A semi-analytical method for predicting temperature distribution in gas injection reservoirs of underground gas storage
By combining a semi-analytical method with linear heat flow mode and radial convection heat transfer, the problem of unpredictable reservoir temperature distribution after gas injection in underground gas storage facilities was solved, and rapid and accurate temperature distribution prediction was achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- PETROCHINA CO LTD
- Filing Date
- 2024-02-26
- Publication Date
- 2026-07-03
AI Technical Summary
The temperature of the gas injected into the underground gas storage is low, and the temperature of the reservoir drops after the gas is injected. The heat exchange mechanism is complex, and existing technologies make it difficult to quickly and accurately predict the temperature distribution of the reservoir.
A semi-analytical method is adopted, which combines the linear heat flow model to consider the unstable heat conduction effect of the upper and lower caprocks. The Laplace space solution of the reservoir temperature distribution is established by combining radial convection heat transfer and Joule-Thomson effect. The temperature distribution is then quickly predicted by Laplace transform and numerical inversion.
It enables rapid and accurate prediction of the temperature distribution pattern of underground gas storage reservoirs, taking into account the complex heat exchange mechanism between the upper and lower caprocks and the reservoir, thus improving prediction efficiency and accuracy.
Smart Images
Figure CN120544726B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of natural gas storage technology, and in particular to a semi-analytical method for predicting the temperature distribution of injection reservoirs in underground gas storage facilities. Background Technology
[0002] The low temperature of the injected gas in underground gas storage facilities leads to a decrease in reservoir temperature after injection. Temperature is correlated with the gas volume coefficient and is one of the main factors affecting gas reservoir engineering analysis. The heat exchange mechanism within the gas storage reservoir is complex, involving fluid convection heat transfer, reservoir heat conduction, and the heat conduction of the upper and lower caprocks. Summary of the Invention
[0003] To address the aforementioned issues, this invention proposes a semi-analytical method for predicting the temperature distribution of underground gas storage reservoirs. This method considers the unstable thermal conduction effects of the upper and lower caprocks using a linear heat flow model, and takes into account radial convection heat transfer within the reservoir and the Joule-Thomson effect near the wellbore. This results in a Laplace space solution for the reservoir temperature distribution under gas injection conditions, which can quickly predict the temperature distribution patterns of underground gas storage reservoirs.
[0004] The technical solution adopted in this invention is as follows:
[0005] A semi-analytical method for predicting the temperature distribution of gas injection reservoirs in underground gas storage facilities includes the following steps:
[0006] S1. Establish a heat transfer model of the caprock of the underground gas storage, perform Laplace transform based on the initial conditions, and calculate the temperature at the interface between the caprock and the reservoir based on the inner and outer boundary conditions.
[0007] S2. Under constant gas injection mass flow rate, the reservoir energy balance equation is established and simplified by combining the convective heat transfer and throttling effect of the fluid, the heat transfer of the caprock, and the heat conduction of the rock skeleton inside the reservoir. The simplified reservoir energy balance equation is then subjected to Laplace transform to calculate the Laplace space solution of the reservoir temperature distribution, and the reservoir temperature distribution is obtained through Laplace numerical inversion.
[0008] Further, in step S1, the heat transfer model of the caprock of the underground gas storage facility includes governing equations, which include:
[0009]
[0010] Wherein, temperature difference ΔT1=T e -T1, temperature difference ΔT2=T e -T2, where T1 is the reservoir temperature, T2 is the caprock temperature, T e t represents the static temperature of the caprock and reservoir, both in Kelvin; t represents the heat transfer time in seconds; η represents the thermal diffusivity of the caprock. k e′ is the thermal conductivity of the cap layer, in W / (mK); ρ e ′ represents the density of the caprock, in kg / m³ 3 c e ′ represents the specific heat of the caprock, expressed in J / (kg·K).
[0011] Furthermore, the heat transfer model of the cap layer of the underground gas storage facility also includes:
[0012] Initial conditions:
[0013] ΔT2| t＝0 =0 (2)
[0014] Inner boundary conditions:
[0015] ΔT2| x＝0 =ΔT1 (3)
[0016] External boundary conditions:
[0017] ΔT2| x→∞ =0 (4).
