Phase field model-based relative permeability curve calculation method

Abstract

A relative permeability curve calculation method based on a phase field model relates to the technical field of oil and gas development, and comprises the following steps: selecting a core sample to establish a pore scale model, carrying out single-phase flow simulation, calculating the absolute permeability of the pore scale model, and carrying out gas and water two-phase flow simulation. Respectively calculating the gas and water phase fluid distribution volume fraction and the fluid total volume flow rate of the pore scale model in each time interval, respectively calculating the gas and water phase fluid total flow of the pore scale model in each time interval, respectively calculating the volume flow of the gas and water phase fluid in each time interval, and calculating the volume flow of the gas and water phase fluid in each time interval; further calculating corresponding relative gas phase permeability and relative water phase permeability in each time interval, and drawing a relative permeability curve by combining with water saturation; according to the invention, the relative permeability curve can better represent the fluid physical properties of the complex porous medium pore model, so that the seepage characteristics of the fluid in the porous medium model can be accurately obtained.

Description

Technical Field

[0001] The invention relates to the technical field of oil and gas development, and in particular to a relative permeability curve calculation method based on a phase field model. Background Art

[0002] Carbonate gas reservoirs generally have edge and bottom water, and the reservoirs are developed with pores, caves, and fractures, with strong heterogeneity, which makes it easy for such gas reservoirs to experience non-uniform water invasion and violent flooding during their development. Research on the two-phase seepage laws of gas and water in carbonate reservoirs from a microscopic pore scale is of great significance for the dynamic prediction of water invasion at a macroscopic scale and the efficient development of gas fields.

[0003] There are two main methods for obtaining relative permeability curves of gas and water phases: direct measurement and indirect calculation. Direct measurement methods include steady-state method and non-steady-state method. The calculation of steady-state method is based on Darcy equation and is suitable for measuring cores with permeability above 0.5mD. The porosity and permeability ratio of reservoirs with developed fractures is much smaller than that of conventional reservoirs, and it is difficult to determine whether a steady state has been reached. The non-steady-state method has a fast measurement speed, but for cores with strong heterogeneity, the production of gas and water phases is difficult to measure accurately, so the accuracy of this method is low. Indirect calculation methods include capillary pressure curves, mine data, and empirical formula calculation methods.

[0004] Since the cost of physical experiments is high and the experimental results are affected by instrument accuracy and experimental conditions, a numerical simulation method with a wide range of applications and low cost is used to study the gas-water two-phase flow mechanism at the pore scale. Commonly used numerical calculation models for capturing the interface of two-phase fluids include the level set method, the fluid volume method, and the phase field method. When using the level set method for simulation, problems such as mass non-conservation are prone to occur, and the fluid volume meter method is difficult to calculate physical quantities related to the curvature of the two-phase interface. The phase field method has the advantages of simple programming and mass conservation, but the phase field function obtained by the solution only characterizes the distribution of the two-phase fluid interface and cannot obtain the specific production of the two-phase fluid.

[0005] In summary, the existing methods for obtaining relative permeability curves of gas and water phases are mainly aimed at the core scale and are not suitable for carbonate reservoirs with developed fractures and caves. For microscopic pore scale models, it is difficult to obtain more accurate relative permeability curves while tracking the migration interface of gas and water phase fluids and exploring the seepage laws, whether by direct or indirect methods. Summary of the invention

[0006] In view of this, the present invention proposes a relative permeability curve calculation method based on a phase field model, which can be used to accurately and quantitatively analyze and study the results of two-phase flow numerical simulation.

[0007] In order to solve at least one of the above technical problems, the technical solution provided by the present invention is:

[0008] The technical solution adopted by the present invention to solve the above problem is a relative permeability curve calculation method based on a phase field model, comprising the following steps:

[0009] Step S1: Select a core sample to establish a pore scale model, set the pressure boundary of the pore scale model, perform a single-phase flow simulation, use a laminar flow field to calculate the gas flow rate when the pore scale model reaches a steady state, and further calculate the absolute permeability K0 of the pore scale model;

[0010] The calculation method of absolute permeability K0 is shown in formula (1):

[0011]

[0012] In formula (1), K0 is the absolute permeability of the pore-scale model, in mD, and Q0 is the gas flow rate when the single-phase flow of the pore-scale model reaches a steady state, in m 3 / s, μ is the gas viscosity in Pa·s, L is the model length in m, and A is the area of ​​the fluid flow outlet in the pore-scale model in m 3 , △p is the pressure difference between the two ends of the model, in Pa;

