change in the rock matrix is:

[0119]

[0120] The volume strain in the initial state is zero, and the matrix porosity can be expressed as:

[0121]

[0122] Where, φ m0 —Initial porosity of rock mass, dimensionless.

[0123] Furthermore, in step S3, establishing the dynamic change equation of formation fracture rate includes the following steps:

[0124] S32. Select the PM model to establish the relationship between formation fracture rate and formation rock strain:

[0125]

[0126] Where, φ f —formation fracture ratio, dimensionless;

[0127] N—axial constraint modulus, N=E(1-v) / (1+v)(1-2v), Pa;

[0128] γ—skeleton compression coefficient, dimensionless;

[0129] S—overburden load, Pa;

[0130] χ—rock skeleton deformation coefficient, dimensionless;

[0131] T—formation temperature, K;

[0132] Taking formation fractures as the research object, during the displacement process of shallow non-coal-bearing gas, the overlying load does not change, competitive adsorption does not occur in the non-coal-bearing strata, and temperature changes are not considered. Based on this, the PM model is improved:

[0133]

[0134] Where, φ f0 —initial formation fracture ratio of the formation, dimensionless;

[0135] p f0 —Initial gas pressure in formation fractures, Pa.

[0136] By taking the partial derivative of Equation (4-36) with respect to time, we can obtain the equation for the dynamic change of formation fracture rate with time:

[0137]

[0138] Furthermore, in step S3, establishing a permeability dynamic change equation includes the following steps:

[0139] S33. The permeability and fracture rate of the formation follow the cubic law, that is:

[0140]

[0141] Where, k is the formation permeability, m 2 ;

[0142] k0—initial permeability of the formation, m 2 ;

[0143] The gas permeability is corrected by the slip coefficient. The corrected gas permeability is:

[0144]

[0145] Where k g —Permeability of the formation considering the slip effect, m 2 ;

[0146] c—slip coefficient, Pa.

[0147] Furthermore, the non-coal-bearing formation natural gas displacement mathematical model includes a gas migration field control equation, a stress field control equation, and a coupling field control equation:

[0148] The gas migration field control equation includes:

[0149]

[0150] Among them, formula (4-41) is the rock matrix migration field, and formula (4-42) is the stratum fracture migration field, which represent diffusion movement and seepage movement respectively. represents the diffusion amount of gas component i in the rock matrix, It represents the seepage rate of gas component i in the formation fracture, and finally seeps to the drainage hole. represents the matrix-formation fracture exchange amount of gas component i;

[0151] The stress field control equations include:

[0152]

[0153] Where, represents the ground stress, Indicates the combined pressure of gas in the matrix and formation fractures, F i Indicates gravity;

[0154] The coupled field control equations include:

[0155]

[0156] The beneficial effects of the present invention are:

[0157] 1) The present invention combines the characteristics of natural gas displacement in non-coal-bearing formations and establishes a mathematical model for natural gas displacement in non-coal-bearing formations, providing a solution with good treatment effect, low cost and high efficiency for natural gas discharge in non-coal-bearing formations.

[0158] 2) Based on the effective stress equation, static equilibrium equation, and geometric equation, the stress and deformation state of the rock mass during natural gas displacement in non-coal-bearing formations was analyzed. At the same time, considering the influence of gas pressure in pores and fractures, the constitutive relationship of the rock mass during natural gas displacement in non-coal-bearing formations was established.

[0159] 3) The porosity model under strain conditions was introduced and combined with the porosity definition to obtain the equation for the change of rock matrix porosity. Based on the PM model and combined with the characteristics of natural gas displacement in non-coal-bearing formations, the PM model was optimized to obtain the dynamic change equation of the formation fracture ratio. Based on the cubic law, the equation for the change of formation permeability was obtained, and the permeability was corrected by considering the gas seepage slippage effect. BRIEF DESCRIPTION OF THE DRAWINGS

[0160] Figure 1 Schematic diagram of the slippage effect during gas seepage in formation fractures in an embodiment of the present invention;

[0161] Figure 2 is the coupling relationship diagram of gas migration field and stress field; DETAILED DESCRIPTION

[0162] The following will clearly and completely describe the technical solutions of the present invention in conjunction with the embodiments. Obviously, the embodiments described are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative work shall fall within the scope of protection of the present invention.