[0018] Furthermore, in step S1, the result of performing the Laplace transform based on the initial conditions includes:
[0019] Governing equations:
[0020]
[0021] Inner boundary conditions:
[0022]
[0023] External boundary conditions:
[0024]
[0025] Where s is the Laplace variable, For the Laplace transform of ΔT2, For the Laplace transform of ΔT1.
[0026] Furthermore, in step S1, the method for calculating the temperature at the interface between the caprock and the reservoir based on the inner and outer boundary conditions includes:
[0027] Calculate the general solution of the governing equation, i.e., equation (5):
[0028]
[0029] Where A and B are undetermined coefficients;
[0030] From the outer boundary condition, i.e., equation (7), we know that the coefficient B = 0;
[0031] From the inner boundary conditions, i.e., equation (6), we get:
[0032]
[0033] Solution:
[0034]
[0035] Therefore, at the interface between the caprock and the reservoir, there are:
[0036]
[0037] Furthermore, in step S2, the method for establishing the reservoir energy balance equation includes:
[0038] Taking a ring-shaped micro-element with the wellbore center as the origin in the radial coordinate system, the reservoir energy balance equation at radius r is established:
[0039]
[0040] Among them, the first term on the left side of equation (12) For convective heat transfer, the second term The third term represents the throttling effect. The fourth term represents the thermal conductivity of the rock skeleton within the reservoir. For the heat transfer terms of the upper and lower cover layers; right side of equation (12) h is the reservoir thickness, r is the radius, both in meters; ρ e This refers to the density of the reservoir rock, expressed in kg / m³. 3 c e ρ represents the specific heat of the reservoir rock, in J / (kg·K); φ represents the reservoir porosity; ρ represents the specific heat of the reservoir rock. g Fluid density, in kg / m³ 3 c p The specific heat of natural gas at constant pressure is expressed in J / (kg·K); c J denoted as the Joule-Thomson coefficient for natural gas, in K / MPa; w is the mass flow rate of the injected natural gas, in kg / s, with the direction of the injected natural gas flow being the same as the direction of coordinate r.
[0041] Furthermore, in step S2, the method for simplifying the reservoir energy balance equation includes: under gas injection conditions, convective heat transfer in the reservoir is dominant, so the radial heat conduction effect inside the reservoir is ignored, and the heat conduction term of the rock skeleton inside the reservoir is taken.
[0042] Furthermore, in step S2, the method for simplifying the reservoir energy balance equation also includes:
[0043] Treating the throttling effect as a thermal skin effect, the throttling temperature drop is estimated using the injection-production pressure difference, and the injected gas temperature is corrected. The reservoir energy balance equation, i.e., equation (12), is simplified as follows:
[0044]
[0045] Modified to the form of temperature difference:
[0046]
[0047] Take b = 4πk e , Equation (14) can be simplified to:
[0048]
[0049] Furthermore, in step S2, the method for performing a Laplace transform on the simplified reservoir energy balance equation includes:
[0050] Taking the Laplace transform of equation (15) yields:
[0051]
[0052] Where s is the Laplace variable, For the Laplace transform of ΔT1, For the Laplace transform of ΔT2.
[0053] Furthermore, in step S2, the method for calculating the Laplace space solution of the reservoir temperature distribution includes:
[0054] Substituting equation (11) into equation (16), we get:
[0055]
[0056] Sorted as:
[0057]
[0058] Integrating both sides, we get:
[0059]
[0060] in, This is the corrected injection gas temperature difference after throttling and temperature drop. Let r be the temperature difference at radius r, and we have:
[0061]
[0062] Expanded to:
[0063]
[0064] The corrected injection gas temperature difference after throttling and temperature drop is:
[0065]
[0066] Where, ΔT inj The bottom gas injection temperature difference, ΔT inj =T e -T inj T inj Δp is the bottom hole injection temperature, in Kelvin; Δp is the injection pressure difference, Δp = p wf -p r p wf p is the bottom pressure of the gas injection well. r The reservoir pressure is expressed in MPa.
