A computational method for quantitative prediction of production boundary of coalbed methane wells in undersaturated reservoirs

Abstract

The present invention discloses a calculation method for quantitatively predicting the production boundary of coalbed methane wells in undersaturated reservoirs, which specifically includes the following steps: (1) setting the assumptions of the calculation method; (2) establishing a quantitative prediction model for the production boundary of coalbed methane wells in undersaturated reservoirs; (3) compiling the calculation process of the model, substituting actual geological and production data to calculate the drainage radius and desorption radius respectively; (4) describing the dynamic changes of the gas phase and water phase radii, and determining the position of the production boundary. The present invention establishes a prediction model and completes the calculation method by introducing a dynamic porosity and water saturation model of the coalbed methane reservoir; by using the calculation method, the dynamic curves of the drainage radius and the desorption radius are described, and finally the production boundary of the coalbed methane well is predicted; based on the calculation method, a sensitivity analysis of pressure propagation is performed to determine the influence of the dynamic porosity and dynamic water saturation of the reservoir on the prediction of the production boundary, and the advancement and practicality of the calculation method are verified.

Description

Technical Field

[0001] The invention belongs to the technical field of coalbed methane development and utilization, and in particular relates to a calculation method for quantitatively predicting the production boundary of a coalbed methane well in an undersaturated reservoir. Background Art

[0002] Due to the unique enrichment mechanism and pressure depletion development mode of coalbed methane, the detailed description of the propagation of the pressure drop funnel has always been the core issue of coalbed methane development. The pressure drop funnel will go through different seepage stages during the propagation process, which can be specifically divided into the unsteady flow stage, the steady flow stage and the quasi-steady flow stage. The division of the seepage stage mainly depends on the external boundary conditions, that is, the production boundary of the coalbed methane well. According to the actual situation of the production site, the production boundary can be divided into the inter-well pressure interference boundary and the damage boundary: for low-permeability and undersaturated coal reservoirs, the well network development mode is usually adopted, that is, the well group development is carried out in a certain area to achieve the coordinated pressure reduction between multiple wells. Therefore, the corresponding production boundary is the area where the pressure drop funnel between the two wells is affected, that is, the inter-well pressure interference boundary; for coalbed methane wells affected by unreasonable factors during the production process, such as unreasonable engineering construction, drainage and production system, the pressure drop funnel cannot reach the far well area and can only reduce pressure in a certain area near the wellbore. Therefore, the production boundary corresponding to this area is the damage boundary.

[0003] The current research on production boundaries still has certain limitations: for the inter-well pressure interference boundary, there is a common misunderstanding at the production site, that is, it is believed that two adjacent wells have the same pressure drop funnel propagation morphology, and thus the inter-well pressure interference boundary is simply estimated to be half of the well spacing. However, the actual pressure drop funnel is affected by multiple factors such as geology, engineering and drainage, and the pressure propagation of different coalbed methane wells is particular. Therefore, an unrealistic and simple estimate of the production boundary will cause a large error in the pressure drop funnel propagation. For the damage boundary, the current main means of predicting the reservoir blockage location are well testing, numerical simulation, etc., but these methods have the disadvantages of small measurement range and strong influence by human factors, and cannot obtain more accurate results. In addition, when solving the reservoir damage problem, the production site often adopts general unblocking measures and cannot realize targeted scheme design, which can easily cause secondary damage to the reservoir. In addition, the current existing mathematical models lack the consideration of the influence of factors such as dynamic porosity and water saturation when predicting the boundary, which makes the prediction results have obvious deviations.

[0004] In summary, there is still a lack of a calculation method for accurately predicting the production boundary of coalbed methane wells that takes into account the dynamic characteristics of the reservoir, which is an urgent problem to be solved for the efficient development of coalbed methane. Summary of the invention

[0005] In order to solve the deficiencies in the prior art, the present invention provides a completely quantitative calculation method for predicting the production boundary of coalbed methane wells. Starting from the pressure state equation and the material balance equation, combined with the dynamic changes of coal reservoir porosity and water saturation, and considering the impact of hydraulic fracturing on pressure propagation, the method establishes a quantitative prediction model for the production boundary of coalbed methane wells, and realizes the refined calculation of data by writing a calculation process, and finally completes the actual production field application.

[0006] To achieve the above object, the present invention proposes a method for calculating the quantitative prediction of the production boundary of a coalbed methane well in an undersaturated reservoir, comprising the following steps:

[0007] (1) The calculation method is based on the following basic assumptions: the formation fluid is a two-phase gas / water phase, the gas is an ideal gas, the water is a slightly compressible fluid, and the instantaneous flow of the fluid in the formation is regarded as a collection of a series of steady-state flows; the reservoir porosity and water saturation change dynamically during the coalbed methane development process;

[0008] (2) Based on the isothermal adsorption and desorption equations of adsorbed gas and the dynamic changes of free gas in the pores, the gas phase material balance equation in coal is derived. Taking into account the compressibility of the formation and the dynamic water saturation, the water phase material balance equation is derived. The gas / water pressure state equation controlled by hydraulic fracturing is established by using the conformal transformation method. The gas / water phase material balance equation and the pressure state equation are combined to establish a prediction model for the pressure drop funnel propagation of coalbed methane wells in undersaturated reservoirs. The inner boundary of the model is the actual bottom hole flow pressure, and the outer boundary is the constant pressure boundary and closed boundary of the instantaneous steady-state flow. The outer boundary pressure of the drainage zone is the initial reservoir pressure, and the outer boundary pressure of the desorption zone is the critical desorption pressure.

[0009] (3) Write the calculation process of the model: Substitute the production data of a certain production time, assume the desorption radius and drainage radius, calculate the corresponding cumulative gas production and cumulative water production in the pressure drop funnel propagation prediction model of undersaturated reservoir coalbed methane wells, and compare the errors between the calculated results and the actual cumulative gas production and cumulative water production. Use the iterative method to continuously optimize the assumed values ​​until the errors reach a reasonable accuracy, and finally determine the optimized desorption radius and drainage radius as the actual values;

[0010] (4) Substitute the complete production data and repeat step 3 to draw the dynamic curves of the desorption radius and drainage radius during the production stage. When the two curves intersect at a point, the radius is determined to be the production boundary range of the undersaturated reservoir coalbed methane well.