[0163] See Figure 1-Figure 2 , the present invention provides a technical solution:

[0164] Example:

[0165] like Figure 1 and Figure 2 As shown in the figure, a method for constructing a mathematical model of natural gas displacement in non-coal-bearing formations is proposed.

[0166] During natural gas displacement, when high-pressure gas is injected into the formation through the injection holes and negative pressure is applied to the drainage holes, a distance-dependent pressure gradient forms within the formation. Natural gas in the formation seeps, driven by the displacement gas pressure gradient. Because the borehole diameter is much smaller than its depth, the effective distance of the drainage influence range after the pressure seepage field is created by drilling can be approximated as a two-dimensional radial flow.

[0167] The displacement of natural gas in formation is essentially the physical migration process of fluid in porous media. The physical and mechanical properties of porous media are closely related to the occurrence and migration of gas. Figure 1 As shown, the natural gas storage and migration space in gas-rich non-coal-bearing formations primarily consists of two aspects: rock pores, which refer to the primary pore space between solid mineral particles in the rock and soil formed during diagenesis; and formation fissures, which are joints and fissures formed after diagenesis due to geological structures, external interference, and other factors. Rock pores typically range in size from tens to hundreds of microns, and pore connectivity is poor. Formation fissures, on the other hand, typically have apertures on the order of millimeters or even centimeters, exhibiting some connectivity. Therefore, gas migration in pores is considered diffusion, while gas migration in fissures is considered seepage. Both rock pores and formation fissures serve as gas storage spaces, but formation permeability and gas migration velocity are primarily controlled by the formation fissure fraction.

[0168] When gas is injected into the injection holes, the injected gas first enters the fracture system under the action of the injection pressure to transfer the flow field. The injected gas and the displacement gas seep toward the low pressure under the combined pressure driving action. The concentration of the injected gas in the fracture system gradually increases, and the concentration of the displacement gas gradually decreases. Subsequently, a concentration difference is formed between the fracture system and the pores, and the injected gas diffuses into the pores, and the displacement gas diffuses into the fractures, forming a diffusion flow transfer field. Ultimately, the concentration and pressure of the displaced gas in the pore and fracture systems are greatly reduced, which is usually called the gas injection drive effect.

[0169] The initiation condition for displacement gas migration is pressure differential. Changes in pore pressure and fracture pressure will cause changes in the effective stress of the rock mass, which in turn causes rock deformation. Rock deformation will in turn affect the rock mass porosity, rock formation fracture rate, and permeability, which in turn affects gas migration. Therefore, non-coal-bearing natural gas displacement should comprehensively consider gas migration, formation rock deformation, and fluid-solid coupling effects.

[0170] The above analysis of the physical processes of displacement ultimately determined that natural gas displacement in non-coal-bearing formations can be primarily attributed to four processes: fracture gas seepage, pore gas diffusion, rock deformation, and porosity and permeability changes. A mathematical model was developed to describe each of these four processes, establishing connections and coupling relationships between them through relevant variables and parameters. The fundamental principles underlying each physical process include: the derivation of the governing equations for gas migration based on Fick's diffusion theorem, Darcy's seepage theorem, and the gas equation of state; the establishment of the rock constitutive relationship during natural gas displacement in non-coal-bearing formations based on the effective stress equation, static equilibrium equation, and geometric equation; and the selection of the PM model, taking into account the characteristics of natural gas displacement in non-coal-bearing formations, to derive the porosity and permeability change equation.