[0067] Substituting equation (22) into equation (21), we obtain the Laplace space solution for the reservoir temperature distribution:
[0068]
[0069] The real space temperature distribution ΔT1(r) is obtained by Laplace numerical inversion of the calculation results of equation (23).
[0070] The beneficial effects of this invention are as follows:
[0071] This invention considers the unstable thermal conduction effects of the upper and lower caprocks in a linear heat flow model, and takes into account radial convection heat transfer and the Joule-Thomson effect in the near-wellbore zone within the reservoir, forming a Laplace space solution for the reservoir temperature distribution under gas injection conditions. This invention can quickly predict the reservoir temperature distribution pattern of underground gas storage facilities. Attached Figure Description
[0072] Figure 1 This is a flowchart of a semi-analytical method for predicting the temperature distribution of gas injection reservoirs in underground gas storage facilities, according to Embodiment 1 of the present invention.
[0073] Figure 2 This is a schematic diagram of the linear heat transfer coordinates of the cap layer cross section in Embodiment 1 of the present invention.
[0074] Figure 3 This is a schematic diagram of the radial flow coordinates of the reservoir plane in Embodiment 1 of the present invention.
[0075] Figure 4 This is the predicted temperature distribution of the gas injection reservoir in Embodiment 1 of the present invention. Detailed Implementation
[0076] To provide a clearer understanding of the technical features, objectives, and effects of the present invention, specific embodiments are now described. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention; that is, the described embodiments are only a part of the embodiments of the invention, not all of them. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.
[0077] Example 1
[0078] During the flow of injected gas (natural gas injected into the underground gas storage facility) from the high-pressure zone at the bottom of the well to the low-pressure zone, there is a Joule-Thomson effect (i.e., a throttling effect) that causes expansion, heat absorption, and cooling, resulting in heat transfer from the upper and lower caprocks to the reservoir. Affected by fluid convection and caprock heat transfer, the reservoir temperature is in an unstable changing process after gas injection.
[0079] Therefore, this embodiment provides a semi-analytical method for predicting the temperature distribution of gas injection reservoirs in underground gas storage facilities, which can quickly predict the temperature distribution patterns of underground gas storage reservoirs. For example... Figure 1 As shown, the method includes the following steps:
[0080] S1. Establish a heat transfer model of the caprock of the underground gas storage, perform Laplace transform based on the initial conditions, and calculate the temperature at the interface between the caprock and the reservoir based on the inner and outer boundary conditions.
[0081] S2. Under constant gas injection mass flow rate, the reservoir energy balance equation is established and simplified by combining the convective heat transfer and throttling effect of the fluid, the heat transfer of the caprock, and the heat conduction of the rock skeleton inside the reservoir. The simplified reservoir energy balance equation is then subjected to Laplace transform to calculate the Laplace space solution of the reservoir temperature distribution, and the reservoir temperature distribution is obtained through Laplace numerical inversion.
[0082] Preferably, such as Figure 2 As shown, the heat transfer model of the cap layer of an underground gas storage facility includes:
[0083] Governing equations:
[0084]
[0085] Initial conditions:
[0086] ΔT2| t＝0 =0 (2)
[0087] Inner boundary conditions:
[0088] ΔT2| x＝0 =ΔT1 (3)
[0089] External boundary conditions:
[0090] ΔT2| x→∞ =0 (4)
[0091] Wherein, temperature difference ΔT1=T e -T1, temperature difference ΔT2=T e -T2, where T1 is the reservoir temperature, T2 is the caprock temperature, T e t represents the static temperature of the caprock and reservoir, both in Kelvin; t represents the heat transfer time in seconds; η represents the thermal diffusivity of the caprock. k e ′ is the thermal conductivity of the cap layer, in W / (mK); ρ e ′ represents the density of the caprock, in kg / m³ 3 c e ′ represents the specific heat of the caprock, expressed in J / (kg·K).
[0092] It should be noted that the caprock heat transfer model assumes the following: the upper and lower caprocks are symmetrical, the caprock is infinitely large, the temperature at the interface between the caprock and the reservoir is the same, the heat transfer from the caprock to the reservoir is a linear heat flow model, the influence of the geothermal gradient is ignored, and the initial temperature of the caprock and the reservoir is the same.
