An analytical solution method for heat transfer performance of a waste oil well converted into a geothermal well

Abstract

This application provides an analytical solution method for the heat transfer performance of abandoned oil wells converted into geothermal wells. The method obtains the geometric parameters, physical property parameters, and calculation parameters of the target oil and gas well, and determines the derived geometric quantities of the borehole and U-tube. Based on the physical property parameters, it determines the thermal diffusivity of multiple segments within the target oil and gas well. Based on the simulation time step in the calculation parameters, it determines the total simulation duration and discretizes this total simulation duration into multiple time steps. Based on the calculation parameters and the linear geothermal gradient formula, it initializes each temperature field. Based on the current local fluid temperature, it uses a temperature-dependent fluid thermophysical property fitting model to determine the temperature parameters. Based on the heat transfer physical process of the buried pipe and the current local fluid temperature, it determines the thermal resistance network. Using a piecewise finite linear heat source model and under-relaxation iterative updates, it finally outputs the outlet water temperature, total heat transfer, heat transfer intensity per unit length, and the temperature distribution along the pipe at the target time, thus solving for the heat transfer performance of the target oil and gas well.

Description

Technical Field

[0001] This application relates to the field of geothermal energy development and utilization technology, and in particular to an analytical solution method for the heat exchange performance of abandoned oil wells converted into geothermal wells. Background Technology

[0002] Globally, numerous abandoned oil and gas wells exist, typically at depths of 1000-5000 meters, with bottom-hole temperatures reaching 60-200°C. Traditionally, these wells are sealed off, a costly and wasteful practice that also wastes significant geothermal resources. Relevant technologies offer the possibility of converting abandoned oil and gas wells into geothermal wells, utilizing circulating fluids within the wellbore to extract underground heat energy, thus achieving both economic and environmental benefits.

[0003] The evaluation of heat transfer performance of abandoned oil and gas wells converted into geothermal wells is the core issue of the engineering of this technology. Existing technologies mainly include the following categories: (1) Numerical simulation methods based on finite element or finite volume methods, which are highly accurate but time-consuming and complex to model, and are not suitable for rapid engineering evaluation and large-scale parameter optimization; (2) Methods based on analytical or semi-analytical solutions, such as infinitely long line heat source models, finitely long line heat source models and cylindrical heat source models, which are fast to calculate but mostly assume uniform thermal properties of rock and soil and cannot consider the layered characteristics of actual strata; (3) Lumped parameter methods based on thermal resistance networks, which are simple but have low accuracy and cannot capture the axial temperature distribution inside the wellbore.

[0004] The main shortcomings of existing technologies are: (1) Most assume that the soil and rock are homogeneous media and ignore the significant differences in thermal conductivity, specific heat capacity and density caused by the change of lithology at different depths in the actual strata; (2) The average physical property parameters are usually used in the linear heat source model, which leads to a large deviation in the calculation of borehole wall temperature; (3) The calculation of thermal resistance of each layer in the wellbore (convection in the pipe, heat conduction in the pipe wall, grouting layer and cement ring) mostly adopts simplified empirical formulas and lacks accurate calculation based on the actual thermal properties and flow state of the fluid; (4) Most analytical models do not consider the thermal short-circuit effect between the two pipes of the U-shaped pipe, which leads to an overestimation of the outlet pipe temperature and an overestimation of the heat exchange. Summary of the Invention

[0005] This application provides an analytical solution method for the heat exchange performance of geothermal wells converted from abandoned oil wells. This method solves the problems of overestimating the outlet pipe temperature and overestimating the heat exchange in existing technologies. By calculating the complete borehole thermal resistance network and comprehensively analyzing the thermal short-circuit effect between U-shaped pipes, the method achieves an accurate solution for the heat exchange performance of geothermal wells.