[0171] The steps include:

[0172] S1. Establishing gas migration equations: This includes establishing the governing equations for gas diffusion in rock pores and establishing the gas seepage equations in formation fractures;

[0173] S11. Establishing the gas mass exchange equation: To facilitate the establishment of the model and the subsequent acquisition of gas parameters, methane, the main component of natural gas, is used as the driven gas and nitrogen is used as the injected gas. During the displacement process, the free natural gas in the pores of the rock matrix acts as a mass source to diffuse outward into the formation fractures, allowing the diffusion and seepage process to continue. At the same time, the injected nitrogen acts as a mass source to diffuse into the rock pores with low nitrogen concentration. The flux formula for the mass exchange between the rock matrix pores and the formation fracture system is expressed as:

[0174]

[0175] Where C m1 、C m2 —respectively, the concentrations of methane and nitrogen in the pores of the matrix rock, kg / m 3 ;

[0176] C f1 、C f2 —respectively, the concentrations of methane and nitrogen in formation fractures, kg / m 3 ;

[0177] Q1, Q2—gas mass exchange rates between matrix rock pores and formation fractures of methane and nitrogen, kg / (m 3 s);

[0178] D1, D2—methane and nitrogen gas diffusion constants, respectively, m 2 / S;

[0179] σ—matrix shape factor, σ=3π 2 / L 2 , m -2 ;

[0180] L—spacing between formation fractures, m;

[0181] The gas concentration is obtained from the gas state equation:

[0182]

[0183] Where, M1 and M2 are the molar masses of methane and nitrogen, kg / mol respectively;

[0184] P m1 、P m2 —pressures of methane and nitrogen in the matrix rock pore system, Pa;

[0185] P f1 、P f2 —pressures of methane and nitrogen in the formation fracture system, Pa, respectively;

[0186] D1, D2—methane and nitrogen gas diffusion constants, respectively, m 2 / s;

[0187] R—gas state constant, J / (mol·K);

[0188] T—temperature of the entire displacement system, K;

[0189] Substituting the gas concentration formula (4-3) into (4-1) yields the mass exchange flux between the rock mass pores and the formation fractures:

[0190]

[0191] S12. Establish the matrix gas diffusion equation: Use Fick's law to describe the diffusion movement of gas in the pores of the rock matrix:

[0192]

[0193] Right now:

[0194]

[0195] Where J— is the diffusion flux, kg / (m 2 s);

[0196] α, β, γ—represent the unit vectors in the x, y, and z directions, respectively, and are dimensionless;

[0197] c—volume concentration of diffusing substances, kg / m 3 ;

[0198] D—is the gas diffusion constant, m 2 / s;

[0199] Combining Fick's diffusion theorem, the mass conservation theorem, and the gas state equation, the continuity equation for the diffusion motion of methane and nitrogen in the rock pore system is obtained based on the fact that the change in gas concentration in the rock pores is the sum of the gas diffusion flux and the mass exchange flux of the rock pore-stratum fracture system:

[0200]

[0201] Substituting equation (4-3) into equation (4-7), we obtain the governing equation for gas diffusion motion in the matrix:

[0202]

[0203] Where, —Matrix porosity.

[0204] S13. Based on Darcy's law, the seepage equation of gas in formation fractures is established:

[0205]

[0206] Where v1 is the seepage rate, m / s; the seepage velocity is usually represented by the letter v (m / s); the Poisson's ratio of the rock mass is generally represented by the Greek letter ν (pronounced "nu"), a dimensionless parameter.

[0207] k—permeability of porous media, m 2 ;

[0208] μ—fluid dynamic viscosity coefficient Pa·s;

[0209] Within Δt, the change in gas mass in any control volume of a porous media formation fracture system is equal to the difference in mass between the fluid flowing into and out of the volume plus the mass generated or absorbed by the control volume itself. The conservation of mass in the formation fracture system is:

[0210]

[0211] Combining the law of conservation of mass and Darcy's seepage theorem, the fracture system in the formation drives seepage under the combined pressure of methane and nitrogen:

[0212]

[0213] Where, — is the formation fracture ratio;

[0214] P f —Total gas pressure in the formation fracture system, Pa;

[0215] μ1, μ2—dynamic viscosity coefficients of methane and nitrogen, Pa·s, respectively.

