A calculation method for dynamic propagation of pressure drop funnel in undersaturated coalbed methane reservoirs

Abstract

The present invention discloses a calculation method for the dynamic propagation of the pressure drop funnel of an undersaturated coalbed methane reservoir, which specifically includes the following steps: (1) based on the development mode of coalbed methane well drainage and pressure reduction, combined with the properties of the undersaturated coal reservoir, the pressure drop funnel propagation stages of the undersaturated coalbed methane reservoir are divided; (2) based on the model assumption that the coal reservoir contains gas / water two-phase fluid and the fluid seepage follows an instantaneous steady-state flow, the model auxiliary equation is introduced, the internal / external boundary conditions are substituted, and a prediction model for the propagation of different types of pressure drop funnels in the undersaturated coalbed methane reservoir is established; (3) the calculation process of the model is compiled: the reservoir geological parameters and the complete production data of the production well are substituted, the dynamic characteristics of the physical properties of the coal reservoir and the actual production boundary of the coalbed methane well are determined, and the dynamic propagation characteristics of the pressure drop funnel of the coalbed methane well are characterized. The present invention completes the characterization of the pressure propagation of the coalbed methane reservoir by combining the calculation method with the prediction model, and applies it to the optimization of the drainage system of the coalbed methane well, verifying its advanced nature and practicality.

Description

Technical Field

[0001] The invention belongs to the technical field of coalbed methane development and utilization, and particularly relates to a method for calculating the dynamic propagation of a pressure drop funnel in an undersaturated coalbed methane reservoir. Background Art

[0002] Unlike conventional oil and gas reservoirs, the unique occurrence mechanism and physical properties of coalbed methane (CBM) reservoirs necessitate a drainage-and-depressurization approach to CBM development: By artificially reducing bottomhole flow pressure, formation water is expelled from coal seam fractures and pores. The reservoir pressure gradually drops below the critical desorption pressure, allowing CBM adsorbed on the coal matrix surface to desorb and be produced. This indicates that CBM production occurs via pressure depletion, and sufficient pressure reduction is crucial for efficient CBM well development. Therefore, accurately describing the propagation patterns of the pressure drop funnel in CBM wells during CBM development is crucial for evaluating the dynamic production potential of coal reservoirs and regulating artificial intervention measures.

[0003] The method for calculating the dynamic propagation of the pressure drop funnel in coalbed methane reservoirs combines the material balance equation with the pressure equation of state, taking into account the influence of factors such as free gas saturation, effective stress effects, matrix shrinkage effects, and fracturing cracks. However, this current method can only characterize the dynamic propagation of the pressure drop funnel in infinitely large coal seams, or simply estimate the production boundary as half the well spacing under well network development conditions, and achieve quantitative characterization of the pressure drop funnel on this basis. In actual production, the production boundary is affected by various factors such as geology, engineering, and drainage, and the production boundary is not always equal to half the well spacing. Inaccurate estimates often mislead the division of coalbed methane well production stages, resulting in large calculation errors. Therefore, the above method for characterizing the dynamic propagation of the pressure drop funnel in coalbed methane reservoirs needs to be improved.

[0004] In summary, there is currently a lack of a calculation method for the dynamic propagation of the pressure drop funnel in undersaturated coalbed methane reservoirs at different production stages that considers the dynamic changes of coal reservoir parameters and combines the actual production boundaries. Summary of the Invention

[0005] To address the shortcomings of existing technologies, the present invention provides a fully quantitative method for calculating the dynamic propagation of the pressure drop funnel in coalbed methane wells. Based on the pressure equation of state and the material balance equation, this method determines the actual location of the coalbed methane production boundary and rationally divides production stages. By combining the dynamic changes in coal reservoir parameters and the impact of hydraulic fracturing on pressure propagation, a prediction model for the dynamic propagation of the pressure drop funnel in undersaturated coalbed methane reservoirs is established. By programming a computational process, refined data calculations are achieved, ultimately enabling field application.

[0006] To achieve the above object, the present invention proposes a method for calculating the dynamic propagation of the pressure drop funnel in an undersaturated coalbed methane reservoir, comprising the following steps:

[0007] (1) Based on the development mode of CBM well dewatering and pressure reduction and the adsorption / desorption properties of undersaturated coal reservoirs, the pressure drop funnel propagation stages of undersaturated CBM reservoirs are divided;

[0008] (2) Based on the model assumptions that the coal reservoir is a dual-porosity medium, the pores contain only gas / water two-phase fluids, and the fluid seepage follows an instantaneous steady-state flow, the dynamic porosity / water saturation model, the fluid pressure state equation / material balance equation, the isothermal adsorption and desorption equation, and the model auxiliary equations of the fracturing fracture control model are introduced. By substituting the internal / external boundary conditions, a prediction model for the propagation of different types of pressure drop funnels in undersaturated coalbed methane reservoirs is established.

[0009] (3) Write the calculation process of the model: substitute the reservoir geological parameters and the complete production data of the production wells, determine the actual production boundary during the coalbed methane production process, and characterize the dynamic propagation characteristics of the pressure drop funnel in different production stages of the coalbed methane well.

[0010] Furthermore, the critical desorption pressure of an undersaturated coal reservoir is lower than the initial reservoir pressure. There are two pressure propagation radii during coalbed methane development: one is the drainage radius, which is the distance reached by the reservoir pressure disturbance, and the corresponding range is the drainage zone; the other is the desorption radius, which is the distance from the reservoir pressure drop to the critical desorption pressure, and the corresponding range is the desorption zone. As the bottomhole flowing pressure continues to drop, the coalbed methane well begins to produce water and gas. At the same time, the drainage radius and desorption radius propagate toward the far well area at different speeds until they reach the production boundary. Therefore, based on the coupling relationship between the drainage radius, desorption radius, production boundary, bottomhole flowing pressure, and boundary pressure, the entire production cycle of the coalbed methane well can be divided into five stages:

[0011] Stage a: The bottom hole flowing pressure is less than the initial reservoir pressure and greater than the critical desorption pressure. The coalbed methane has not yet started to desorb, and the desorption radius has not expanded. However, the drainage radius gradually expands with water production but has not reached the boundary.