[0093] Preferably, in step S1, the result of performing the Laplace transform in conjunction with the initial conditions includes:
[0094] Governing equations:
[0095]
[0096] Inner boundary conditions:
[0097]
[0098] External boundary conditions:
[0099]
[0100] Where s is the Laplace variable, For the Laplace transform of ΔT2, For the Laplace transform of ΔT1.
[0101] Preferably, in step S1, the method for calculating the temperature at the interface between the caprock and the reservoir based on the inner and outer boundary conditions includes:
[0102] Calculate the general solution of the governing equation, i.e., equation (5):
[0103]
[0104] Where A and B are undetermined coefficients;
[0105] From the outer boundary condition, i.e., equation (7), we know that the coefficient B = 0;
[0106] From the inner boundary conditions, i.e., equation (6), we get:
[0107]
[0108] Solution:
[0109]
[0110] Therefore, at the interface between the caprock and the reservoir, there are:
[0111]
[0112] Preferably, in step S2, the method for establishing the reservoir energy balance equation includes:
[0113] like Figure 3 As shown, an annular micro-element is considered in the radial coordinate system with the wellbore center as the origin, and an energy balance equation is established at radius r. Under a constant gas injection mass flow rate, considering the convective heat transfer and throttling effect of the fluid, the heat transfer of the caprock, and the heat conduction of the internal rock skeleton of the reservoir, a reservoir energy balance equation is established:
[0114]
[0115] Among them, the first term on the left side of equation (12) For convective heat transfer, the second term The third term represents the throttling effect. The fourth term represents the thermal conductivity of the rock skeleton within the reservoir. For the heat transfer terms of the upper and lower cover layers; right side of equation (12) h is the reservoir thickness, r is the radius, both in meters; ρ e This refers to the density of the reservoir rock, expressed in kg / m³. 3 c e ρ represents the specific heat of the reservoir rock, in J / (kg·K); φ represents the reservoir porosity; ρ represents the specific heat of the reservoir rock. g Fluid density, in kg / m³ 3 c p The specific heat of natural gas at constant pressure is expressed in J / (kg·K); c J denoted as the Joule-Thomson coefficient for natural gas, in K / MPa; w is the mass flow rate of the injected natural gas, in kg / s, with the direction of the injected natural gas flow being the same as the direction of coordinate r.
[0116] Specifically, the mass flow rate w and the standard volumetric flow rate q inj The relationship can be expressed as: w = 0.11574q inj γ g , where q inj This is the standard volumetric flow rate for gas injection, 10 4 m3 / d;γ g This represents the relative density of natural gas.
[0117] It should be noted that the reservoir energy balance equation assumes that the reservoir is horizontally uniform in thickness and homogeneous, the injected gas flows radially in steady state within the reservoir, the reservoir temperature is uniform in the vertical direction, and the influence of reservoir thickness and fluid kinetic energy changes are ignored.
[0118] Preferably, in step S2, the method for simplifying the reservoir energy balance equation includes:
[0119] First, the radial heat conduction effect within the reservoir is ignored. Since convective heat transfer dominates in the reservoir under gas injection conditions, this is taken as...
[0120] Secondly, the throttling effect term is treated as a supplementary boundary condition. Since the throttling effect only has a significant impact within a few meters of the wellbore, it is regarded as a thermal skin effect. The throttling temperature drop is estimated using the injection-production pressure difference, and the gas temperature injected into the reservoir is corrected.
[0121] Therefore, the reservoir energy balance equation, i.e., equation (12), is simplified to:
[0122]
[0123] Modified to the form of temperature difference:
[0124]
[0125] Take b = 4πk e , Equation (14) can be simplified to:
[0126]
[0127] Preferably, in step S2, the method for performing a Laplace transform on the simplified reservoir energy balance equation includes:
[0128] Taking the Laplace transform of equation (15) yields:
[0129]
[0130] Where s is the Laplace variable, For the Laplace transform of ΔT1, For the Laplace transform of ΔT2.