[0216] S2. Establish the governing equation for rock mass deformation: S21. The effective stress principle states that under external loads, stress in the rock mass is shared by the rock mass skeleton and the water vapor in the rock mass pores. However, only stress transmitted through the rock mass skeleton causes rock mass deformation; force transmitted through the water vapor in the pores does not contribute to the strength and deformation of the rock mass. This force that affects rock mass strength, rock mass porosity, stratum cracks, and permeability is the effective stress. The effective stress equation for rock mass deformation is as follows:

[0217] σ′ ij =σ ij -P (4-12)

[0218] Where σ′ ij —effective normal stress on the plane, Pa;

[0219] σ ij —total normal stress on the plane, Pa;

[0220] P—pore pressure of formation rock mass, Pa;

[0221] S22. Consider the natural gas-bearing formation as a rock pore-formation fracture porous medium composed of rock solid particles, matrix rock pores, and rock formation fractures. Both the rock pores and formation fractures serve as gas storage sites, and the formation fracture rate controls the rock permeability. Considering that the rock particle size is too large for methane molecules, a correction is introduced based on formula (4-12) to meet accuracy requirements. The corrected dual rock pore medium effective stress law is:

[0222] σ′ ij =σ ij -(β f P f +β m P m )δ ij (4-13)

[0223]

[0224] Where, β f —Correction coefficient of effective stress in formation fractures, dimensionless;

[0225] β m —Correction coefficient of effective stress in rock mass pores, dimensionless;

[0226] P f —Gas pressure in formation fractures, Pa;

[0227] P m —Gas pressure in rock pores, Pa;

[0228] K—rock mass bulk modulus, Pa;

[0229] K m —rock matrix bulk modulus, Pa;

[0230] K s —rock skeleton bulk modulus, Pa;

[0231] E—elastic modulus of rock mass, Pa;

[0232] E m —rock matrix elastic modulus, Pa;

[0233] v—Poisson’s ratio of rock mass, dimensionless;

[0234] φ m —rock porosity, dimensionless;

[0235] δ ij —Kirsten's constant, expressed as dimensionless;

[0236] S23. Establish an equilibrium equation for rock deformation: Ignore the inertial force generated by natural gas flow and rock skeleton deformation, and neglect the gravity of the natural gas itself. Select a small parallelepiped unit from the natural gas-bearing rock mass, and set its three edges to be dx, dy, and dz. Assume that the stresses acting on each face of the unit are uniformly distributed. The stresses on this unit are in equilibrium, and it satisfies the static equilibrium equation:

[0237]

[0238] The above equation is the differential form of the equilibrium equation. Formula (4-17) is expressed in the form of a tensor:

[0239] σ ij，j +F i =0(i,j=1,2,3) (4-18)

[0240] Where, F i —body force component, Pa;

[0241] σ ij，j —total stress component, Pa;

[0242] At the same time, the deformation of the rock mass skeleton follows the effective stress principle, and formula (4-17) is expressed in the form of effective stress:

[0243] σ′ ij，j +[(β f P f +β m P m )δ ij ] ，j +F i =0(i,j=1,2,3)(4-19);

[0244] S24. Establish the geometric equation of rock mass deformation: Rock mass deformation satisfies the Cauchy equation:

[0245]

[0246] Its tensor form is:

[0247]

[0248] Where, ε ij — volumetric strain tensor;

[0249] u—deformation displacement;

[0250] S25. Establish constitutive relations: Assume that the rock mass is a homogeneous isotropic material. During displacement, the rock mass deformation is in the elastic deformation stage. The rock mass deformation in three dimensions follows the generalized Hooke's law:

[0251] σ′ ij =λδ ij ε v +2Gε ij (i, j = 1, 2, 3) (4-22)

[0252] Where, λ is the Mellar constant of the rock mass, λ = Ev / (1+v)(1-2v), Pa;

[0253] ε v — rock mass volume strain;

[0254] G—rock shear modulus, G=E / 2(1+v);

[0255] Considering the effects of rock pore gas pressure and formation fracture gas pressure in dual-porosity media, and substituting formula (4-22) into effective stress formula (4-13), we obtain:

[0256] σ ij -(β f P f +β m P m )δ ij -λδij ε v +2Gε ij =0 (4-23)

[0257] Substituting formula (4-22) into the stress balance equation formula (4-19), we can obtain the rock deformation control equation during natural gas displacement:

[0258]

[0259] S3. Establish dynamic porosity and permeability change control equations: This includes S31: Establish dynamic change equations for rock mass porosity: The pores in the non-coal-bearing rock matrix are the main location for gas storage. The porosity of the rock matrix deforms under external loads and gas pressure, affecting the displacement process. The rock matrix is ​​considered a porous medium, and its porosity is expressed as follows:

[0260]

[0261] V=V P +V S (4-26)

[0262] Where, φ is the rock matrix porosity, dimensionless;

[0263] V—rock matrix volume, m 3 ;

[0264] V P —pore volume, m 3 ;

[0265] V S —Solid particle volume, m 3 ;

[0266] The following expression is obtained based on the relationship between rock mass porosity and the components of rock mass matrix volume:

[0267]

[0268] Using the rock mass porosity model under strain conditions, the porosity in the rock matrix is ​​expressed as:

[0269]

[0270] Where, α is the effective stress coefficient, dimensionless;

[0271] β—effective stress coefficient, dimensionless;

[0272] K—rock mass bulk modulus, Pa;

[0273] K p —pore bulk modulus, Pa;

[0274] K s —bulk modulus of solid particles, Pa;

[0275] ε s —desorption adsorption strain;

[0276] The natural gas displacement process in non-coal-bearing strata does not consider gas desorption and adsorption, and the expression for the porosity change in the rock matrix is:

[0277]

[0278] The volume strain in the initial state is zero, and the matrix porosity can be expressed as:

[0279]

[0280] Where, φ m0 —Initial porosity of rock mass, dimensionless.

[0281] S32. Establishing the dynamic change equation of formation fracture ratio: The PM model can better represent the change of formation fracture permeability with gas pressure, and the parameters of the PM model are more highly coupled with the rock deformation and gas migration equations, which conforms to the principle of easy parameter acquisition. Therefore, the PM model is selected as the theoretical basis for the dynamic change of formation fracture ratio, and the relationship between formation fracture ratio and formation rock strain is established:

[0282]

[0283] Where, φ f —formation fracture ratio, dimensionless;

[0284] N—axial constraint modulus, N=E(1-v) / (1+v)(1-2v), Pa;

[0285] γ—skeleton compression coefficient, dimensionless;

[0286] S—overburden load, Pa;

[0287] χ—rock skeleton deformation coefficient, dimensionless;

[0288] T—formation temperature, K;

[0289] Taking formation fissures as the research object, during the displacement process of shallow gas in non-coal-bearing strata, gas seepage mainly occurs in the near-horizontal layer through the formation fissures of the rock strata. The overlying load does not change, and the skeleton compression deformation is considered in the rock mass porosity equation. There is no competitive adsorption in non-coal-bearing strata, and the temperature change is small. Based on this, the PM model is improved:

[0290]

[0291] Where, φ f0 —initial formation fracture ratio of the formation, dimensionless;

[0292] p f0 —Initial gas pressure in formation fractures, Pa.

[0293] By taking the partial derivative of Equation (4-36) with respect to time, we can obtain the equation for the dynamic change of formation fracture rate with time:

[0294]

[0295] S33. Establish a dynamic permeability change equation. The permeability and fracture rate of the formation follow the cubic law, that is:

[0296]

[0297] Where, k is the formation permeability, m 2 ;

[0298] k0—initial permeability of the formation, m 2 ;

[0299] There is a slippage effect during the seepage of gas in formation fractures. In porous media, the gas permeability is greater than the liquid permeability. The gas permeability should be corrected by the slippage coefficient. According to the model proposed by Klinkenberg for gas permeability in porous media, the gas permeability is corrected by the slippage coefficient. The corrected gas permeability is:

[0300]

[0301] Where k g —Permeability of the formation considering the slip effect, m 2 ;

[0302] c—slip coefficient, Pa.

[0303] S4. By establishing the above formulas and combining equations (4-4), (4-8), (4-11), (4-24), (4-32), (4-36), (4-39), and (4-40), a mathematical model for natural gas displacement in non-coal-bearing formations is established.