[0006] To achieve the above objectives, embodiments of this application provide an analytical solution method for determining the heat exchange performance of abandoned oil wells converted into geothermal wells. The method includes: Obtain the geometric parameters, physical property parameters, and calculation parameters of the target oil and gas well, and determine the derived geometric quantities of the borehole and U-tube based on the geometric parameters; The thermal diffusivity of multiple segments in the target oil and gas well is determined based on the physical property parameters; the total simulation duration is determined based on the simulation time step in the calculation parameters, and the total simulation duration is discretized into multiple time steps; Based on the calculation parameters, each temperature field is initialized using the linear geothermal gradient formula. Based on the current local fluid temperature, temperature parameters are determined using a temperature-dependent fluid thermal property fitting model; and a thermal resistance network is determined based on the heat transfer physical process of the buried pipe and the current local fluid temperature. A piecewise finite linear heat source model is used to construct the temperature response matrix of the borehole wall of the target oil and gas well, and the borehole wall temperature is obtained. Based on the borehole wall temperature and the fluid energy balance equation, the friction temperature distribution along the downpipe and riser is iteratively solved in each time step. The borehole wall temperature is updated using under-relaxation iteration, and the change in wellhead water temperature is used as the convergence criterion. If convergence is determined, the water temperature, total heat transfer, heat transfer intensity per unit length, and temperature distribution along the borehole at each time point are output.

[0007] In one possible implementation, the method further includes: Obtain the layered physical property parameters from the layered data file; The determination of the thermal diffusivity of multiple segments in the target oil and gas well based on the aforementioned physical property parameters includes: The layered physical property parameters are mapped to multiple segments by linear interpolation, and the thermal diffusivity of each segment is determined.

[0008] In one possible implementation, the geometric parameters are well depth, borehole diameter, casing outer diameter, casing wall thickness, U-tube outer diameter, U-tube wall thickness, tube center distance, and U-tube type; the U-tube type is a single U-tube, double U-tube, triple U-tube, or quadruple U-tube downhole heat exchanger. The physical properties are thermal conductivity of soil and rock, specific heat capacity of soil and rock, density of soil and rock, thermal conductivity of cement, thermal conductivity of PE pipe and thermal conductivity of steel; the calculation parameters are inlet water temperature, circulation flow rate, surface temperature, geothermal gradient, simulation time step, time step, number of segments and convergence tolerance. The derived geometric quantities are casing inner diameter, casing inner radius, borehole radius, cement sheath thickness, U-tube inner diameter, U-tube inner radius, U-tube outer radius, single-tube flow area, discrete depth coordinates, and segment length.

[0009] In one possible implementation, the method further includes: When the U-tube type is a double U-tube, triple U-tube, or quadruple U-tube downhole heat exchanger, the total flow rate of the target oil and gas well is evenly distributed to multiple equivalent single U-tubes using the equivalent pipe method. The target heat exchange performance of each of the multiple equivalent single U-tubes is solved, and the total heat exchange performance is obtained by superimposing the multiple target heat exchange performances. The outlet water temperature is determined based on the mixing temperature of the multiple equivalent single U-tubes.

[0010] In one possible implementation, the temperature parameters are density, specific heat capacity, thermal conductivity, and dynamic viscosity; the thermal resistance network consists of convection thermal resistance inside the pipe, thermal conductivity thermal resistance of the pipe wall, thermal resistance of the grouting layer, thermal resistance of the cement ring, and thermal short-circuit thermal resistance between pipes.

[0011] In one possible implementation, the fluid thermal property fitting model is a polynomial function of temperature; wherein the density is a cubic polynomial of temperature, the specific heat capacity is a cubic polynomial of temperature, the thermal conductivity is a quadratic polynomial of temperature, and the dynamic viscosity is an Arrhenius exponential function.

[0012] In one possible implementation, the piecewise finite line heat source model determines the response of each segment using a series expansion of an exponential integral function, and approximates the responses of adjacent segments using a finite line heat source midpoint based on a complementary error function; and the non-uniformity of the layered soil and rock is handled by the cross-layer geometric mean thermal conductivity and geometric mean thermal diffusivity.

[0013] In one possible implementation, the under-relaxation iterative update includes the following steps: solving for the temperature distribution in the downcomer; applying a temperature continuity condition at the bottom; solving for the temperature distribution in the riser; updating the heat transfer per unit length of each segment based on the current fluid temperature; updating the borehole wall temperature by calling the segmented finite line heat source model; and using under-relaxation techniques to update the borehole wall temperature to improve iterative stability.

[0014] In one possible implementation, the relaxation factor of the under-relaxation iteration ranges from 0.5 to 0.9; the convergence tolerance in the convergence criterion ranges from [missing value]. to .

[0015] In one possible implementation, the method is implemented using MATLAB or Python programming; the method includes a graphical user interface for parameter input, real-time calculation progress display, result visualization, and data export.