[0131] Preferably, in step S2, the method for calculating the Laplace space solution of the reservoir temperature distribution includes:
[0132] Substituting equation (11) into equation (16), we get:
[0133]
[0134] Sorted as:
[0135]
[0136] Integrating both sides, we get:
[0137]
[0138] in, This is the corrected injection gas temperature difference after throttling and temperature drop. Let r be the temperature difference at radius r, and we have:
[0139]
[0140] Expanded to:
[0141]
[0142] The corrected injection gas temperature difference after throttling and temperature drop is:
[0143]
[0144] Where, ΔT inj The bottom gas injection temperature difference, ΔT inj =T e -T inj T inj Δp is the bottom hole injection temperature, in Kelvin; Δp is the injection pressure difference, Δp = p wf -p r p wf p is the bottom pressure of the gas injection well. r The reservoir pressure is expressed in MPa.
[0145] Substituting equation (22) into equation (21), we obtain the Laplace space solution for the reservoir temperature distribution:
[0146]
[0147] The real space temperature distribution ΔT1(r) is obtained by Laplace numerical inversion of the calculation results of equation (23).
[0148] Specifically, this embodiment uses a gas storage facility as an example to verify the semi-analytical method for predicting the temperature distribution of the gas injection reservoir in an underground gas storage facility. The basic parameters of the gas storage facility are shown in Table 1, and the gas injection flow rate q... inj 150×10 4 m 3 / d, the injection temperature at the bottom of the well is 35℃, and the cumulative injection temperature over 220 days is 3.3×10 8 m 3 .
[0149] Table 1 Basic Parameters of Gas Storage Facility
[0150]
[0151] The reservoir temperature distribution predicted based on the method in this embodiment is as follows: Figure 4 As shown, the radius of the region where the temperature changes significantly under this gas injection condition is less than 120m.
[0152] Example 2
[0153] This embodiment is based on embodiment 1:
[0154] This embodiment provides a computer device, including a memory and a processor. The memory stores a computer program, and the processor executes the computer program to implement the steps of the semi-analytical method for predicting the temperature distribution of the gas injection reservoir in an underground gas storage facility as described in Embodiment 1. The computer program can be in the form of source code, object code, executable file, or some intermediate form.
[0155] Example 3
[0156] This embodiment is based on embodiment 1:
[0157] This embodiment provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the steps of the semi-analytical method for predicting the temperature distribution of an underground gas storage injection reservoir as described in Embodiment 1. The computer program can be in the form of source code, object code, executable file, or some intermediate form. The storage medium includes any entity or device capable of carrying computer program code, a recording medium, a computer memory, a read-only memory (ROM), a random access memory (RAM), an electrical carrier signal, a telecommunication signal, and a software distribution medium, etc. It should be noted that the content contained in the storage medium can be appropriately added to or subtracted according to the requirements of legislation and patent practice in a jurisdiction. For example, in some jurisdictions, according to legislation and patent practice, the storage medium does not include electrical carrier signals and telecommunication signals.
[0158] It should be noted that, for the sake of simplicity, the foregoing method embodiments are described as a series of actions. However, those skilled in the art should understand that this application is not limited to the described order of actions, as some steps may be performed in other orders or simultaneously according to this application. Furthermore, those skilled in the art should also understand that the embodiments described in the specification are preferred embodiments, and the actions and modules involved are not necessarily essential to this application.