[0011] Based on the dynamic porosity and dynamic water saturation model given in step (1), the dynamic change curve of coal reservoir porosity and water saturation during coalbed methane production is drawn, which specifically includes the following assumptions: the coal reservoir is a dual-porosity medium, the tiny pores in the coal matrix are the main storage space for gas, the cleats and fissures are the main migration space for fluid, the fluid in the coal reservoir can be divided into gas phase and water phase, the gas is an ideal gas, the water is a slightly compressible fluid, and the instantaneous flow of the fluid in the formation is regarded as a collection of a series of steady-state flows; the reservoir Porosity and water saturation change dynamically during coalbed methane development: the pressure propagation of undersaturated coal reservoirs is divided into gas desorption zone and water drainage zone with the critical desorption pressure as the boundary. The dynamic change mechanism of reservoir porosity in these two zones is different: in the drainage zone, the coal reservoir is compacted by the effective stress effect and the porosity is reduced; while in the desorption zone, due to the desorption of coalbed methane from the surface of the coal matrix, the porosity is simultaneously damaged by the effective stress effect and restored by the matrix shrinkage effect.

[0012] Furthermore, the dynamic porosity of coal reservoirs in different regions can be expressed as:

[0013]

[0014] In the formula, is the initial reservoir porosity, dimensionless; is the dynamic porosity in the drainage area, dimensionless; is the dynamic porosity in the desorption zone, dimensionless; is the porosity corresponding to the critical desorption pressure of the reservoir pressure, MPa; P g is the gas phase pressure state equation, MPa; P w is the water phase pressure equation of state, MPa; P i is the initial reservoir pressure, MPa; P cd is the critical desorption pressure, MPa; P L is the Langmuir pressure, MPa; ε max is the maximum volumetric strain, dimensionless; C f is the coal rock compression coefficient, MPa -1 , changes dynamically with the reservoir pressure and can be expressed by the following dynamic equation:

[0015] C f =0.0026×P n 2 -0.0252×P n +0.1631 (2)

[0016] Furthermore, coalbed methane desorption not only causes changes in reservoir porosity, but also occupies a portion of the pore volume, causing the formation water in the pores to be discharged. Therefore, the water saturation of the formation and the reservoir pressure conform to the following relationship:

[0017]

[0018] in,

[0019] A=WGMR(C f +C g +C d ) (4)

[0020] B=WGMR(C f +C g )+C f +C w (5)

[0021] C=WGMR+1 (6)

[0022]

[0023]

[0024] Where P n and P n+1 Represent the reservoir microelement pressure at the nth and n+1th steps, MPa; the corresponding S w n and S w n+1 Represent the water saturation at the nth and n+1th steps, dimensionless; S w is the dynamic water saturation, dimensionless; C w is the compressibility of water, MPa -1 ; C d is the desorption compressibility coefficient, MPa -1 ;P sc is the pressure under standard conditions, MPa; Z sc T is the gas deviation factor under standard conditions, dimensionless; sc is the temperature under standard conditions, K; Z is the deviation factor under reservoir conditions, dimensionless; T is the reservoir temperature, K; V L is the Langmuir volume, m 3 / t; ρ is reservoir density, t / m 3 ; WGMR is the mobility ratio of gas and water phases, dimensionless; S wc is the bound water saturation, dimensionless; K rg * is the relative permeability of the gas phase at irreducible water saturation, dimensionless; l and m are the Corey exponents of the gas phase and water phase, dimensionless; μ gand μ w are respectively the gas phase viscosity and the water phase viscosity, mPa·s; C g is the gas compressibility coefficient, MPa -1 , since the gas is an ideal gas, Z = 1, and the gas compressibility coefficient can be expressed as C g =1 / P n .

[0025] Based on the coal reservoir material balance equation and the gas / water phase pressure state equation, combined with the impact of hydraulic fracturing on pressure propagation, the dynamic porosity and dynamic water saturation models are used to establish a prediction model for the pressure drop funnel propagation of undersaturated reservoir coalbed methane wells. Specifically:

[0026] When the reservoir pressure is greater than the critical desorption pressure, the production stage of the coalbed methane well is the single-phase water flow stage. At this time, the water phase pressure state equation in the drainage area can be expressed as:

[0027]

[0028] When the pressure drops below the critical desorption pressure, the production stage of the coalbed methane well is the gas-water two-phase flow stage, and the gas phase pressure state equation in the desorption zone can be expressed in the form of pressure square:

[0029]

[0030] With the desorption of coalbed methane, the range of the desorption zone gradually increases, and the boundary conditions in the drainage area increase with the increase of the desorption radius. Therefore, the water phase pressure state equation is further transformed as follows.

[0031]

[0032] Furthermore, the above gas / water phase pressure state equation is applicable to radial seepage without pressure fractures, but low permeability coal reservoirs usually use artificial measures such as hydraulic fracturing to improve reservoir permeability. The extension of hydraulic fractures along the principal stress direction will cause the overall pressure drop to propagate elliptically to the far well area. Therefore, for the convenience of calculation, the conformal transformation method is used to convert the elliptical domain Z(x, y) into the linear domain ζ(ξ, η). The pressure state equations of the gas phase and the water phase can be expressed as follows:

[0033]

[0034]

[0035] Where P wf is the bottom hole flowing pressure, MPa; L f is the half length of the fracture, m; r wf is the wellbore radius, m; r i is the drainage radius, m; r cdis the desorption radius, m; r is the pressure propagation radius, m; ξ wf is the wellbore radius in the linear domain, which is infinitesimal compared to the drainage radius and desorption radius, and is set to 0 and dimensionless; ξ i is the drainage radius in the linear domain, dimensionless; ξ cd is the desorption radius in the linear domain and is dimensionless.