[0013] Step S2: For the pore-scale geometric model, the volume of the pore-scale geometric model is calculated as V p , coupling the laminar flow field and phase field model, simulating the gas-water two-phase flow in the pore-scale model, and obtaining the volumes V1 and V2 of the gas-water two-phase fluids in the pore-scale model and the total volume flow rate Q of the fluid in each time interval t based on the simulation results, and calculating the gas phase fluid distribution volume fraction V in each time interval t of the pore-scale model respectively g and the volume fraction V of the aqueous phase fluid distribution in each time interval t w ;

[0014] Among them, the gas phase fluid distribution volume fraction V g and the volume fraction of the aqueous phase V w The calculation formulas are shown in formula (2) and (3) respectively:

[0015]

[0016] In equations (2) and (3), V1 and V2 are the volumes of gas and water phase fluids in the pore-scale model, respectively, in m 3 , V p is the total volume of the pore-scale model, in m 3 , V g and V w is the volume fraction of gas and water two-phase fluid distribution in the pore scale model at each time interval t, dimensionless;

[0017] Step S3: Integrate the total volume flow rate Q of the fluid for each time interval t, and calculate the total flow rate Q1 of the gas and water two-phase fluids in the pore scale model within each time interval t;

[0018] Among them, the calculation formula of the total flow rate Q1 of the gas and water two-phase fluid is shown in formula (4):

[0019] Q1=∫Qdt (4)

[0020] Where Q is the total volume flow rate of the fluid in the pore-scale model in each time interval t, in m 3 / s, Q1 is the total flow rate of gas and water two-phase fluid in the pore scale model in each time interval t, unit is m 3 ;

[0021] Step S4: Calculate the volume flow rate Q of the gas phase fluid in each time interval t g ' and the volume flow rate Q of the water phase fluid in each time interval t w ';

[0022] Step S5: Based on the volume flow rate Q of the gas phase fluid in each time interval t g ' and the volume flow rate Q of the water phase fluid in each time interval t w ', calculate the gas permeability K of the pore-scale model in each time interval t g and water permeability K w , and further calculate the relative gas permeability K corresponding to each time interval t rg and relative water permeability K rw ;

[0023] Step S6: Using the water saturation S corresponding to each time interval t w , relative gas permeability K rg , relative water permeability K rw The relative permeability curve is drawn.

[0024] The technical effects of the present invention are:

[0025] 1. The present invention overcomes the difficulty that the flow values ​​of the gas and water phase fluids cannot be obtained in the phase field model simulation results, and thus the relative permeability of the gas and water phases cannot be calculated. It solves the problem that the method can only qualitatively describe the two-phase flow distribution when simulating the gas and water two-phase flow, but cannot perform quantitative analysis. The obtained relative permeability curve can better characterize the fluid physical properties of the complex porous medium pore model, and thus accurately derive the seepage characteristics of the fluid in the porous medium model.

[0026] 2. In addition to being used to calculate relative permeability curves in microscopic pore scale models, the present invention can also be extended to calculate larger scale models, such as the study of two-phase fluid permeability characteristics in mesoscopic models, and has wide application value. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings required for use in the embodiments will be briefly introduced below. It should be understood that the following drawings only show certain embodiments of the present invention and therefore should not be regarded as limiting the scope. For ordinary technicians in this field, other related drawings can be obtained based on these drawings without paying creative work.

[0028] Figure 1 It is a graph showing the relative permeability of gas and water phases in the present invention. DETAILED DESCRIPTION

[0029] The present invention will be further described in detail below in conjunction with the embodiments and drawings.

[0030] In order to make the purpose, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention. Therefore, the following detailed description of the embodiments of the present invention provided in the drawings is not intended to limit the scope of the invention claimed for protection, but merely represents selected embodiments of the present invention.

[0031] A relative permeability curve calculation method based on a phase field model comprises the following steps:

[0032] Step S1: Select gas as the simulated fluid, refer to the pressure range when measuring the permeability of the actual core sample, and make the pressure condition equivalent according to the ratio of the pore scale model to the actual core sample size, that is, according to the ratio of the pore scale model to the core sample size, proportionally adjust and set the pressure boundary of the numerical simulation model on the basis of the pressure condition range when testing the permeability of the core sample, perform single-phase flow simulation, use laminar flow field calculation to solve the gas flow rate when the pore scale model reaches a steady state, and determine the absolute permeability K0 of the pore scale model according to the Darcy formula;

[0033] The calculation method of absolute permeability K0 is shown in formula (1):

[0034]