[0304] The mathematical model of natural gas displacement in non-coal-bearing formations includes the gas migration field control equation, the stress field control equation, and the coupling field control equation:

[0305] The gas migration field control equation includes:

[0306]

[0307] Among them, formula (4-41) is the rock matrix migration field, and formula (4-42) is the stratum fracture migration field, which represent diffusion movement and seepage movement respectively. represents the diffusion amount of gas component i in the rock matrix, It represents the seepage rate of gas component i in the formation fracture, and finally seeps to the drainage hole. represents the matrix-formation fracture exchange amount of gas component i;

[0308] The stress field control equations include:

[0309]

[0310] Where, represents the ground stress, Indicates the combined pressure of gas in the matrix and formation fractures, F i Indicates gravity;

[0311] The coupled field control equations include:

[0312] σ′ ij =σ ij -(β f P f +β m P m )δ ij (4-44)

[0313]

[0314] like Figure 2 As shown in the figure, the coupling relationship between gas migration field and stress field is manifested in effective stress, rock porosity, formation fracture rate and permeability. During the natural gas displacement process, when the gas pressure in the rock formation changes, it will cause stress and strain in the rock formation, which in turn will cause changes in the rock porosity, formation fracture rate and permeability of the rock formation, which in turn affects the migration of gas.

[0315] The foregoing description is merely a preferred embodiment of the present invention. It should be understood that the present invention is not limited to the form disclosed herein and should not be construed as excluding other embodiments. Rather, the present invention can be used in various other combinations, modifications, and environments and can be modified within the scope of the concept described herein through the above teachings or techniques or knowledge in the relevant field. Modifications and variations made by those skilled in the art that do not depart from the spirit and scope of the present invention are intended to be protected by the appended claims.

Claims

1. A method for constructing a mathematical model for natural gas displacement in non-coal-bearing formations, characterized by: The steps include: S1. Establishing gas migration equations: This includes establishing the governing equations for gas diffusion in rock pores and establishing the gas seepage equations in formation fractures; S2. Establish the governing equation for formation rock deformation; S3. Establish dynamic porosity and permeability change control equations: including establishing dynamic change equations for rock mass porosity, formation fracture rate, and permeability; S4. Based on steps S1-S3, a mathematical model for natural gas displacement in non-coal-bearing formations is established.

2. The method for constructing a mathematical model for natural gas displacement in non-coal-bearing formations according to claim 1, characterized in that: In step S1, establishing the governing equation for gas diffusion motion in rock pores includes the following steps: S11. Establish a gas mass exchange equation: Taking methane as the driven gas and nitrogen as the injected gas, the flux formula for the mass exchange between rock pores and formation fractures is expressed as: Where C m1 、C m2 —respectively, the concentrations of methane and nitrogen in rock pores, kg / m 3 ; C f1 、C f2 —respectively, the concentrations of methane and nitrogen in formation fractures, kg / m 3 ; Q1, Q2—gas mass exchange rates of methane and nitrogen between rock pores and formation fractures, kg / (m 3 s); D1, D2—methane and nitrogen gas diffusion constants, respectively, m 2 / s; σ—matrix shape factor, σ=3π 2 / L 2 , m -2 ; L—spacing between formation fractures, m; The gas concentration is obtained from the gas state equation: Where, M1 and M2 are the molar masses of methane and nitrogen, kg / mol respectively; P m1 、P m2 —pressures of methane and nitrogen in rock pores, Pa; P f1 、P f2 —pressures of methane and nitrogen in formation fractures, Pa, respectively; D1, D2—methane and nitrogen gas diffusion constants, respectively, m 2 / s; R—gas state constant, J / (mol·K); T—temperature of the entire displacement system, K; Substituting the gas concentration formula (4-3) into (4-1) yields the mass exchange flux between rock pores and formation fractures: S12. Establish the gas diffusion equation in rock mass pores: Use Fick's law to describe the diffusion movement of gas in the pores of rock matrix: Right now: Where J— is the diffusion flux, kg / (m 2 s); α, β, γ—represent the unit vectors in the x, y, and z directions, respectively, and are dimensionless; c—volume concentration of diffusing substances, kg / m 3 ; D—is the gas diffusion constant, m 2 / s; According to the fact that the change of gas concentration in rock pores is the sum of gas diffusion flux and mass exchange flux between rock pores and formation fractures, the diffusion and continuity equations of methane and nitrogen in rock pores are obtained: Substituting equation (4-3) into equation (4-7), we obtain the governing equation for gas diffusion movement in rock pores: Where, —Porosity of rock matrix.