[0036] The material balance equation of coal reservoir can also be divided into gas phase material balance equation and water phase material balance equation. In the gas phase material balance equation, the cumulative gas production is equal to the desorption amount of coalbed methane plus the initial free gas amount minus the amount of coalbed methane remaining in the pores, which can be expressed as:

[0037]

[0038]

[0039]

[0040] For the material balance equation of the water phase, it is divided into water production in the drainage area and water production in the desorption area. The water production mechanism in different areas is different: the reduction of porosity and the compressible elastic expansion of water in the drainage area lead to water production, which can be expressed as:

[0041]

[0042] The water production in the desorption zone also comes from the reduction of water saturation in the pores caused by gas desorption. Therefore, the water production in the desorption zone can be expressed as:

[0043]

[0044] In the formula, G P is the cumulative gas production under ground conditions, m 3 ; W P1 is the water yield of the drainage area, m 3 ; W P2 is the water production in the desorption zone, m3; S wi is the initial water saturation, dimensionless; h is the coal seam thickness, m; B g is the gas volume coefficient, dimensionless; B gi is the initial gas volume coefficient, dimensionless; B w is the volume coefficient of formation water. Since formation water is a slightly compressible fluid, it is set to 1 and dimensionless. V is the volume of the coal reservoir affected by the pressure drop funnel, which is converted into the swept volume in the linear domain according to the Jacobian matrix.

[0045]

[0046] By introducing the dynamic porosity and water saturation in the coalbed methane development process, combining the pressure state equation and material balance equation at different coalbed methane drainage stages and integrating them, the pressure propagation model can be obtained:

[0047]

[0048]

[0049] Furthermore, the calculation process of the model is written, and the production data of a certain production time is substituted. By assuming the desorption radius and drainage radius, the corresponding cumulative gas production and cumulative water production are calculated in the pressure drop funnel propagation prediction model of the undersaturated reservoir coalbed methane well, and the errors between the calculated results and the actual cumulative gas production and cumulative water production are compared. The assumed value is continuously optimized by the iterative method until the error reaches a reasonable accuracy, and finally the optimized desorption radius and drainage radius are determined as the actual value. Specifically, the following steps are included:

[0050] (1) Determine the dynamic characteristics of the coalbed methane reservoir: Substitute the coalbed methane well geological parameters defined above into the dynamic model to characterize the relationship between the dynamic porosity and dynamic water saturation of the coal reservoir and the reservoir pressure, and apply the dynamic curves to the subsequent calculation process;

[0051] (2) Determine the basic mathematical method used in model calculation: Since geological parameters and production data such as bottom hole flowing pressure and cumulative gas / water production are known quantities, while drainage radius and desorption radius are unknown quantities, an iterative algorithm is used for calculation. The cumulative gas / water production is calculated by substituting the unknown quantities into the method, and the assumed values ​​are continuously optimized until the error between the calculated results and the actual production data is less than 1%, which is defined as reasonable accuracy;

[0052] (3) Calculation of the drainage radius in the single-phase water flow stage: Substitute the actual production data and assume a drainage radius ξ when the bottom hole pressure is greater than the critical desorption pressure. i The value is substituted into the model to describe the dynamic characteristics of the pressure drop funnel in the drainage area and calculate the cumulative water production W P1 , if the calculation result W P1 The actual cumulative water production W Preal If the error between them reaches a reasonable accuracy, the assumed value corresponds to the actual drainage radius, otherwise another drainage radius is assumed and recalculated;

[0053] (4) Calculate the drainage radius and desorption radius in the gas-water two-phase flow stage: Substitute the actual production data. When the bottom hole pressure is less than the critical desorption pressure, first assume that the desorption radius ξ cd , substituted into the model to characterize the dynamic characteristics of the gas phase pressure drop funnel in the desorption zone and calculate the cumulative gas production G P , by continuously optimizing the assumed value until the calculated result G PThe actual cumulative gas production G Preal The error between them reaches a reasonable accuracy; secondly, based on the desorption radius ξ cd Calculate the water production W in the desorption zone using the gas phase pressure drop funnel P2 ; Finally, assume that the drainage radius ξ in the drainage area i , continue to optimize the assumed value until the calculated water production in the drainage area and the water production in the desorption area (W P1 +W P2 ) and the actual cumulative water production W Preal The error between them reaches a reasonable accuracy;

[0054] (5) Convert the calculation results in the linear domain into the elliptical domain: In order to intuitively obtain the morphological characteristics of pressure propagation in the elliptical domain, define x i and i 、x cd and cd is the maximum and minimum semi-axis lengths of the drainage and desorption zones in the elliptical domain, following:

[0055]

[0056]

[0057] Furthermore, the complete production data is substituted and the above steps are repeated to draw the dynamic curves of the desorption radius and the drainage radius in the production stage. When the two curves intersect at one point, the radius is determined to be the production boundary range of the coalbed methane well in the undersaturated reservoir. Specifically, the complete production data is substituted into the calculation process, and the dynamic curves of the desorption radius and the drainage radius in the production stage are drawn. With the continuous production of coalbed methane, the desorption radius curve gradually increases. As for the drainage radius, since the production boundary is a value to be solved, the quasi-steady-state flow pressure state equation cannot be defined in the model, resulting in the calculated drainage radius curve first increasing and then decreasing. When the drainage radius curve intersects with the desorption radius curve at one point, the range corresponding to the intersection is the production boundary of the coalbed methane well.

[0058] By adopting the above technical solution, the present invention has the following technical effects:

[0059] (1) By introducing the dynamic porosity and water saturation model of coalbed methane reservoirs, combining the gas / water phase pressure state equation and material balance equation in different regions, and considering the impact of hydraulic fracturing on the propagation of the pressure drop funnel, a quantitative prediction model for the production boundary of coalbed methane wells in undersaturated reservoirs was established, and the calculation method was established by compiling a calculation process;

[0060] (2) Through the established method, the dynamic curves of drainage radius and desorption radius are characterized, and finally the production boundary of coalbed methane wells is predicted, such as the pressure interference boundary between wells under the well network development mode and the coal reservoir damage boundary under unreasonable development conditions;

[0061] (3) Based on this calculation method, a sensitivity analysis of pressure propagation is conducted to determine the impact of reservoir dynamic porosity and dynamic water saturation on production boundary prediction, thus verifying the advancement and practicality of this calculation method. BRIEF DESCRIPTION OF THE DRAWINGS

[0062] Figure 1 Quantitative prediction calculation method for production boundary of coalbed methane wells in undersaturated reservoirs;

[0063] Figure 2 Schematic diagram of the calculation process;

[0064] Figure 3 Actual drainage curve of case well A;

[0065] Figure 4 Actual drainage curve of case well B;

[0066] Figure 5 Water saturation dynamic curves of two case wells;

[0067] Figure 6 Porosity dynamic curves of two case wells;

[0068] Figure 7 Calculation results of drainage radius / desorption radius of well A; Figure 8 Calculation results of drainage radius / desorption radius of well B;

[0069] Fig. 9 Actual dynamic propagation diagram of the drainage radius / desorption radius linear domain of well A;

[0070] Fig.10 Actual dynamic propagation diagram of drainage radius / desorption radius linear domain in well B;

[0071] Fig.11 Actual dynamic propagation diagram of the drainage radius / desorption radius ellipse domain of well A;

[0072] Fig.12 Actual dynamic propagation diagram of the drainage radius / desorption radius ellipse domain of well B;

[0073] Fig.13 Sensitivity analysis diagram of water saturation to pressure propagation calculation results. DETAILED DESCRIPTION

[0074] To make the objectives, technical solutions and advantages of the present invention more clear, the embodiments of the present invention will be further described in detail below with reference to the accompanying drawings.