[0012] Stage b: The bottomhole pressure is less than the initial reservoir pressure and greater than the critical desorption pressure. The drainage radius reaches the boundary and the boundary pressure gradually decreases. This situation can usually be achieved when the bottomhole pressure decreases slowly in the early stage or the coal seam has good conductivity.

[0013] Stage c: The bottomhole flowing pressure drops rapidly to below the critical desorption pressure, causing the coalbed methane to begin desorption before the drainage radius reaches the production boundary. In this stage, the drainage radius and the desorption radius expand outward simultaneously. According to the relative permeability theory, gas and water compete with each other for seepage channels in the fracture system, and the pressure propagation of the gas and water phases will restrict each other.

[0014] Stage d: The bottomhole pressure decreases slowly. When the drainage radius reaches the production boundary and the boundary pressure gradually decreases, the bottomhole pressure drops below the critical desorption pressure. After that, the desorption radius expands and the gas production increases accordingly.

[0015] Stage e: The desorption radius propagates to the production boundary, the overall reservoir pressure drops below the critical desorption pressure, the inter-well pressure interference is formally formed, and the reservoir pressure drops rapidly, promoting the desorption and production of coalbed methane;

[0016] Since stage b and stage c are the result of the bottom hole pressure decreasing at different rates, the production cycle of a coalbed methane well does not go through all five stages. Therefore, the production types of coalbed methane wells can be divided into two types: abde type and acde type.

[0017] Furthermore, the assumptions of the prediction model are as follows: (1) the coal reservoir is a dual-porosity medium, the pores are the main adsorption space for coalbed methane, and the cleats are the main seepage channels for the fluid; (2) the fluid in the coal reservoir consists of two phases: gas and water, the coalbed methane is an ideal gas, and the coalbed water is a slightly compressible fluid; (3) the instantaneous flow of formation fluid can be regarded as a collection of a series of steady-state flows.

[0018] The outer boundary conditions of the model are constant pressure boundaries and closed boundaries based on the drainage radius, desorption radius, and boundary pressure. The inner boundary conditions are boundaries corresponding to the bottomhole flow pressure. During the development of coalbed methane, the reservoir porosity and water saturation will change dynamically. The dynamic porosity can be expressed as:

[0019]

[0020] C f =0.0026×P n 2 -0.0252×P n +0.1631 (2)

[0021] Dynamic water saturation can be expressed as:

[0022]

[0023] in,

[0024] A=WGMR(C f +C g +C d ) (4)

[0025] B=WGMR(C f +C g )+C f +C w (5)

[0026] C=WGMR+1 (6)

[0027]

[0028]

[0029] Where, 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, MPa; ε max is the maximum volumetric strain, dimensionless; 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; P represents the reservoir pressure, MPa; P n and P n+1 Represent the reservoir microelement pressure at step n and step n+1, 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 d is the desorption compressibility coefficient, MPa -1 ; C f is the coal rock compression coefficient, MPa -1 ; C w is the compressibility of water, MPa -1 ; C g is the gas compressibility coefficient, MPa -1 , under ideal gas conditions, it can be expressed as C g =1 / P n ;P sc is the pressure under standard conditions, MPa; Z sc is the gas deviation factor under standard conditions, dimensionless; T sc is the temperature under standard conditions, K; Z is the deviation factor under reservoir conditions, which is equal to 1 for ideal gas and 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 gas-water two-phase mobility ratio, 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, respectively; μ g and μ w are the gas phase viscosity and water phase viscosity, mPa·s, respectively.

[0030] Furthermore, the state of pressure propagation is different in different production stages: in stages a and b, the pressure propagates only in the drainage area; in stages c and d, the pressure propagates simultaneously in the drainage and desorption areas; and in stage e, the pressure propagates only in the desorption area. In addition, hydraulic fractures cause the pressure drop to propagate toward the far wellbore in an elliptical shape with the principal stress direction as the axis. To facilitate calculation, the conformal transformation method is used to convert the elliptical domain Z(x, y) into the linear domain ζ(ξ, η). In summary, based on the coupling relationship between the drainage radius, desorption radius, bottomhole flowing pressure, and production boundary, the pressure state equations in different production stages can be expressed as:

[0031] Phase a:

[0032]

[0033] Phase b:

[0034]

[0035] Phase c:

[0036]

[0037] Phase d:

[0038]

[0039] Phase e:

[0040]

[0041] Where, P wf is the bottom hole pressure, MPa; P e is the boundary pressure, MPa; ξ i is the drainage radius in the linear domain, dimensionless; ξ is the pressure drop funnel propagation radius in the linear domain, dimensionless; cd is the desorption radius in the linear domain, dimensionless; ξ wf is the wellbore radius in the linear domain, which is infinitesimal compared to the drainage radius / desorption radius, and is set to 0 and dimensionless; R e is the production boundary range in the linear domain and is dimensionless.