3. The method for constructing a mathematical model for natural gas displacement in non-coal-bearing formations according to claim 2, characterized in that: In step S1, establishing the control equation of gas seepage movement in formation fractures includes the following steps: S13. Establish the seepage equation of gas in formation fractures: Where, v1—seepage rate, m / s; k—permeability of porous media, m 2 ; μ—dynamic viscosity of the fluid Pa·s; Within Δt, the change in gas mass in any control volume of a porous media formation fracture is equal to the difference in mass between the fluid flowing into and out of the volume plus the mass generated or absorbed by the control volume itself. The conservation of mass in formation fractures is: Combining the law of conservation of mass and Darcy's seepage theorem, formation fractures drive seepage under the combined pressure of methane and nitrogen: Where, — is the formation fracture ratio; P f —Total gas pressure in formation fractures, Pa; μ1, μ2—dynamic viscosity coefficients of methane and nitrogen, Pa·s, respectively.

4. The method for constructing a mathematical model for natural gas displacement in non-coal-bearing formations according to claim 3, characterized in that: In step S2, establishing the governing equation for formation rock mass deformation includes the following steps: S21. Establish the effective stress equation for rock mass deformation: in ij =s ij -P (4-12) Where σ′ ij —effective normal stress on the plane, Pa; σ ij —total normal stress on the plane, Pa; P—pore pressure of formation rock mass, Pa; S22. Based on formula (4-12), the correction is introduced. The effective stress equation after the correction is: in ij =s ij -(b f P f +b m P m )d ij (4-13) Where, β f —Correction coefficient of effective stress in formation fractures, dimensionless; β m —Correction coefficient of effective stress in rock mass pores, dimensionless; P f —Gas pressure in formation fractures, Pa; P m —Gas pressure in rock pores, Pa; K—rock mass bulk modulus, Pa; K m —rock matrix bulk modulus, Pa; K s —rock skeleton bulk modulus, Pa; E—elastic modulus of rock mass, Pa; E m —rock matrix elastic modulus, Pa; ν—Poisson's ratio of rock mass, dimensionless; φ m —rock porosity, dimensionless; δ ij —Kirsten's constant, expressed as dimensionless; S23. Establish an equilibrium equation for rock mass deformation: Ignore the inertial forces generated by natural gas flow and rock mass skeleton deformation, and neglect the gravity of the natural gas itself. Select a small parallelepiped unit from the natural gas-bearing rock mass, and define its three edges as dx, dy, and dz. Assume that the stresses acting on each face of the unit are uniformly distributed. The stresses on this unit are in equilibrium, satisfying the static equilibrium equation: Express formula (4-17) in the form of a tensor: s ij，j +F i =0(i,j=1,2,3) (4-18) Where, F i —body force component, Pa; σ ij，j —total stress component, Pa; Express formula (4-17) in the form of effective stress: in ij，j +[(β f P f +b m P m )d ij ] ，j +F i =0(i,j=1,2,3)(4-19); S24. Establish the geometric equation of rock mass deformation: Rock mass deformation satisfies the Cauchy equation: Its tensor form is: Where, ε ij — volumetric strain tensor; u—deformation displacement; S25. Establish constitutive relations: Assume that the rock mass is a homogeneous isotropic material. During displacement, the rock mass deformation is in the elastic deformation stage. The rock mass deformation in three dimensions follows the generalized Hooke's law: in ij =λδ ij e v +2Gε ij (i, j=1, 2, 3) (4-22) Where, λ is the Mellar constant of the rock mass, λ = Ev / (1+v)(1-2v), Pa; ε v — rock mass volume strain; G—rock shear modulus, G=E / 2(1+v); Considering the effects of rock pore gas pressure and formation fracture gas pressure in dual-porosity media, and substituting formula (4-22) into effective stress formula (4-13), we obtain: s ij -(b f P f +b m P m )d ij -ld ij e v +2Gε ij =0 (4-23) Substituting formula (4-22) into formula (4-19), we can obtain the governing equation for rock deformation during natural gas displacement:

5. The method for constructing a mathematical model for natural gas displacement in non-coal-bearing formations according to claim 4, characterized in that: In step S3, establishing the dynamic change equation of rock mass porosity includes the following steps: S31. The rock matrix is ​​considered as a porous medium, and its porosity is expressed as follows: V=V P +V S (4-26) Where, φ m — rock matrix porosity, dimensionless; V—rock matrix volume, m 3 ; V P —pore volume, m 3 ; V S —Solid particle volume, m 3 ; The following expression is obtained based on the relationship between rock mass porosity and the components of rock mass matrix volume: Using the rock mass porosity model under strain conditions, the porosity in the rock matrix is ​​expressed as: Where, α is the effective stress coefficient, dimensionless; β—effective stress coefficient, dimensionless; K—rock mass bulk modulus, Pa; K p —pore bulk modulus, Pa; K s —bulk modulus of solid particles, Pa; ε s —desorption adsorption strain; The natural gas displacement process in non-coal-bearing strata does not consider gas desorption and adsorption, and the expression for the porosity change in the rock matrix is: The volume strain in the initial state is zero, and the matrix porosity can be expressed as: Where, φ m0 —Initial porosity of rock mass, dimensionless.

6. The method for constructing a mathematical model for natural gas displacement in non-coal-bearing formations according to claim 5, characterized in that: In step S3, establishing the dynamic change equation of formation fracture rate includes the following steps: S32. Select the PM model to establish the relationship between formation fracture rate and formation rock strain: Where, φ f —formation fracture ratio, dimensionless; N—axial constraint modulus, N=E(1-v) / (1+v)(1-2v), Pa; γ—skeleton compression coefficient, dimensionless; S—overburden load, Pa; χ—rock skeleton deformation coefficient, dimensionless; T—formation temperature, K; Taking formation fractures as the research object, during the displacement process of shallow non-coal-bearing gas, the overlying load does not change, competitive adsorption does not occur in the non-coal-bearing strata, and temperature changes are not considered. Based on this, the PM model is improved: Where, φ f0 —initial formation fracture ratio of the formation, dimensionless; p f0 —Initial gas pressure in formation fractures, Pa. By taking the partial derivative of Equation (4-36) with respect to time, we can obtain the equation for the dynamic change of formation fracture rate with time:

7. The method for constructing a mathematical model for natural gas displacement in non-coal-bearing formations according to claim 6, characterized in that: In step S3, establishing the permeability dynamic change equation includes the following steps: S33. The permeability and fracture rate of the formation follow the cubic law, that is: Where, k is the formation permeability, m 2 ; k0—initial permeability of the formation, m 2 ; The gas permeability is corrected by the slip coefficient. The corrected gas permeability is: Where k g —Permeability of the formation considering the slip effect, m 2 ; c—slip coefficient, Pa.

8. The method for constructing a mathematical model for natural gas displacement in non-coal-bearing formations according to claim 7, characterized in that: The mathematical model of natural gas displacement in non-coal-bearing formations includes the gas migration field control equation, the stress field control equation, and the coupling field control equation: The gas migration field control equation includes: Among them, formula (4-41) is the rock matrix migration field, and formula (4-42) is the stratum fracture migration field, which represent diffusion movement and seepage movement respectively. represents the diffusion amount of gas component i in the rock matrix, It represents the seepage rate of gas component i in the formation fracture, and finally seeps to the drainage hole. represents the matrix-formation fracture exchange amount of gas component i; The stress field control equations include: Where, represents the ground stress, Indicates the combined pressure of gas in the matrix and formation fractures, F i Indicates gravity; The coupled field control equations include: in ij =s ij -(b f P f +b m P m )d ij (4-44)

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