[0075] Due to the development method of coalbed methane pressure depletion, coalbed methane blocks are mostly developed by well network deployment. The propagation of pressure drop funnel can form pressure interference between multiple wells, which is conducive to the coordinated pressure reduction of the reservoir. In addition, there are often unreasonable engineering construction and drainage systems in the production process, and the coal reservoir will be irreversibly blocked, which is the damage boundary. The existence of the production boundary will change the seepage state of the fluid in the reservoir, so it is necessary to calculate and characterize the production boundary in detail. Figure 1 The quantitative prediction and calculation method for the production boundary of a coalbed methane well in an undersaturated reservoir provided by the present invention comprises:

[0076] (1) The calculation method is based on the following basic assumptions: the formation fluid is a two-phase gas / water phase, the gas is an ideal gas, the water is a slightly compressible fluid, and the instantaneous flow of the fluid in the formation is regarded as a collection of a series of steady-state flows; the reservoir porosity and water saturation change dynamically during the coalbed methane development process;

[0077] (2) Based on the isothermal adsorption and desorption equations of adsorbed gas and the dynamic changes of free gas in the pores, the gas phase material balance equation in coal is derived. Taking into account the compressibility of the formation and the dynamic water saturation, the water phase material balance equation is derived. The gas / water pressure state equation controlled by hydraulic fracturing is established by using the conformal transformation method. The gas / water phase material balance equation and the pressure state equation are combined to establish a prediction model for the pressure drop funnel propagation of coalbed methane wells in undersaturated reservoirs. The inner boundary of the model is the actual bottom hole flow pressure, and the outer boundary is the constant pressure boundary and closed boundary of the instantaneous steady-state flow. The outer boundary pressure of the drainage zone is the initial reservoir pressure, and the outer boundary pressure of the desorption zone is the critical desorption pressure.

[0078] (3) Write the calculation process of the model: Substitute the production data of a certain production time, assume the desorption radius and drainage radius, calculate the corresponding cumulative gas production and cumulative water production in the pressure drop funnel propagation prediction model of undersaturated reservoir coalbed methane wells, and compare the errors between the calculated results and the actual cumulative gas production and cumulative water production. Use the iterative method to continuously optimize the assumed values ​​until the errors reach a reasonable accuracy, and finally determine the optimized desorption radius and drainage radius as the actual values;

[0079] (4) Substitute the complete production data and repeat step 3 to draw the dynamic curves of the desorption radius and drainage radius during the production stage. When the two curves intersect at a point, the radius is determined to be the production boundary range of the undersaturated reservoir coalbed methane well.

[0080] Based on the dynamic porosity and dynamic water saturation model, the dynamic change curve of coal reservoir porosity and water saturation during coalbed methane production is drawn, including the following assumptions: the coal reservoir is a dual-porosity medium, the tiny pores in the coal matrix are the main storage space for gas, the cleat fractures are the main migration space for fluid, the fluid in the coal reservoir can be divided into gas phase and water phase, the gas is an ideal gas, the water is a slightly compressible fluid, and the instantaneous flow of the fluid in the formation is regarded as a collection of a series of steady-state flows; the reservoir porosity and water saturation are important in coalbed methane development. Dynamic changes occur during the process: the pressure propagation of the undersaturated coal reservoir is divided into the gas phase desorption zone and the water phase drainage zone with the critical desorption pressure as the boundary. The dynamic change mechanism of reservoir porosity in these two zones is different: in the drainage zone, the coal reservoir is compacted by the effective stress effect, and the porosity is reduced; in the desorption zone, due to the desorption of coalbed methane from the surface of the coal matrix, the porosity is simultaneously damaged by the effective stress effect and restored by the matrix shrinkage effect. The dynamic porosity of coal reservoirs in different zones can be expressed as:

[0081]

[0082] In the formula, is the initial reservoir porosity, dimensionless; is the dynamic porosity in the drainage area, dimensionless; is the dynamic porosity in the desorption zone, dimensionless; is the porosity corresponding to the critical desorption pressure of the reservoir pressure, MPa; P g is the gas phase pressure state equation, MPa; P w is the water phase pressure equation of state, MPa; P i is the initial reservoir pressure, MPa; P cd is the critical desorption pressure, MPa; P L is the Langmuir pressure, MPa; ε max is the maximum volumetric strain, dimensionless; C f is the coal rock compression coefficient, MPa -1 , changes dynamically with the reservoir pressure and can be expressed by the following dynamic equation:

[0083] C f =0.0026×P n 2 -0.0252×P n +0.1631 (25)

[0084] In addition, coalbed methane desorption not only causes changes in reservoir porosity, but also occupies a portion of the pore volume, causing the formation water in the pores to be discharged. Therefore, the water saturation of the formation and the reservoir pressure conform to the following relationship:

[0085]

[0086] in,

[0087] A=WGMR(C f +C g +C d ) (27)

[0088] B=WGMR(C f +C g )+C f +C w (28)

[0089] C=WGMR+1 (29)

[0090]

[0091]

[0092] Where P n and P n+1 Represent the reservoir microelement pressure at the nth and n+1th steps, MPa; the corresponding S w n and S w n+1 Represent the water saturation at the nth and n+1th steps, dimensionless; S w is the dynamic water saturation, dimensionless; C w is the compressibility of water, MPa -1 ; C d is the desorption compressibility coefficient, MPa -1 ;P sc is the pressure under standard conditions, MPa; Z sc T is the gas deviation factor under standard conditions, dimensionless; sc is the temperature under standard conditions, K; Z is the deviation factor under reservoir conditions, dimensionless; T is the reservoir temperature, K; V L is the Langmuir volume, m 3 / t; ρ is reservoir density, t / m 3 ; WGMR is the mobility ratio of gas and water phases, dimensionless; S wc is the bound water saturation, dimensionless; K rg * is the relative permeability of the gas phase at irreducible water saturation, dimensionless; l and m are the Corey exponents of the gas phase and water phase, dimensionless; μ g and μ w are respectively the gas phase viscosity and the water phase viscosity, mPa·s; C g is the gas compressibility coefficient, MPa -1 , since the gas is an ideal gas, Z = 1, and the gas compressibility coefficient can be expressed as Cg =1 / P n .