[0042] Furthermore, the unique enrichment mechanism of coalbed methane leads to different seepage and production mechanisms of the gas / water phase: the source of gas production in coalbed methane wells is mainly the desorption of adsorbed gas plus the production of free gas (dissolved gas is not considered); the source of water production in coalbed methane wells can be divided into water production in the drainage area and water production in the desorption area. Water production in the drainage area comes from the reduction of porosity and elastic expansion of formation water, while water production in the desorption area comes not only from the dynamic change of porosity and elastic expansion of formation water, but also from the desorption of gas and the occupation of pore space, resulting in a decrease in water saturation and the expulsion of some formation water. Therefore, combined with the dynamic porosity / water saturation model and the isothermal adsorption and desorption equation, the gas / water phase material balance equation at different production stages can be expressed as follows:

[0043] Phase a, b:

[0044]

[0045] Stages c, d, e:

[0046]

[0047] in:

[0048]

[0049]

[0050] Considering the influence of hydraulic fracturing, the volume in the linear domain is expressed as:

[0051]

[0052] Where G P is the cumulative gas production under surface conditions, m 3 ;W P is the cumulative water production, 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; L f is the half length of the fracture, m; h is the thickness of the coal seam, 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, which is set to 1 and dimensionless because formation water is a slightly compressible fluid; A is the area of ​​coal reservoir affected by the pressure drop funnel, m 2 ; V is the volume of coal reservoir affected by the pressure drop funnel, m 3 .

[0053] Furthermore, by substituting the pressure state balance equation into the corresponding material balance equation, a dynamic propagation model of the pressure drop funnel in different production stages of undersaturated coalbed methane reservoirs can be established:

[0054] Phase a, b:

[0055]

[0056] Stages c, d, e:

[0057]

[0058] in:

[0059]

[0060] Furthermore, the dynamic curve of the pressure drop funnel is further characterized by writing a calculation process, which mainly includes the following steps:

[0061] (1) Determine the mathematical method for model calculation: Since the drainage radius, desorption radius, and boundary pressure are unknown quantities, while the bottom hole pressure and cumulative gas / water production are known quantities, the iterative method is the optimal calculation method. The assumed unknown quantities are continuously optimized and substituted into the model for calculation until the error between the calculated cumulative gas / water production and the actual known quantity is less than 1%, which is defined as reasonable accuracy;

[0062] (2) Determine the dynamic changes of reservoir parameters: characterize dynamic curves based on dynamic porosity and water saturation models and apply them to subsequent calculation processes;

[0063] (3) Determine the actual production boundary: Since the production boundary is a parameter to be solved, the complete production data is substituted into stages a and c. The drainage radius and desorption radius are solved by iteration and the curves are drawn. The intersection of the two curves is the production boundary position and formation time. For the convenience of observation, x is defined as i and y i 、x cd and y cd are the maximum and minimum semi-axis lengths of the drainage and desorption zones in the elliptical domain. The following formula is used to convert the linear domain parameters into parameters in the elliptical domain:

[0064]

[0065]

[0066] (4) Based on the calculated production boundary and the coupling relationship between the bottom hole flow pressure and the critical desorption pressure, the production stages and the corresponding pressure drop funnel types are finely divided, and the drainage radius, desorption radius and boundary pressure parameters of different production stages are calculated using the iterative method: in stages a and b, the drainage radius ξ is assumed to be iand boundary pressure R e , substitute into the calculation process to obtain the cumulative water production W in the drainage area P1 And compared with the actual cumulative water production W Preal Compare and continuously optimize the assumed value until the error meets reasonable accuracy; in stages c and d, first assume the desorption radius ξ cd , substitute into the calculation process to obtain the cumulative water production W in the desorption zone P2 and cumulative gas production G P , compared with G P The actual cumulative gas production G Preal , continuously optimize the assumed value until the error meets the reasonable accuracy, and then assume the drainage radius ξ i and boundary pressure R e , calculate the cumulative water production W in the drainage area P1 , compared with the total water production in the whole area (W P1 +W P2 ) and the actual cumulative water production W Preal , continuously optimize the assumed value until the error meets reasonable accuracy; in stage e, assume that the boundary pressure R e , substitute into the calculation process to obtain the cumulative gas production G in the desorption zone P and the actual cumulative gas production G Preal By comparison, the assumed value is continuously optimized until the error meets reasonable accuracy; finally, the calculated results are substituted into the pressure state equation to characterize the dynamic propagation characteristics of the pressure drop funnel.

[0067] By adopting the above technical solution, the present invention has the following technical effects:

[0068] (1) By introducing the dynamic porosity and water saturation model of coalbed methane reservoirs, considering the influence of hydraulic fracturing cracks on the propagation of pressure drop funnel, and combining the characteristics of the gas / water phase pressure state equation and material balance equation at different drainage stages of coalbed methane wells, a prediction model for the propagation of pressure drop funnel in undersaturated coalbed methane reservoirs was established. The calculation process was compiled to complete the establishment of the calculation method.

[0069] (2) By using the established method and substituting actual geological and production data, the production boundaries of CBM wells are precisely predicted, the production stages of CBM wells are accurately divided, and the dynamic propagation of the pressure drop funnel in each production stage is characterized;

[0070] (3) Based on this calculation method, the coalbed methane well production system is optimized to verify the advancement and practicality of this calculation method. BRIEF DESCRIPTION OF THE DRAWINGS

[0071] Figure 1 Schematic diagram of the dynamic propagation calculation method for the pressure drop funnel in undersaturated coalbed methane reservoirs;

[0072] Figure 2 Schematic diagram of pressure drop funnel propagation in stage a of coalbed methane well;

[0073] Figure 3 Schematic diagram of pressure drop funnel propagation in stage b of coalbed methane well;

[0074] Figure 4 Schematic diagram of pressure drop funnel propagation in stage c of coalbed methane well;

[0075] Figure 5 Schematic diagram of pressure drop funnel propagation in stage d of coalbed methane well;

[0076] Figure 6 Schematic diagram of pressure drop funnel propagation in stage e of coalbed methane well;