[0016] The technical solutions provided in this application embodiment have at least the following technical effects or advantages: (1) Significantly improved computational efficiency: Based on the piecewise finite line heat source analytical model, the calculation time for full-depth, long-cycle heat transfer simulation is reduced from several hours in numerical simulation to tens of seconds, making large-scale parameter sensitivity analysis and design optimization feasible; (2) Layered soil and rock adaptability; By supporting the loading and interpolation of layered soil and rock thermal property data, it can accurately reflect the influence of lithological changes at different depths on heat transfer performance under actual geological conditions. The geometric mean processing of cross-layer thermal properties ensures the physical consistency of the response matrix. (3) Complete borehole thermal resistance network; taking into account convection inside the pipe, heat conduction of the pipe wall, thermal resistance of the grouting layer (including heat flow deflection effect), thermal resistance of the cement ring and thermal short-circuit effect between U-shaped pipes, the thermal resistance calculation adopts an accurate formula based on local fluid temperature and flow state, avoiding the error caused by empirical constants; (4) Temperature dependence of fluid properties; The temperature-dependent fluid thermal property fitting model is adopted to automatically reflect the property changes caused by temperature changes during the flow of fluid in the wellbore, which improves the calculation accuracy of deep well (large temperature change) conditions. (5) Multi-U tube expansion capability; Supports single-U to four-U tube downhole heat exchangers through the equivalent pipe method, with a unified solution framework and good scalability; (6) Good numerical stability; by adopting logarithmic time steps and under-relaxed iteration strategy, the numerical stability and computational efficiency in the long-term quasi-steady-state stage are ensured while guaranteeing the accuracy of early transient solutions. (7) Good engineering practicality; it can be implemented through programming on a computing platform, providing a graphical user interface for easy operation by engineering technicians, supporting parameterized batch calculation, result visualization, and data export functions in multiple formats (CSV, Excel, MAT), and can be directly embedded into the geothermal engineering design workflow. Attached Figure Description

[0017] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the description of the embodiments of this application or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0018] Figure 1 A flowchart illustrating an analytical solution method for determining the heat exchange performance of a geothermal well converted from an abandoned oil well, as provided in an embodiment of this application. Figure 2 A schematic diagram of a wellbore structure for converting an abandoned oil well into a geothermal well, provided in an embodiment of this application; Figure 3 A water temperature distribution curve along the pipeline is provided as an embodiment of this application; Figure 4 A real-time inlet and outlet water temperature change graph provided in this application embodiment; Figure 5 A heat exchange rate curve over time is provided as an embodiment of this application; Figure 6 This is a graph showing the change of heat transfer intensity over time, provided as an embodiment of this application. Detailed Implementation

[0019] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of this application. All other embodiments obtained by those skilled in the art based on the embodiments of this application without creative effort are within the scope of protection of this application.

[0020] In the description of the embodiments of this application, it should be noted that the terms "center," "upper," "lower," "left," "right," "vertical," "horizontal," "inner," and "outer," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are used only for the convenience of describing the embodiments of this application and for simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on this application. The terms "first," "second," and "third" are used for descriptive purposes only and should not be construed as indicating or implying relative importance. Furthermore, the terms "installed," "connected," and "linked" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; a mechanical connection or an electrical connection; a direct connection or an indirect connection through an intermediate medium; or a connection within two components. Those skilled in the art can understand the specific meaning of the above terms in the embodiments of this application according to the specific circumstances.

[0021] Figure 1 This is a flowchart illustrating an analytical solution method for determining the heat exchange performance of a geothermal well converted from an abandoned oil well, as provided in an embodiment of this application. Figure 1 As shown, the method may include the following steps.

[0022] S101. Obtain the geometric parameters, physical property parameters, and calculation parameters of the target oil and gas well, and determine the derived geometric quantities of the borehole and U-tube based on the geometric parameters.

[0023] In some embodiments, the geometric parameter is the well depth. Drilling diameter Outer diameter of the casing Casing wall thickness U-shaped tube outer diameter U-shaped pipe wall thickness Pipe center distance And the U-tube type; this U-tube type is a single U-tube, double U-tube, triple U-tube, or quadruple U-tube downhole heat exchanger. The default parameters are suitable for typical abandoned oil well renovation scenarios with 177.8mm casing and 32mm PE-U-tube. This physical property parameter is the thermal conductivity of the soil / rock. Specific heat capacity of rock and soil Density of soil and rock Cement thermal conductivity PE pipe thermal conductivity and the thermal conductivity of steel The calculation parameter is the inlet water temperature. Circulation flow Surface temperature geothermal gradient Simulation time step Time step Number of segments N and convergence tolerance The derived geometric quantity is the inner diameter of the casing. Inner radius of the casing Drilling radius Cement ring thickness U-shaped tube inner diameter U-shaped tube inner radius U-shaped tube outer radius Single-tube flow area Discrete depth coordinates and segment length .