[0093] Based on the coal reservoir material balance equation and the gas / water phase pressure state equation, combined with the impact of hydraulic fracturing on pressure propagation, the dynamic porosity and dynamic water saturation models are used to establish a prediction model for the pressure drop funnel propagation of undersaturated reservoir coalbed methane wells. Specifically:

[0094] When the reservoir pressure is greater than the critical desorption pressure, the production stage of the coalbed methane well is the single-phase water flow stage. At this time, the water phase pressure state equation in the drainage area can be expressed as:

[0095]

[0096] When the pressure drops below the critical desorption pressure, the production stage of the coalbed methane well is the gas-water two-phase flow stage, and the gas phase pressure state equation in the desorption zone can be expressed in the form of pressure square:

[0097]

[0098] As the coalbed methane desorbs, the desorption zone gradually increases, and the boundary conditions in the drainage area increase with the increase of the desorption radius. Therefore, the water phase pressure state equation is further transformed into:

[0099]

[0100] The above gas / water phase pressure state equation is applicable to radial seepage without pressure fractures, but low permeability coal reservoirs usually use artificial measures such as hydraulic fracturing to improve reservoir permeability. The extension of hydraulic fractures along the principal stress direction will cause the overall pressure drop to propagate to the far well area in an elliptical shape. Therefore, for the convenience of calculation, the conformal transformation method is used to convert the elliptical domain Z(x, y) into the linear domain ζ(ξ, η). The pressure state equations of the gas phase and water phase can be expressed as follows:

[0101]

[0102]

[0103] Where P wf is the bottom hole flowing pressure, MPa; L f is the half length of the fracture, m; r wf is the wellbore radius, m; r i is the drainage radius, m; r cd is the desorption radius, m; r is the pressure propagation radius, m; ξ wf is the wellbore radius in the linear domain, which is infinitesimal compared to the drainage radius and desorption radius, and is set to 0 and dimensionless; ξ i is the drainage radius in the linear domain, dimensionless; ξ cdis the desorption radius in the linear domain and is dimensionless.

[0104] The material balance equation of coal reservoir can also be divided into gas phase material balance equation and water phase material balance equation. In the gas phase material balance equation, the cumulative gas production is equal to the desorption amount of coalbed methane plus the initial free gas amount minus the amount of coalbed methane remaining in the pores, which can be expressed as:

[0105]

[0106]

[0107]

[0108] For the material balance equation of the water phase, it is divided into water production in the drainage area and water production in the desorption area. The water production mechanism in different areas is different: the reduction of porosity and the compressible elastic expansion of water in the drainage area lead to water production, which can be expressed as:

[0109]

[0110] The water production in the desorption zone also comes from the reduction of water saturation in the pores caused by gas desorption. Therefore, the water production in the desorption zone can be expressed as:

[0111]

[0112] In the formula, G P is the cumulative gas production under ground conditions, m 3 ; W P1 is the water yield of the drainage area, m 3 ; W P2 is the water production in the desorption zone, m 3 ; S wi is the initial water saturation, dimensionless; h is the coal seam thickness, m; B g is the gas volume coefficient, dimensionless; B gi is the initial gas volume coefficient, dimensionless; B w is the volume coefficient of formation water. Since formation water is a slightly compressible fluid, it is set to 1 and dimensionless. V is the volume of the coal reservoir affected by the pressure drop funnel, which is converted into the swept volume in the linear domain according to the Jacobian matrix:

[0113]

[0114] By introducing the dynamic porosity and water saturation in the coalbed methane development process, combining the pressure state equation and material balance equation at different coalbed methane drainage stages and integrating them, the pressure propagation model can be obtained:

[0115]

[0116]

[0117] like Figure 2 As shown in the figure, the calculation process of the calculation method is written, the production data of a certain production time is substituted, and the corresponding cumulative gas production and cumulative water production are calculated in the pressure drop funnel propagation prediction model of the undersaturated reservoir coalbed methane well by assuming the desorption radius and drainage radius, and the errors between the calculation results and the actual cumulative gas production and cumulative water production are compared. The assumed value is continuously optimized by the iteration method until the error reaches a reasonable accuracy, and finally the optimized desorption radius and drainage radius are determined as the actual value, which specifically includes the following steps:

[0118] (1) Determine the dynamic characteristics of the coalbed methane reservoir: Substitute the coalbed methane well geological parameters defined above into the dynamic model to characterize the relationship between the dynamic porosity and dynamic water saturation of the coal reservoir and the reservoir pressure, and apply the dynamic curves to the subsequent calculation process;

[0119] (2) Determine the basic mathematical method used in model calculation: Since geological parameters and production data such as bottom hole flowing pressure and cumulative gas / water production are known quantities, while drainage radius and desorption radius are unknown quantities, an iterative algorithm is used for calculation. The cumulative gas / water production is calculated by substituting the unknown quantities into the method, and the assumed values ​​are continuously optimized until the error between the calculated results and the actual production data is less than 1%, which is defined as reasonable accuracy;

[0120] (3) Calculation of the drainage radius in the single-phase water flow stage: Substitute the actual production data and assume a drainage radius ξ when the bottom hole pressure is greater than the critical desorption pressure. i The value is substituted into the model to describe the dynamic characteristics of the pressure drop funnel in the drainage area and calculate the cumulative water production W P1 , if the calculation result W P1 The actual cumulative water production W Preal If the error between them reaches a reasonable accuracy, the assumed value corresponds to the actual drainage radius, otherwise another drainage radius is assumed and recalculated;