[0077] Figure 7 Model calculation flow chart;

[0078] Figure 8 Actual drainage curve of case well A;

[0079] Figure 9 Actual drainage curve of case well B;

[0080] Figure 10 Water saturation dynamic curves of two case wells;

[0081] Figure 11 Porosity dynamic curves of two case wells;

[0082] Figure 12 Schematic diagram of the dynamic propagation of the pressure drop funnel in case well A;

[0083] Figure 13 Schematic diagram of the dynamic propagation of the pressure drop funnel in case well B;

[0084] Figure 14 Comparison chart of quantitative optimization of drainage and production system of case well B;

[0085] Figure 15 Numerical simulation history matching and production capacity prediction results of case well B;

[0086] Figure 16 Production boundary prediction curve in the elliptical domain after optimizing the drainage and production system of case well B;

[0087] Figure 17 Schematic diagram of the dynamic propagation of the pressure drop funnel after optimizing the drainage and production system in Case Well B. DETAILED DESCRIPTION

[0088] To make the objectives, technical solutions and advantages of the present invention more clear, the embodiments of the present invention will be described in further detail below with reference to the accompanying drawings.

[0089] Due to the development method of coalbed methane well drainage and pressure reduction, the coal reservoir pressure is the core element of the "occurrence-drainage-desorption-diffusion-seepage-production" mass transfer mechanism during coalbed methane development. The pressure drop funnel is a formal representation of the dynamic propagation of reservoir pressure. Therefore, a detailed description of the propagation characteristics of the coal reservoir pressure drop funnel is the key to evaluating the production capacity potential of production wells and realizing efficient coalbed methane development. Figure 1 The present invention proposes a method for calculating the dynamic propagation of the pressure drop funnel of an undersaturated coalbed methane reservoir, which specifically includes the following steps:

[0090] (1) Based on the development mode of CBM well dewatering and pressure reduction and the adsorption / desorption properties of undersaturated coal reservoirs, the pressure drop funnel propagation stages of undersaturated CBM reservoirs are divided;

[0091] (2) Based on the model assumptions that the coal reservoir is a dual-porosity medium, the pores contain only gas / water two-phase fluids, and the fluid seepage follows an instantaneous steady-state flow, the dynamic porosity / water saturation model, the fluid pressure state equation / material balance equation, the isothermal adsorption and desorption equation, and the model auxiliary equations of the fracturing fracture control model are introduced. By substituting the internal / external boundary conditions, a prediction model for the propagation of different types of pressure drop funnels in undersaturated coalbed methane reservoirs is established.

[0092] (3) Write the calculation process of the model: substitute the reservoir geological parameters and the complete production data of the production wells, determine the actual production boundary during the coalbed methane production process, and characterize the dynamic propagation characteristics of the pressure drop funnel in different production stages of the coalbed methane well.

[0093] The critical desorption pressure of an undersaturated coal reservoir is lower than the initial reservoir pressure. During coalbed methane development, there are two pressure propagation radii: one is the drainage radius, which is the distance reached by the reservoir pressure disturbance, and the corresponding range is the drainage zone; the other is the desorption radius, which is the distance from the reservoir pressure drop to the critical desorption pressure, and the corresponding range is the desorption zone. As the bottomhole flow pressure continues to drop, the coalbed methane well begins to produce water and gas. At the same time, the drainage radius and desorption radius propagate toward the far well area at different speeds until they reach the production boundary. After that, the boundary pressure gradually decreases. Therefore, based on the coupling relationship between the drainage radius, desorption radius, production boundary, bottomhole flow pressure, and boundary pressure, the entire production cycle of the coalbed methane well can be divided into five stages:

[0094] Stage a: The bottom hole pressure is less than the initial reservoir pressure and greater than the critical desorption pressure. The coalbed methane has not yet begun to desorb, and the desorption radius has not expanded. However, the drainage radius gradually expands with water production but has not reached the boundary. Figure 2 As shown;

[0095] Stage b: The bottom hole pressure is less than the initial reservoir pressure and greater than the critical desorption pressure. The drainage radius reaches the boundary and the boundary pressure gradually decreases. This situation can usually be achieved when the bottom hole pressure decreases slowly in the early stage or the coal seam has good conductivity. Figure 3 As shown;

[0096] Stage c: The bottomhole pressure drops rapidly to below the critical desorption pressure, resulting in the desorption of coalbed methane before the drainage radius reaches the production boundary. In this stage, the drainage radius and the desorption radius expand outward at the same time. According to the relative permeability theory, gas / water compete with each other for seepage channels in the fracture system, and the pressure propagation of gas / water phases will restrict each other, such as Figure 4 As shown;

[0097] Stage d: The bottom hole pressure drops slowly. When the drainage radius reaches the production boundary and the boundary pressure gradually decreases, the bottom hole pressure drops below the critical desorption pressure. After that, the desorption radius expands and the gas production increases accordingly. Figure 5 As shown;

[0098] (5) Stage e: The desorption radius propagates to the production boundary, the overall reservoir pressure drops below the critical desorption pressure, the inter-well pressure interference is formally formed, and the reservoir pressure drops rapidly, promoting the desorption and production of coalbed methane. Figure 6 As shown;

[0099] Since stage b and stage c are the result of the bottom hole pressure decreasing at different rates, the production cycle of a coalbed methane well does not go through all five stages. Therefore, the production types of coalbed methane wells can be divided into two types: abde type and acde type.