Claims
1. A semi-analytical method for predicting the temperature distribution of gas injection reservoirs in underground gas storage facilities, characterized in that, Includes the following steps: S1. Establish a heat transfer model of the caprock of the underground gas storage, perform Laplace transform based on the initial conditions, and calculate the temperature at the interface between the caprock and the reservoir based on the inner and outer boundary conditions. S2. Under constant gas injection mass flow rate, the reservoir energy balance equation is established and simplified by combining the convective heat transfer and throttling effect of the fluid, the heat transfer of the caprock, and the heat conduction of the rock skeleton inside the reservoir. The simplified reservoir energy balance equation is then subjected to Laplace transform to calculate the Laplace space solution of the reservoir temperature distribution, and the reservoir temperature distribution is obtained through Laplace numerical inversion. In step S1, the heat transfer model of the caprock of the underground gas storage facility includes governing equations, which include: （1） Among them, temperature difference , The temperature of the cap layer, The static temperatures of the caprock and reservoir are given in units of 1. ; Heat transfer time, in units of ; The thermal diffusivity of the cap layer is... , The thermal conductivity of the capping layer is given in units of 1000 ppm. ; This refers to the density of the caprock, in units of... ; Specific heat of caprock, in units of ; The cap layer heat transfer model for underground gas storage facilities also includes: Initial conditions: （2） Inner boundary conditions: （3） External boundary conditions: （4） Among them, temperature difference , Reservoir temperature, all units are... ; In step S1, the result of the Laplace transform based on the initial conditions includes: Governing equations: （5） Inner boundary conditions: （6） External boundary conditions: （7） in, For Laplace variables, for Laplace transform, for The Laplace transform of; In step S1, the method for calculating the temperature at the interface between the caprock and the reservoir based on the inner and outer boundary conditions includes: Calculate the general solution of the governing equation, i.e., equation (5): （8） in, , These are coefficients to be determined; From the outer boundary condition, i.e., equation (7), we know that the coefficients ; From the inner boundary conditions, i.e., equation (6), we get: （9） Solution: （10） Therefore, at the interface between the caprock and the reservoir, there are: （11）； In step S2, the methods for establishing the reservoir energy balance equation include: Taking a ring-shaped infinitesimal element with the center of the wellbore as the origin in the radial coordinate system, and establishing the radius... The reservoir energy balance equation at the location is: （12） Among them, the first term on the left side of equation (12) [ [ ] represents the convective heat transfer term, and the second term [ ] ] represents the throttling effect term, the third term [ [This refers to the thermal conductivity term of the rock skeleton within the reservoir, the fourth term] [This refers to the heat transfer term between the upper and lower cover layers; the right side of equation (12)] ; For reservoir thickness, The radius is , and the unit is . ; Density of reservoir rock, in units of ; Specific heat of reservoir rock, in units of ; Reservoir porosity; Fluid density, in units of ; The specific heat of natural gas at constant pressure is expressed in units of 1. ; The Thomson-Joule coefficient for natural gas is given in units of 1000 joules. / MPa ; The mass flow rate of injected natural gas, in units of The direction of natural gas flow and coordinates Same direction; In step S2, the method for simplifying the reservoir energy balance equation includes: under gas injection conditions, convective heat transfer in the reservoir is dominant, so the radial heat conduction effect inside the reservoir is ignored, and the heat conduction term of the internal rock skeleton of the reservoir is taken. ; In step S2, the method for simplifying the reservoir energy balance equation also includes: Treating the throttling effect as a thermal skin effect, the throttling temperature drop is estimated using the injection-production pressure difference, and the gas temperature injected into the reservoir is corrected. The reservoir energy balance equation, i.e., equation (12), is simplified as follows: （13） Modified to the form of temperature difference: （14） Pick , Equation (14) can be simplified to: （15）。 2. The semi-analytical method for predicting the temperature distribution of gas injection reservoirs in underground gas storage facilities according to claim 1, characterized in that, In step S2, the method for performing a Laplace transform on the simplified reservoir energy balance equation includes: Taking the Laplace transform of equation (15) yields: （16） in, For Laplace variables, for Laplace transform, for The Laplace transform of .
3. The semi-analytical method for predicting the temperature distribution of gas injection reservoirs in underground gas storage facilities according to claim 2, characterized in that, In step S2, the method for calculating the Laplace space solution of the reservoir temperature distribution includes: Substituting equation (11) into equation (16), we get: （17） Sorted as: （18） Integrating both sides, we get: （19） in, This is the corrected injection gas temperature difference after throttling and temperature drop. radius The temperature difference at the location, and the following: （20） Expanded to: （21） The corrected injection gas temperature difference after throttling and temperature drop is: （22） in, For the temperature difference of gas injection at the bottom of the well, , This refers to the bottom hole gas injection temperature, in units of... ; For the gas injection pressure difference, , This refers to the bottom pressure of the gas injection well. The reservoir pressure is expressed in units of 1 / 2. ; Substituting equation (22) into equation (21), we obtain the Laplace space solution for the reservoir temperature distribution: （23） The real-space temperature distribution is obtained by performing Laplace numerical inversion on the calculation results of equation (23). .