[0121] (4) Calculate the drainage radius and desorption radius in the gas-water two-phase flow stage: Substitute the actual production data. When the bottom hole pressure is less than the critical desorption pressure, first assume that the desorption radius ξ cd , substituted into the model to characterize the dynamic characteristics of the gas phase pressure drop funnel in the desorption zone and calculate the cumulative gas production G P , by continuously optimizing the assumed value until the calculated result G P The actual cumulative gas production G Preal The error between them reaches a reasonable accuracy; secondly, based on the desorption radius ξ cd Calculate the water production W in the desorption zone using the gas phase pressure drop funnel P2 ; Finally, assume that the drainage radius ξ in the drainage area i, continue to optimize the assumed value until the calculated water production in the drainage area and the water production in the desorption area (W P1 +W P2 ) and the actual cumulative water production W Preal The error between them reaches a reasonable accuracy;

[0122] (5) Convert the calculation results in the linear domain into the elliptical domain: In order to intuitively obtain the morphological characteristics of pressure propagation in the elliptical domain, define x i and i 、x cd and cd is the maximum and minimum semi-axis lengths of the drainage and desorption zones in the elliptical domain, following:

[0123]

[0124]

[0125] The complete production data is substituted into the calculation process, and the dynamic curves of the desorption radius and drainage radius in the production stage are drawn. With the continuous production of coalbed methane, the desorption radius curve gradually increases. As for the drainage radius, since the production boundary is a value to be solved, the quasi-steady-state flow pressure state equation cannot be defined in the model, resulting in the calculated drainage radius curve first increasing and then decreasing. When the drainage radius curve intersects the desorption radius curve at one point, the range corresponding to the intersection is the production boundary of the coalbed methane well.

[0126] The following is an example of two coalbed methane wells with a well spacing of about 250 meters in the Shizhuang South Block in the southern Qinshui Basin to quantitatively predict the production boundary of coalbed methane wells in undersaturated reservoirs. Figure 3 , 4 As shown in the figure, Well A is a high-yield well with an average daily gas production of more than 1500m 3 / d, which can maintain high and stable production. Its water production and bottom hole pressure gradually decreased in the early stage and remained at a low value in the later stage. The productivity characteristic curve of well B shows that the well was affected by an unreasonable working system. The bottom hole pressure dropped rapidly in the early stage, causing the gas production to quickly reach the peak and then drop rapidly to 1000m 3 / d or less. The basic geology of the two wells was accurately obtained through well logging and testing data, literature review, experimental testing, and numerical simulation history matching adjustment. w =1, C w 0.00045MPa -1 , ε max is 0.7%, h is 6m, L f is 60m, r wf is 0.1m, T is 300K, μ w 0.656mPa·s, μ gis 0.01134 mPa·s, ρ is 1.4 t / m 3 , and other geological parameters are shown in Table 1.

[0127] Table 1. Geological parameters of target wells

[0128]

[0129]

[0130] First, the geological parameters are substituted into the model to quantitatively calculate the dynamic water saturation and dynamic porosity of the two wells, such as Figure 5 , 6 As shown in the figure, when the pressure is less than the critical desorption pressure, the water saturation of Well A decreases continuously with the decrease of pressure, and the decreasing gradient also gradually slows down; compared with Well A, the water saturation of Well B drops sharply when it is lower than the critical desorption pressure, and then tends to decrease gently; the porosity of both wells has a trend of decreasing in the early stage and gradually increasing in the later stage during the depressurization process.

[0131] By combining the static geological parameters and dynamic characteristics of the reservoir, substituting the actual production data, and using the calculation method proposed in this patent, the change curve of the drainage radius and desorption radius during the coalbed methane production process is calculated. Figure 7 , 8 It can be seen intuitively that the propagation characteristics of the desorption radius and the drainage radius are obviously different: with the desorption and production of coalbed methane, the desorption radius gradually increases, but the drainage radius shows a trend of first increasing and then decreasing. It is worth noting that the drainage radius cannot decrease in the actual production process. The main reason for the analysis is that the seepage state of the fluid in the formation after the pressure propagates to the production boundary should follow the quasi-steady-state flow, but the model is aimed at calculating the production boundary of the coalbed methane well. Therefore, only the instantaneous steady-state flow is considered in the model and the quasi-steady-state flow cannot be considered. Finally, when the drainage radius and the desorption radius intersect, the corresponding range is the production boundary of the coalbed methane well. It can be seen from the figure that the two wells reached the production boundary at about 550 days and 200 days respectively. The actual expansion curves of the drainage radius and the desorption radius are as follows. Fig. 9 , 10 For the convenience of observation and analysis, formula (45,46) is used to convert Fig. 9 , 10 The calculation results of are converted from the linear domain to the elliptical domain, and we can get Fig.11 , 12The curve shows that the long semi-axis of the ellipse of well A can reach about 120 meters, which is approximately equal to half of the well spacing, while the long semi-axis of the ellipse of well B is only 80 meters, which is much less than half of the well spacing. In summary, the boundary reached by well A is the well pressure interference boundary. The farther production boundary and pressure interference enable the well to achieve high and stable production in the later period. The unreasonable working system leads to the boundary reached by well B as the damage boundary. The closer production boundary limits the production capacity of the well in the later period and can only maintain low-level production.

[0132] The impact of dynamic water saturation on pressure propagation and production boundary prediction is further evaluated. Fig.13 It can be seen that the influence of dynamic water saturation on the propagation of gas / water phase pressure is obviously different: in the early stage of production, since the water saturation in the drainage area is equal to the initial value and does not change, the dynamic water saturation has no effect on the water phase pressure state equation in this area; but if the dynamic change of water saturation is not considered in the desorption area, it means that the water saturation in the pores of the coal reservoir will directly drop from the initial value to the irreducible water saturation during the pressure drop, overestimating the recoverable water in the pores, resulting in more formation water discharge, and thus a smaller drainage radius is calculated under the same cumulative water production; in contrast, not considering the dynamic water saturation will make the gas saturation larger, that is, more desorbed gas remains in the pores, and ultimately leads to a larger desorption radius calculated under the same cumulative gas production. In addition, since the formation water production in the desorption area accounts for a large proportion of the cumulative water production, and the volume of coalbed methane remaining in the pores is much smaller than the coalbed methane production, the influence of dynamic water saturation on the expansion of the drainage radius is much greater than that on the desorption radius. More importantly, if the dynamic change of water saturation is not considered, the predicted production boundary will be reduced in range and advanced in time, which will cause huge errors in the quantitative calculation of the pressure drop funnel. Therefore, considering the dynamic water saturation will make the calculation results of the production boundary more accurate.