[0035] In formula (1), K0 is the absolute permeability of the pore-scale model, in mD, and Q0 is the gas flow rate when the single-phase flow of the pore-scale model reaches a steady state, in m 3 / s, μ is the gas viscosity in Pa·s, L is the model length in m, and A is the area of ​​the fluid flow outlet in the pore-scale model in m 3 , △p is the pressure difference between the two ends of the model, in Pa;

[0036] Step S2: For the pore-scale geometric model, the volume of the pore-scale geometric model is calculated as V p The COMSOL Multiphysics multi-physics simulation software is used to couple the laminar flow field and phase field model to simulate the gas-water two-phase flow in the pore-scale model. Based on the simulation results, the volumes V1 and V2 of the gas-water two-phase fluids in the pore-scale model, as well as the total volume flow rate Q of the fluid in each time interval t, are obtained. The gas phase fluid distribution volume fraction V in the pore-scale model in each time interval t is calculated. g and the volume fraction V of the aqueous phase fluid distribution in each time interval t w ;

[0037] Among them, the gas phase fluid distribution volume fraction V g and the volume fraction of the aqueous phase V w The calculation formulas are shown in formula (2) and (3) respectively:

[0038]

[0039] In equations (2) and (3), V1 and V2 are the volumes of gas and water phase fluids in the pore-scale model, respectively, in m 3 , V p is the total volume of the pore-scale model, in m 3 , V g and V w is the volume fraction of gas and water two-phase fluid distribution in the pore scale model at each time interval t, dimensionless;

[0040] Step S3: Integrate the total volume flow rate Q of the fluid for each time interval t, and calculate the total flow rate Q1 of the gas and water two-phase fluids in the pore scale model within each time interval t;

[0041] Among them, the calculation formula of the total flow rate Q1 of the gas and water two-phase fluid is shown in formula (4):

[0042] Q1=∫Qdt (4)

[0043] Where Q is the total volume flow rate of the fluid in the pore-scale model in each time interval t, in m 3 / s, Q1 is the total flow rate of gas and water two-phase fluid in the pore scale model in each time interval t, unit is m 3 ;

[0044] Step S4: Calculate the volume flow rate Q of the gas phase fluid in each time interval t g ' and the volume flow rate Q of the water phase fluid in each time interval t w ';

[0045] The volume flow rate Q of the gas phase fluid in each time interval t is g ' and the volume flow rate Q of the water phase fluid in each time interval t w The calculation method of ' is shown in formula (5) and (6):

[0046]

[0047] In formula (5) and (6), t is the time interval in seconds, Q g is the total production of gas phase fluid in each time interval t, in m 3 , Q w is the total production of water phase fluid in each time interval t, in m 3 .

[0048] On this basis, the total production of water phase fluid in each time interval t is Q w The calculation formula is shown in formula (7):

[0049] Q w =Q w0 -Q w1 =V p ×(1-V w )=V p ×V g (7)

[0050] In formula (7), V p is the total volume of the pore-scale model, in m 3 , V g is the volume fraction of gas phase fluid distribution, dimensionless, V w is the volume fraction of the aqueous phase, dimensionless, Q w0 is the initial water content of the model, in m 3 ;Q w1 is the water content at each time interval t in the pore-scale model, in m 3 ;

[0051] The total production of gas phase fluid in each time interval t is Q g The calculation formula is shown in formula (8):

[0052] Qg =Q1-Q w (8)

[0053] In formula (8), Q1 is the total flow rate of gas and water two-phase fluid in the pore scale model in each time interval t.

[0054] Step S5: Based on the volume flow rate Q of the gas phase fluid in each time interval t g ' and the volume flow rate Q of the water phase fluid in each time interval t w ', calculate the gas permeability K of the pore-scale model in each time interval t g and water permeability K w , further calculate the relative gas permeability K corresponding to each time interval t rg and relative water permeability K rw ;

[0055] Among them, the pore-scale model gas permeability K g and water permeability K w The calculation formulas are shown in formula (9) and (10) respectively:

[0056]

[0057] In formula (9) and (10), K g and K w are the gas permeability and water permeability of the pore-scale model, in mD, μ is the gas viscosity, in Pa·s, L is the model length, in m, and A is the area of ​​the fluid flow outlet in the pore-scale model, in m 3 , △p is the pressure difference between the two ends of the model, in Pa;

[0058] The relative gas permeability K corresponding to each time interval t rg and relative water permeability K rw The calculation formulas are shown in formula (11) and (12) respectively:

[0059]

[0060] In formula (11) and (12), K rg and K rw are the relative gas permeability and relative water permeability, dimensionless.

[0061] Step S6: Using the water saturation S corresponding to each time interval t w , relative gas permeability K rg , relative water permeability K rw The relative permeability curve is drawn, where the water saturation S wIt is numerically equal to the volume fraction V of the aqueous phase fluid distribution in each time interval t. w .