[0100] The assumptions of the prediction model are as follows: (1) the coal reservoir is a dual-porosity medium, where the pores are the main adsorption space for coalbed methane and the cleats are the main seepage channels for the fluid; (2) the fluid in the coal reservoir consists of two phases, gas and water, with coalbed methane being an ideal gas and coalbed water being a slightly compressible fluid; (3) the instantaneous flow of formation fluid can be regarded as a collection of a series of steady-state flows; the outer boundary conditions of the model are a constant pressure boundary and a closed boundary based on the drainage radius, desorption radius, and boundary pressure, and the inner boundary conditions are the boundaries corresponding to the bottomhole flow pressure; during the development of coalbed methane, the reservoir porosity and water saturation will change dynamically, where the dynamic porosity can be expressed as:

[0101]

[0102] C f =0.0026×P n 2 -0.0252×P n +0.1631 (25)

[0103] Dynamic water saturation can be expressed as:

[0104]

[0105] in,

[0106] A=WGMR(C f +C g +C d ) (27)

[0107] B=WGMR(C f +C g )+C f +C w (28)

[0108] C=WGMR+1 (29)

[0109]

[0110]

[0111] Where, 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, MPa; ε max is the maximum volumetric strain, dimensionless; 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; P represents the reservoir pressure, MPa; P n and P n+1 Represent the reservoir microelement pressure at step n and step n+1, 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 d is the desorption compressibility coefficient, MPa -1 ; C f is the coal rock compression coefficient, MPa -1 ; C w is the compressibility of water, MPa -1 ; C g is the gas compressibility coefficient, MPa -1 , under ideal gas conditions, it can be expressed as C g =1 / P n ;P sc is the pressure under standard conditions, MPa; Z scis the gas deviation factor under standard conditions, dimensionless; T sc is the temperature under standard conditions, K; Z is the deviation factor under reservoir conditions, which is equal to 1 for ideal gas and 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 gas-water two-phase mobility ratio, 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, respectively; μ g and μ w are the gas phase viscosity and water phase viscosity, mPa·s, respectively.

[0112] The pressure propagation state varies across production stages: in stages a and b, pressure propagates only within the drainage zone; in stages c and d, pressure propagates simultaneously within the drainage and desorption zones; and in stage e, pressure propagates only within the desorption zone. Furthermore, hydraulic fracturing causes the pressure drop to propagate elliptically toward the far-away wellbore, centered on the principal stress direction. For ease of calculation, a conformal transformation is used to convert the elliptical domain Z(x,y) into the linear domain ζ(ξ,η). Based on the coupling relationship between the drainage radius, desorption radius, bottomhole pressure, and production boundary, the pressure state equations for different production stages can be expressed as:

[0113] Phase a:

[0114]

[0115] Phase b:

[0116]

[0117] Phase c:

[0118]

[0119] Phase d:

[0120]

[0121] Phase e:

[0122]

[0123] Where, P wf is the bottom hole pressure, MPa; P e is the boundary pressure, MPa; ξ is the pressure drop funnel propagation radius in the linear domain, dimensionless; i is the drainage radius in the linear domain, dimensionless; ξ cdis the desorption radius in the linear domain, dimensionless; ξ wf is the wellbore radius in the linear domain, which is infinitesimal compared to the drainage radius / desorption radius, and is set to 0 and dimensionless; R e is the production boundary range in the linear domain and is dimensionless.

[0124] The unique enrichment mechanism of coalbed methane leads to different seepage and production mechanisms of gas / water phases: the source of gas production in coalbed methane wells is mainly the desorption of adsorbed gas plus the production of free gas (dissolved gas is not considered); the source of water production in coalbed methane wells can be divided into water production in the drainage zone and water production in the desorption zone. Water production in the drainage zone comes from the reduction of porosity and elastic expansion of formation water, while water production in the desorption zone comes not only from the dynamic change of porosity and elastic expansion of formation water, but also from the desorption of gas and the occupation of pore space, resulting in a decrease in water saturation and the expulsion of some formation water. Therefore, combined with the dynamic porosity / water saturation model and the isothermal adsorption and desorption equation, the gas / water phase material balance equation at different production stages can be expressed as follows:

[0125] Phase a, b:

[0126]

[0127] Stages c, d, e:

[0128]

[0129] in:

[0130]

[0131]

[0132] Considering the influence of hydraulic fracturing, the volume in the linear domain is expressed as:

[0133]

[0134] Where G P is the cumulative gas production under surface conditions, m 3 ;W P is the cumulative water production, 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; L f is the half length of the fracture, m; h is the thickness of the coal seam, m; B g is the gas volume coefficient, dimensionless; B gi is the initial gas volume coefficient, dimensionless; B wis the volume coefficient of formation water, which is set to 1 and dimensionless because formation water is a slightly compressible fluid; A is the area of ​​coal reservoir affected by the pressure drop funnel, m 2 ; V is the volume of coal reservoir affected by the pressure drop funnel, m 3 .

[0135] Substituting the pressure state balance equation into the corresponding material balance equation, a dynamic propagation model of the pressure drop funnel in different production stages of undersaturated coalbed methane reservoirs can be established:

[0136] Phase a, b:

[0137]

[0138] Stages c, d, e:

[0139]

[0140] in:

[0141]

[0142] The CBM pressure equation of state and material balance equations were derived considering the impact of hydraulic fracturing fractures and different production stages. Combined with dynamic porosity and water saturation models, a dynamic propagation model of the pressure drop funnel in undersaturated CBM reservoirs was established. The dynamic curve of the pressure drop funnel was further characterized by writing a calculation process, which mainly includes the following steps:

[0143] (1) Determine the mathematical method for model calculation: Since the drainage radius, desorption radius, and boundary pressure are unknown quantities, while the bottom hole pressure and cumulative gas / water production are known quantities, the iterative method is the optimal calculation method. The assumed unknown quantities are continuously optimized and substituted into the model for calculation until the error between the calculated cumulative gas / water production and the actual known quantity is less than 1%, which is defined as reasonable accuracy;