[0133] This embodiment does not impose any formal limitation on the shape, material, structure, etc. of the present invention. Any simple modification, equivalent change and modification made to the above embodiment based on the technical essence of the present invention shall fall within the protection scope of the technical solution of the present invention.

Claims

1. A method for quantitatively predicting the production boundary of coalbed methane wells in undersaturated reservoirs, characterized in that: The method comprises the following steps: (1) The calculation method is based on the following basic assumptions: the formation fluid is a two-phase gas / water phase, the gas is an ideal gas, the water is a slightly compressible fluid, and the instantaneous flow of the fluid in the formation is regarded as a collection of a series of steady-state flows; the reservoir porosity and water saturation change dynamically during the coalbed methane development process; (2) The gas phase material balance equation in coal is derived based on the isothermal adsorption and desorption equation of adsorbed gas and the dynamic change of free gas in the pores; the water phase material balance equation is derived by considering the formation compressibility and dynamic water saturation; the gas / water pressure state equation controlled by hydraulic fracturing is established by using conformal transformation; the gas / water phase material balance equation and the pressure state equation are combined to establish a prediction model for the pressure drop funnel propagation of coalbed methane wells in undersaturated reservoirs; the inner boundary of the model is the actual bottom hole flow pressure, and the outer boundary is the constant pressure boundary and closed boundary of the instantaneous steady-state flow, where the outer boundary pressure of the drainage zone is the initial reservoir pressure, and the outer boundary pressure of the desorption zone is the critical desorption pressure; (3) Write the calculation process of the model: substitute the production data of the production time, assume the desorption radius and drainage radius, calculate the corresponding cumulative gas production and cumulative water production in the pressure drop funnel propagation prediction model of the undersaturated reservoir coalbed methane well, and compare the errors between the calculated results and the actual cumulative gas production and cumulative water production. Use the iterative method to continuously optimize the assumed value until the error reaches a reasonable accuracy, and finally determine the optimized desorption radius and drainage radius as the actual value; (4) Substitute the complete production data and repeat step (3) to draw the dynamic curves of the desorption radius and the drainage radius during the production stage. When the two curves intersect at a point, the radius is determined to be the production boundary range of the undersaturated reservoir coalbed methane well.

2. The method for quantitatively predicting the production boundary of a coalbed methane well in an undersaturated reservoir according to claim 1, characterized in that: The quantitative prediction calculation method for the production boundary of coalbed methane wells in undersaturated reservoirs is based on the assumptions put forward in step (1), as follows: Coal reservoirs are dual-porosity media. The tiny pores in the coal matrix are the main storage space for gas, and the cleats and fissures are the main migration space for fluids. The fluids in the coal reservoir can be divided into gas phase and water phase. The gas is an ideal gas, and the water is a slightly compressible fluid. The instantaneous flow of the fluid in the formation is regarded as a collection of a series of steady-state flows. The reservoir porosity and water saturation change dynamically during the development of coalbed methane: the critical desorption pressure is used as the boundary for undersaturated coal reservoirs, and the pressure propagation of the coal reservoir is divided into a gas phase desorption zone and a water phase drainage zone. In the drainage zone, the coal reservoir is compacted by the effective stress effect, and the porosity is reduced. In the desorption zone, since coalbed methane is desorbed from the surface of the coal matrix, the porosity is simultaneously damaged by the effective stress effect and restored by the matrix shrinkage effect. Therefore, the dynamic porosity of coal reservoirs in different regions can be expressed as: In the formula, is the initial reservoir porosity, dimensionless; is the dynamic porosity in the drainage area, dimensionless; is the dynamic porosity in the desorption zone, dimensionless; is the porosity corresponding to the critical desorption pressure of the reservoir pressure, MPa; P g is the gas phase pressure equation of state, MPa; P w is the water phase pressure equation of state, MPa; P i is the initial reservoir pressure, MPa; P cd is the critical desorption pressure, MPa; P L is the Langmuir pressure, MPa; ε max is the maximum volumetric strain, dimensionless; C f is the coal rock compression coefficient, MPa -1 , changes dynamically with the reservoir pressure and can be expressed by the following dynamic equation: C f =0.0026×P n 2 -0.0252×P n +0.1631 (2) Coalbed methane desorption not only causes changes in reservoir porosity, but also occupies a portion of the pore volume, causing the formation water in the pores to be discharged. Therefore, the water saturation of the formation and the reservoir pressure conform to the following relationship: in, A=WGMR(C f +C g +C d ) (4) B=WGMR(C f +C g )+C f +C w (5) C=WGMR+1 (6) Where P n and P n+1 Represent the reservoir microelement pressure at the nth and n+1th steps, MPa; the corresponding S w n and S w n+1 Represent the water saturation at the nth and n+1th steps, dimensionless; S w is the dynamic water saturation, dimensionless; C w is the compressibility of water, MPa -1 ; C d is the desorption compressibility coefficient, MPa -1 ;P sc is the pressure under standard conditions, MPa; Z sc T is the gas deviation factor under standard conditions, dimensionless; sc is the temperature under standard conditions, K; Z is the deviation factor under reservoir conditions, dimensionless; T is the reservoir temperature, K; V L is the Langmuir volume, m 3 / t; ρ is reservoir density, t / m 3 ; WGMR is the mobility ratio of gas and water phases, dimensionless; S wc is the bound water saturation, dimensionless; K rg * is the relative permeability of the gas phase at irreducible water saturation, dimensionless; l and m are the Corey exponents of the gas phase and water phase, dimensionless; μ g and μ w are respectively the gas phase viscosity and the water phase viscosity, mPa·s; C g is the gas compressibility coefficient, MPa -1 , since the gas is an ideal gas, Z = 1, and the gas compressibility coefficient can be expressed as C g =1 / P n .