[0062] Embodiment 1:

[0063] First, the experimental core was selected to establish a pore scale model in the holder, and the absolute permeability K0 of the pore scale model was measured. The length of the pore scale model was 0.001 m, and the area of ​​the outlet was 1.6×10 -9 m 2 , the simulation gas is methane, the gas viscosity is 0.01 mPa·s, and the given pressure boundary is 1×10 4 Pa, single-phase flow simulation was performed, and the gas flow rate when the pore-scale model reached steady state was 2.63×10 -10 m 3 / s, and the absolute permeability K0 of the pore-scale model is determined to be 163.96 mD according to Darcy's formula.

[0064] Then, the gas-water two-phase flow simulation is performed, and the total volume V of the pore-scale model is calculated for the same pore-scale model as above. p is 2.81×10 -12 m 3 The model was saturated with water, the water viscosity was 1 mPa·s, and methane was injected into the model inlet, the gas viscosity was 0.01 mPa·s, and the given pressure boundary was 1×10 4 Pa, two-phase flow simulation is performed, coupling the laminar flow field and phase field model. The volumes V1 and V2 of the gas and water phase fluids in the pore-scale model, as well as the total volume flow rate Q of the fluid in each time interval t, are obtained from the simulation results. The gas phase fluid distribution volume fraction V in the pore-scale model in each time interval t is calculated respectively. g and the volume fraction V of the aqueous phase fluid distribution in each time interval t w The total volume flow rate Q of the fluid is integrated for each time interval t to obtain the total flow rate Q1 of the gas and water phase fluids corresponding to each time interval t in the model. The total flow rate Q1 is divided and calculated to obtain the total production Q of the gas and water phase fluids corresponding to each time interval t in the model. g and Q w , and further obtain the volume flow rate Q of the gas and water two-phase fluid in each time interval t g 'and Q w '. Among them, the total flow rate Q1 of the gas and water two-phase fluid in the pore model and the volume flow rate of the gas and water two-phase fluid corresponding to different measuring points and the calculation results are shown in Table 1. The measuring points in the table are the data calculation points corresponding to each time interval t:

[0065] Table 1 Volume flow calculation results

[0066]

[0067]

[0068] Next, the permeability K of the gas and water phases in the pore-scale model is determined according to the Darcy formula: g and K w , calculate the relative permeability K of gas and water phases rg and K rw , and calculate the water saturation S corresponding to each time interval t w , water saturation S corresponding to different data points w And the relative permeability K of gas and water phases respectively rg , K rw As shown in Table 2, on this basis, using the data in Table 2 combined with water saturation S w The relative permeability curves of gas and water phases are shown in Figure 1 shown.

[0069] Table 2 Phase permeability calculation results

[0070]

[0071]

[0072] In the description of the present invention, it should be pointed out that the terms "upper", "lower", "front", "back", "left", "right", "top", "bottom", "inside" and "outside" etc. indicating directions or positional relationships are based on the directions or positional relationships shown in the accompanying drawings, and are only for the convenience of describing the present invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific direction, be constructed and operated in a specific direction, and should not be understood as a limitation on the present invention.

[0073] The above is only a preferred specific embodiment of the present invention, but the protection scope of the present invention is not limited thereto. Any changes or substitutions that can be easily thought of by a person skilled in the art within the technical scope disclosed in the embodiments of the present invention should be included in the protection scope of the present invention. Therefore, the protection scope of the present invention should be based on the protection scope of the claims.