[0144] (2) Determine the dynamic changes of reservoir parameters: characterize dynamic curves based on dynamic porosity and water saturation models and apply them to subsequent calculation processes;

[0145] (3) Determine the actual production boundary: Since the production boundary is a parameter to be solved, the complete production data is substituted into stages a and c. The drainage radius and desorption radius are solved by iteration and the curves are drawn. The intersection of the two curves is the production boundary position and formation time. For the convenience of observation, x is defined as i and y i 、x cd and y cd are the maximum and minimum semi-axis lengths of the drainage and desorption zones in the elliptical domain. The following formula is used to convert the linear domain parameters into parameters in the elliptical domain:

[0146]

[0147]

[0148] (4) Based on the calculated production boundary and the coupling relationship between the bottom hole flow pressure and the critical desorption pressure, the production stages and the corresponding pressure drop funnel types are finely divided. The geological parameters and production data are substituted and the drainage radius, desorption radius and boundary pressure parameters of different production stages are calculated using the iterative method, such as Figure 7 As shown: Stage a and b assume drainage radius ξ i and boundary pressure R e , substitute into the calculation process to obtain the cumulative water production W in the drainage area P1 And compared with the actual cumulative water production W Preal Compare and continuously optimize the assumed value until the error meets reasonable accuracy; in stages c and d, first assume the desorption radius ξ cd , substitute into the calculation process to obtain the cumulative water production W in the desorption zone P2 and cumulative gas production G P , compared with G P The actual cumulative gas production G Preal , continuously optimize the assumed value until the error meets the reasonable accuracy, and then assume the drainage radius ξ i and boundary pressure R e , calculate the cumulative water production W in the drainage area P1 , compared with the total water production in the whole area (W P1 +W P2 ) and the actual cumulative water production W Preal , continuously optimize the assumed value until the error meets reasonable accuracy; in stage e, assume that the boundary pressure R e , substitute into the calculation process to obtain the cumulative gas production G in the desorption zone P and the actual cumulative gas production G Preal By comparison, the assumed value is continuously optimized until the error meets reasonable accuracy; finally, the calculated results are substituted into the pressure state equation to characterize the dynamic propagation characteristics of the pressure drop funnel.

[0149] Two CBM wells with a spacing of about 250 meters in the Shizhuang South block in the southern Qinshui Basin were selected as examples to conduct a precise prediction of the dynamic propagation of the pressure drop funnel in the undersaturated CBM reservoir. Well A has a production time of about 2500 days and an average daily gas production of 1500m3. 3 / d, CBM production has gone through the stages of increasing production, stable production, and exhaustion. Bottom hole pressure and water production gradually decreased in the early stage of production, and remained stable at a low level in the later stage. Figure 8 As shown in Figure 2, the rapid drop in bottom hole pressure in Well B in the early stage caused the daily gas production to quickly reach 2000m3 after the start of gas production. 3 / d, and then as the bottom hole pressure and water production remained at a low level, the daily gas production dropped rapidly to 1000m3 / d or less, indicating that the drainage system of the well is unreasonable, resulting in the inability of the coalbed methane to maintain high and stable production. Figure 9 Through well logging / testing data, literature review, experimental testing, and numerical simulation history matching adjustments, the basic geological parameters of the two wells were accurately obtained, as shown in Table 1.

[0150] Table 1. Basic geological parameters of target wells

[0151]

[0152] Based on the established model, the geological and production data are substituted into the calculation process, and the dynamic propagation of the pressure drop funnel of the two case wells is characterized according to the model calculation steps. First, the dynamic water saturation and dynamic porosity of the two case wells are calculated respectively, as shown in the following example: Figure 10 、 11 As shown. The dynamic curves of the drainage radius and desorption radius were further solved to obtain production boundary data. In the linear domain, the production boundary range of Well A can reach 1.25, while the production boundary range of Well B is only 0.99. According to Equations 19 and 20, the production boundary in the linear domain is converted to the elliptical domain. The long semi-axis of Well A reaches 120m, approximately equal to half the well spacing, indicating that the production boundary of Well A is a pressure interference boundary. However, the long semi-axis of Well B is only 80m, far less than half the well spacing, indicating that the boundary of Well B is a damage boundary.

[0153] like Figure 12 、 13 As shown in the figure, the model is substituted into the calculation process to characterize the pressure drop funnel characteristics of the case wells. For Well A, the drainage radius gradually expands outward until it reaches the production boundary before 40 days of production. The boundary pressure then gradually decreases. During this period, the bottomhole flowing pressure remains above the critical desorption pressure. After 100 days, the bottomhole flowing pressure drops below the critical desorption pressure, and the boundary pressure has dropped to near the critical desorption pressure. Thereafter, the desorption radius gradually expands outward. After 600 days, the desorption radius propagates to the production boundary, the boundary pressure gradually decreases, and the overall reservoir pressure within the well control range drops below the critical desorption pressure, and the coalbed methane reaches a stable production stage. The production stage of Well A is an abde type. For Well B, during the drainage radius expansion stage, the bottomhole pressure had dropped below the critical desorption pressure, and coalbed methane began to desorb. The competition between the gas / water phases in the seepage channel limited the full expansion of the drainage radius, and the pressure spread to the damage boundary. After 40 days, the bottomhole pressure and the boundary pressure dropped synchronously, and the desorption radius gradually expanded outward. After 300 days, the desorption radius spread to the damage boundary, and the overall reservoir pressure within the well control range dropped below the critical desorption pressure. The production stage of Well B is acde type.