3. The calculation method for quantitatively predicting the production boundary of a coalbed methane well in an undersaturated reservoir according to claim 1 is characterized in that: The establishment of the prediction model for the pressure drop funnel propagation of undersaturated reservoir coalbed methane wells in step (2) is based on the pressure state equation and material balance equation, combined with the influence of hydraulic fracturing on pressure propagation, and citing the dynamic porosity and dynamic water saturation model: When the reservoir pressure is greater than the critical desorption pressure, the production stage of the coalbed methane well is the single-phase water flow stage. At this time, the water phase pressure state equation in the drainage area can be expressed as: When the pressure drops below the critical desorption pressure, the production stage of the coalbed methane well is the gas-water two-phase flow stage, and the gas phase pressure state equation in the desorption zone can be expressed in the form of pressure square: As the coalbed methane desorbs, the desorption zone gradually increases, and the boundary conditions in the drainage area increase with the increase of the desorption radius. Therefore, the water phase pressure state equation is further transformed into: The above gas / water phase pressure state equation is applicable to radial seepage without pressure fractures, but low permeability coal reservoirs usually use artificial measures such as hydraulic fracturing to improve reservoir permeability. The extension of hydraulic fractures along the principal stress direction will cause the overall pressure drop to propagate to the far well area in an elliptical shape. Therefore, for the convenience of calculation, the conformal transformation method is used to convert the elliptical domain Z(x, y) into the linear domain ζ(ξ, η). The pressure state equations of the gas phase and water phase can be expressed as follows: Where P wf is the bottom hole flowing pressure, MPa; L f is the half length of the fracture, m; r wf is the wellbore radius, m; r i is the drainage radius, m; r cd is the desorption radius, m; r is the pressure propagation radius, m; ξ wf is the wellbore radius in the linear domain, which is infinitesimal compared to the drainage radius and desorption radius, and is set to 0 and dimensionless; ξ i is the drainage radius in the linear domain, dimensionless; ξ cd is the desorption radius in the linear domain, dimensionless; The material balance equation of coal reservoir can also be divided into gas phase material balance equation and water phase material balance equation. In the gas phase material balance equation, the cumulative gas production is equal to the desorption amount of coalbed methane plus the initial free gas amount minus the amount of coalbed methane remaining in the pores, which can be expressed as: B gi =B g |P g =P cd (16) For the material balance equation of the water phase, it is divided into water production in the drainage area and water production in the desorption area. The water production mechanism in different areas is different. The reduction of porosity and elastic expansion of water compressibility in the drainage area lead to water production, which can be expressed as: The water production in the desorption zone also comes from the reduction of water saturation in the pores caused by gas desorption. Therefore, the water production in the desorption zone can be expressed as: In the formula, G P is the cumulative gas production under ground conditions, m 3 ; W P1 is the water yield of the drainage area, m 3 ; W P2 is the water production in the desorption zone, m 3 ; S wi is the initial water saturation, dimensionless; h is the coal seam thickness, m; B g is the gas volume coefficient, dimensionless; B gi is the initial gas volume coefficient, dimensionless; B w is the volume coefficient of formation water. Since formation water is a slightly compressible fluid, it is set to 1 and dimensionless. V is the volume of the coal reservoir affected by the pressure drop funnel, which is converted into the swept volume in the linear domain according to the Jacobian matrix: By introducing the dynamic porosity and water saturation in the coalbed methane development process, combining the pressure state equation and material balance equation at different coalbed methane drainage stages and integrating them, the pressure propagation model can be obtained: 。 4. The calculation method for quantitatively predicting the production boundary of a coalbed methane well in an undersaturated reservoir according to claim 3 is characterized in that: The pressure drop funnel propagation prediction model for undersaturated reservoir coalbed methane wells is given. The drainage radius and desorption radius are further quantitatively predicted by writing a calculation process, which mainly includes the following steps: (1) Determine the dynamic characteristics of the coalbed methane reservoir: Substitute the coalbed methane well geological parameters defined above into the dynamic model to characterize the relationship between the dynamic porosity and dynamic water saturation of the coal reservoir and the reservoir pressure, and apply the dynamic curves to the subsequent calculation process; (2) Determine the basic mathematical method used in the model calculation: use an iterative algorithm to calculate the cumulative gas / water production by substituting the assumed unknown quantities into the method, and continuously optimize the assumed values ​​until the error between the calculated results and the actual production data is less than 1%, which is defined as reasonable accuracy; (3) Calculation of the drainage radius in the single-phase water flow stage: Substitute the actual production data and assume a drainage radius ξ when the bottom hole pressure is greater than the critical desorption pressure. i The value is substituted into the model to describe the dynamic characteristics of the pressure drop funnel in the drainage area and calculate the cumulative water production W P1 , if the calculation result W P1 The actual cumulative water production W Preal If the error between them reaches a reasonable accuracy, the assumed value corresponds to the actual drainage radius, otherwise another drainage radius is assumed and recalculated; (4) Calculate the drainage radius and desorption radius in the gas-water two-phase flow stage: Substitute the actual production data. When the bottom hole pressure is less than the critical desorption pressure, first assume that the desorption radius ξ cd , substituted into the model to characterize the dynamic characteristics of the gas phase pressure drop funnel in the desorption zone and calculate the cumulative gas production G P , by continuously optimizing the assumed value until the calculated result G P The actual cumulative gas production G Preal The error between them reaches a reasonable accuracy; secondly, based on the desorption radius ξ cd Calculate the water production W in the desorption zone using the gas phase pressure drop funnel P2 ; Finally, assume that the drainage radius ξ in the drainage area i , continue to optimize the assumed value until the calculated water production in the drainage area and the water production in the desorption area (W P1 +W P2 ) and the actual cumulative water production W Preal The error between them reaches a reasonable accuracy; (5) Convert the calculation results in the linear domain into the elliptical domain: In order to intuitively obtain the morphological characteristics of pressure propagation in the elliptical domain, define x i and i 、x cd and cd is the maximum and minimum semi-axis lengths of the drainage and desorption zones in the elliptical domain, following: 。 5. The calculation method for quantitatively predicting the production boundary of a coalbed methane well in an undersaturated reservoir according to claim 1 is characterized in that: In the step (4), the complete production data is substituted into the calculation process to draw the dynamic curves of the desorption radius and the drainage radius in the production stage. Specifically, as the coalbed methane is continuously produced, the desorption radius curve gradually increases. As for the drainage radius, since the production boundary is a value to be solved, the quasi-steady-state flow pressure state equation cannot be defined in the model, resulting in the calculated drainage radius curve increasing first and then decreasing. When the drainage radius curve intersects the desorption radius curve at a point, the range corresponding to the intersection is the production boundary of the coalbed methane well.

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