Claims

1. A relative permeability curve calculation method based on a phase field model, characterized in that: The following steps are involved: Step S1: Select a core sample to establish a pore scale model, set the pressure boundary of the pore scale model, perform a single-phase flow simulation, use a laminar flow field to calculate the gas flow rate when the pore scale model reaches a steady state, and further calculate the absolute permeability K0 of the pore scale model; The calculation method of absolute permeability K0 is shown in formula (1): In formula (1), K0 is the absolute permeability of the pore-scale model, in mD, and Q0 is the gas flow rate when the single-phase flow of the pore-scale model reaches a steady state, in m 3 / s, μ is the gas viscosity in Pa·s, L is the model length in m, and A is the area of ​​the fluid flow outlet in the pore-scale model in m 3 , △p is the pressure difference between the two ends of the model, in Pa; Step S2: For the pore-scale geometric model, the volume of the pore-scale geometric model is calculated as V p , coupling the laminar flow field and phase field model, simulating the gas-water two-phase flow in the pore-scale model, and obtaining the volumes V1 and V2 of the gas-water two-phase fluids in the pore-scale model and the total volume flow rate Q of the fluid in each time interval t based on the simulation results, and calculating the gas phase fluid distribution volume fraction V in each time interval t of the pore-scale model respectively g and the volume fraction V of the aqueous phase fluid distribution in each time interval t w ; Among them, the gas phase fluid distribution volume fraction V g and the volume fraction of the aqueous phase V w The calculation formulas are shown in formula (2) and (3) respectively: In equations (2) and (3), V1 and V2 are the volumes of gas and water phase fluids in the pore-scale model, respectively, in m 3 , V p is the total volume of the pore-scale model, in m 3 , V g and V w is the volume fraction of gas and water two-phase fluid distribution in the pore-scale model at each time interval t, dimensionless; Step S3: Integrate the total volume flow rate Q of the fluid for each time interval t, and calculate the total flow rate Q1 of the gas and water two-phase fluids in the pore scale model within each time interval t; Among them, the calculation formula of the total flow rate Q1 of the gas and water two-phase fluid is shown in formula (4): Q1=∫Qdt (4) Where Q is the total volume flow rate of the fluid in the pore-scale model in each time interval t, in m 3 / s, Q1 is the total flow rate of gas and water two-phase fluid in the pore scale model in each time interval t, unit is m 3 ; Step S4: Calculate the volume flow rate Q of the gas phase fluid in each time interval t g ' and the volume flow rate Q of the water phase fluid in each time interval t w '; Step S5: Based on the volume flow rate Q of the gas phase fluid in each time interval t g ' and the volume flow rate Q of the water phase fluid in each time interval t w ', calculate the gas permeability K of the pore-scale model in each time interval t g and water permeability K w , and further calculate the relative gas permeability K corresponding to each time interval t rg and relative water permeability K rw ; Step S6: Using the water saturation S corresponding to each time interval t w , relative gas permeability K rg , relative water permeability K rw The relative permeability curve is drawn.

2. A relative permeability curve calculation method based on a phase field model according to claim 1, characterized in that: The volume flow rate Q of the gas phase fluid in each time interval t g ' and the volume flow rate Q of the water phase fluid in each time interval t w The calculation method of ' is shown in formula (5) and (6): In formula (5) and (6), t is the time interval in seconds, Q g is the total production of gas phase fluid in each time interval t, in m 3 , Q w is the total production of water phase fluid in each time interval t, in m 3 .

3. A relative permeability curve calculation method based on a phase field model according to claim 2, characterized in that: The total production Q of the aqueous phase fluid in each time interval t w The calculation formula is shown in formula (7): Q w =Q w0 -Q w1 =V p ×(1-V w )=V p ×V g (7) In formula (7), V p is the total volume of the pore-scale model, in m 3 , V g is the volume fraction of gas phase fluid distribution, dimensionless, V w is the volume fraction of the aqueous phase, dimensionless, Q w0 is the initial water content of the model, in m 3 ;Q w1 is the water content at each time interval t in the pore-scale model, in m 3 ; The total production of gas phase fluid in each time interval t is Q g The calculation formula is shown in formula (8): Q g =Q1-Q w (8) In formula (8), Q1 is the total flow rate of gas and water two-phase fluid in the pore scale model in each time interval t.

4. A relative permeability curve calculation method based on a phase field model according to claim 1, characterized in that: The pore-scale model gas permeability K g and water permeability K w The calculation formulas are shown in formula (9) and (10) respectively: In formula (9) and (10), K g and K w are the gas permeability and water permeability of the pore-scale model, in mD, μ is the gas viscosity, in Pa·s, L is the model length, in m, and A is the area of ​​the fluid flow outlet in the pore-scale model, in m 3 , △p is the pressure difference between the two ends of the model, in Pa; The relative gas permeability K corresponding to each time interval t rg and relative water permeability K rw The calculation formulas are shown in formula (11) and (12) respectively: In formula (11) and (12), K rg and K rw are the relative gas permeability and relative water permeability, dimensionless.

5. A relative permeability curve calculation method based on a phase field model according to claim 1, characterized in that: The laminar flow field and phase field models are solved by coupling using COMSOL Multiphysics multi-physics field simulation software.

6. A relative permeability curve calculation method based on a phase field model according to claim 1, characterized in that: The pressure boundary of the pore scale model is adjusted and set in proportion to the size ratio of the pore scale model and the core sample, based on the range of pressure conditions when testing the permeability of the core sample.

7. A relative permeability curve calculation method based on a phase field model according to claim 1, characterized in that: The water saturation S w It is numerically equal to the volume fraction V of the aqueous phase fluid distribution in each time interval t. w .

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