[0154] Unreasonable working system restricts the high and stable production of coalbed methane wells. In order to fully expand the pressure drop funnel, the drainage system was optimized by reducing the rate of decline of the bottom hole pressure of Well B 300 days ago, such as Figure 14 The production and drainage systems before and after optimization are substituted into the COMET numerical simulation software for historical matching and capacity prediction, as shown in Figure 15 As shown in the figure, the production capacity characteristics of Well B under different working systems are completely different: although the daily gas production increases slowly and the peak value is lower than the actual value after the optimized working system, the daily gas production exceeds the actual daily gas production after 200 days of production. In addition, the daily gas production in the later period is generally high, which can maintain high and stable production. The production boundary range is larger after optimization, with a production boundary of 1.2 in the linear domain and a major semi-axis of up to 108m in the elliptical domain. However, it takes longer to reach the boundary, about 600 days, as shown in the figure. Figure 16 As shown. Substitute the simulated production data into the model to further calculate the dynamic propagation of the coal reservoir pressure drop funnel after optimizing the working system. Figure 17 As shown in the figure, after the optimization of the working system, the production stage of the coalbed methane well changed from the ABDE type to the ACDE type, indicating that the propagation of the pressure drop funnel of the ACDE type is more conducive to the high and stable production of the coalbed methane well. When artificially intervening in the drainage system, the decline rate of the bottom hole flowing pressure should be slowed down in the early stage.

[0155] This embodiment does not impose any formal restrictions on the shape, material, structure, etc. of the present invention. Any simple modifications, equivalent changes and modifications made to the above embodiments based on the technical essence of the present invention are within the scope of protection of the technical solution of the present invention.

Claims

1. A method for calculating the dynamic propagation of the pressure drop funnel in an undersaturated coalbed methane reservoir, characterized in that: The method comprises the following steps: (1) Based on the development mode of CBM well dewatering and pressure reduction and the adsorption / desorption properties of undersaturated coal reservoirs, the pressure drop funnel propagation stages of undersaturated CBM reservoirs are divided; (2) Based on the model assumptions that the coal reservoir is a dual-porosity medium, the pores contain only gas / water two-phase fluids, and the fluid seepage follows an instantaneous steady-state flow, the dynamic porosity / water saturation model, the fluid pressure state equation / material balance equation, the isothermal adsorption and desorption equation, and the model auxiliary equations of the fracturing fracture control model are introduced. By substituting the internal / external boundary conditions, a prediction model for the propagation of different types of pressure drop funnels in undersaturated coalbed methane reservoirs is established. (3) Compile the calculation process of the model: Substitute the reservoir geological parameters and the complete production data of the production wells to determine the dynamic characteristics of the coal reservoir physical properties and the actual production boundaries of the coalbed methane wells, and characterize the dynamic propagation characteristics of the pressure drop funnel in different production stages of the coalbed methane wells; The prediction model for the propagation of different types of pressure drop funnels in undersaturated coalbed methane reservoirs is based on the following assumptions: (1) Coal reservoirs are dual-porosity media, where pores are the main adsorption space for coalbed methane and cleats are the main seepage channels for fluids; (2) The fluid in the coal reservoir consists of gas and water phases, with coalbed methane being an ideal gas and coalbed water being a slightly compressible fluid; (3) The instantaneous flow of formation fluid can be regarded as a collection of a series of steady-state flows; The outer boundary conditions of the model are constant pressure boundaries and closed boundaries based on the drainage radius, desorption radius, and boundary pressure, while the inner boundary conditions are boundaries corresponding to the bottomhole flowing pressure. Based on the coupling relationship between the drainage radius, desorption radius, bottomhole flowing pressure, and production boundary, combined with the fracturing fracture control model, the pressure state equations at different production stages can be expressed as follows: Phase a: Phase b: Phase c: Phase d: Phase e: Where, 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 wf is the bottom hole pressure, MPa; P e is the boundary pressure, MPa; ξ is the pressure drop funnel propagation radius in the linear domain, dimensionless; i is the drainage radius in the linear domain, dimensionless; ξ cd is the desorption radius in the linear domain, dimensionless; ξ wf is the wellbore radius in the linear domain, which is infinitesimal compared to the drainage radius / desorption radius, and is set to 0 and dimensionless; R e is the production boundary range in the linear domain, dimensionless; The unique enrichment mechanism of coalbed methane leads to different seepage and production mechanisms of the gas / water phase: the source of gas production in coalbed methane wells is mainly the desorption of adsorbed gas plus the production of free gas, without considering dissolved gas; the source of water production in coalbed methane wells can be divided into water production in the drainage zone and water production in the desorption zone. Water production in the drainage zone comes from the reduction of porosity and elastic expansion of formation water, while water production in the desorption zone comes not only from the dynamic change of porosity and elastic expansion of formation water, but also from the desorption of gas and the occupation of pore space, resulting in a decrease in water saturation and the expulsion of some formation water. Therefore, combined with the dynamic porosity / water saturation model and the isothermal adsorption and desorption equation, the gas / water phase material balance equation at different production stages can be expressed as follows: Phase a, b: Stages c, d, e: in: Considering the influence of hydraulic fracturing, the volume in the linear domain is expressed as: Where, is the initial reservoir porosity, dimensionless; is the dynamic porosity in the drainage area, dimensionless; is the dynamic porosity in the desorption zone, dimensionless; C w is the compressibility of water, MPa -1 ; G P is the cumulative gas production under surface conditions, m 3 ;W P is the cumulative water production, 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; S w is the dynamic water saturation, dimensionless; Z is the deviation factor under reservoir conditions, Z for ideal gas is equal to 1, dimensionless; T is the reservoir temperature, K; P L is the Langmuir pressure, MPa; V L is the Langmuir volume, m 3 / t; ρ is reservoir density, t / m 3 ;L f is the half length of the fracture, m; h is the thickness of the coal seam, 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, which is set to 1 and dimensionless because formation water is a slightly compressible fluid; A is the area of ​​coal reservoir affected by the pressure drop funnel, m 2 ; V is the volume of coal reservoir affected by the pressure drop funnel, m 3 ; Substituting the pressure state balance equation into the corresponding material balance equation, a dynamic propagation model of the pressure drop funnel in different production stages of undersaturated coalbed methane reservoirs can be established: Phase a, b: Stages c, d, e: in:

2. The method for calculating the dynamic propagation of the pressure drop funnel of an undersaturated coalbed methane reservoir according to claim 1 is characterized in that: The division of the coalbed methane reservoir pressure drop funnel propagation stage proposed in step (1) is as follows: The critical desorption pressure of an undersaturated coal reservoir is lower than the initial reservoir pressure. During coalbed methane development, there are two pressure propagation radii: one is the drainage radius, which is the distance reached by the reservoir pressure disturbance, and the corresponding range is the drainage zone; the other is the desorption radius, which is the distance from the reservoir pressure drop to the critical desorption pressure, and the corresponding range is the desorption zone. As the bottomhole flowing pressure continues to drop, the coalbed methane well begins to produce water and gas. At the same time, the drainage radius and desorption radius propagate toward the far well area at different speeds until they reach the production boundary. After that, the boundary pressure gradually decreases. Therefore, based on the coupling relationship between the drainage radius, desorption radius, production boundary, bottomhole flowing pressure, and boundary pressure, the entire production cycle of the coalbed methane well can be divided into five stages: Stage a: The bottom hole flowing pressure is less than the initial reservoir pressure and greater than the critical desorption pressure. The coalbed methane has not yet started to desorb, and the desorption radius has not expanded. However, the drainage radius gradually expands with water production but has not reached the boundary. Stage b: The bottomhole pressure is less than the initial reservoir pressure and greater than the critical desorption pressure. The drainage radius reaches the boundary and the boundary pressure gradually decreases. This situation can usually be achieved when the bottomhole pressure decreases slowly in the early stage or the coal seam has good conductivity. Stage c: The bottomhole flowing pressure drops rapidly to below the critical desorption pressure, causing the coalbed methane to begin desorption before the drainage radius reaches the production boundary. In this stage, the drainage radius and the desorption radius expand outward simultaneously. According to the relative permeability theory, gas and water compete with each other for seepage channels in the fracture system, and the pressure propagation of the gas and water phases will restrict each other. Stage d: The bottomhole pressure decreases slowly. When the drainage radius reaches the production boundary and the boundary pressure gradually decreases, the bottomhole pressure drops below the critical desorption pressure. After that, the desorption radius expands and the gas production increases accordingly. Stage e: The desorption radius propagates to the production boundary, the overall reservoir pressure drops below the critical desorption pressure, the inter-well pressure interference is formally formed, and the reservoir pressure drops rapidly, promoting the desorption and production of coalbed methane.

3. The method for calculating the dynamic propagation of the pressure drop funnel of an undersaturated coalbed methane reservoir according to claim 1 is characterized in that: The undersaturated coalbed methane reservoir pressure drop funnel propagation prediction model describes the pressure drop funnel dynamic curve by writing a calculation process, and mainly includes the following steps: (1) Determine the mathematical method for model calculation: bottom hole flowing pressure and cumulative gas / water production are known quantities, while drainage radius, desorption radius, and boundary pressure are unknown quantities. An iterative method is used to optimize the assumed unknown quantities and substitute them into the model for calculation until the error between the calculated cumulative gas / water production and the actual known quantities is less than 1%; (2) Determine the dynamic changes of reservoir parameters: characterize dynamic curves based on dynamic porosity and water saturation models and apply them to subsequent calculation processes; (3) Determine the actual production boundary: The production boundary is the parameter to be solved; substitute the complete production data into stages a and c, use the iterative method to solve the drainage radius and desorption radius respectively and draw the curves. The intersection of the two curves is the production boundary position and formation time; define the maximum semi-axis length and minimum semi-axis length of the drainage area and desorption area in the elliptical domain respectively, and convert the linear domain parameters into parameters within the elliptical domain; (4) Based on the calculated production boundary and the coupling relationship between bottomhole flowing pressure and critical desorption pressure, the production stages and corresponding pressure drop funnel types are finely divided, and the drainage radius, desorption radius and boundary pressure parameters of different production stages are calculated using an iterative method: In stages a and b, the drainage radius ξ is assumed to be i and boundary pressure R e , substitute into the calculation process to obtain the cumulative water production W in the drainage area P1 And compared with the actual cumulative water production W Preal Compare and continuously optimize the assumed value until the error meets reasonable accuracy; In stages c and d, we first assume that the desorption radius ξ cd , substitute into the calculation process to obtain the cumulative water production W in the desorption zone P2 And the cumulative gas production G P , compared with G P The actual cumulative gas production G Preal , continuously optimize the assumed value until the error meets the reasonable accuracy, and then assume the drainage radius ξ i and boundary pressure R e , calculate the cumulative water production W in the drainage area P1 , compared with the total water production of the whole area (W P1 +W P2 ) and the actual cumulative water production W Preal , continuously optimize the hypothesis value until the error meets reasonable accuracy; Stage e assumes the boundary pressure R e , substitute into the calculation process to obtain the cumulative gas production G in the desorption zone P and the actual cumulative gas production G Preal By comparison, the assumed value is continuously optimized until the error meets reasonable accuracy; finally, the calculated results are substituted into the pressure state equation to characterize the dynamic propagation characteristics of the pressure drop funnel.

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