[0024] S102. Determine the thermal diffusivity of multiple segments in the target oil and gas well based on the physical property parameter; determine the total simulation duration based on the simulation time step in the calculation parameter, and divide the total simulation duration into multiple time steps.

[0025] For example, this physical property parameter can be a uniform physical property parameter. The total simulation time can be expressed as: ;in, Total simulation duration, This is for simulating the time step.

[0026] S103. Based on the calculation parameters, initialize each temperature field using the linear geothermal gradient formula.

[0027] For example, the original geothermal temperature of the strata can be used as the initial value for each temperature field, and calculations can be performed based on a linear geothermal gradient: ;in, For depth The original temperature of the strata at that location (°C); Surface temperature (°C); The geothermal gradient is expressed in °C / 100m.

[0028] S104. Based on the current local fluid temperature, determine the temperature parameters using a temperature-dependent fluid thermal property fitting model; and determine the thermal resistance network based on the heat transfer physical process of the buried pipe and the current local fluid temperature.

[0029] S105. Using a piecewise finite line heat source model, construct the temperature response matrix of the borehole wall of the target oil and gas well to obtain the borehole wall temperature; based on the borehole wall temperature and the fluid energy balance equation, iteratively solve the temperature distribution along the downcomer and riser in each time step.

[0030] S106. The borehole wall temperature is updated using under-relaxation iteration, and the change in wellhead water temperature is used as the convergence criterion. If convergence is determined, the water temperature, total heat transfer, heat transfer intensity per unit length, and temperature distribution along the borehole at the target time are output.

[0031] For example, the total heat exchange (Unit: W): ; Heat transfer intensity per unit length (Unit: W / m): The outlet water temperature at each moment is expressed as follows: The water temperature along the downcomer is expressed as follows: The riser pipe friction water temperature is expressed as follows: The borehole wall temperature is expressed as: .

[0032] Based on the above technical solution, the convection inside the pipe, the heat conduction of the pipe wall, the thermal resistance of the grouting layer (including the heat flow deflection effect), the thermal resistance of the cement ring, and the thermal short-circuit effect between the U-shaped pipes can be comprehensively considered. The thermal resistance calculation adopts an accurate formula based on the local fluid temperature and flow state, avoiding the error caused by empirical constants. The logarithmic time step and under-relaxation iteration strategy are adopted to ensure the numerical stability and computational efficiency in the long-term quasi-steady-state stage while ensuring the accuracy of the early transient solution.

[0033] In one possible implementation, the method further includes: acquiring layered physical property parameters from a layered data file; mapping the layered physical property parameters to multiple segments using linear interpolation, and determining the thermal diffusivity of each segment. For example, layer depth, thermal conductivity, density, and specific heat capacity data can be read from an Excel or CSV file, and the layered physical properties can be mapped to each segment using linear interpolation to determine the thermal diffusivity of each segment. The layered data file may contain four columns of data: depth (m), thermal conductivity (W / (m·K)), density (kg / m³), and specific heat capacity (J / (kg·K)). The program automatically sorts the data by depth and performs interlayer linear interpolation. The thermal diffusivity is determined using the following formula: ;in, Where is the thermal diffusivity, The thermal conductivity of the soil and rock is... For soil and rock density, This refers to the specific heat capacity of the rock and soil.

[0034] In one possible implementation, the method may further include: when the U-tube type is a double U-tube, triple U-tube, or quadruple U-tube downhole heat exchanger, using the equivalent pipe method to evenly distribute the total flow of the target oil and gas well to multiple equivalent single U-tubes, solving the target heat exchange performance of each of the multiple equivalent single U-tubes, and superimposing the multiple target heat exchange performances to obtain the total heat exchange; and determining the outlet water temperature based on the mixing temperature of the multiple equivalent single U-tubes.

[0035] In one possible implementation, the temperature parameters are density, specific heat capacity, thermal conductivity, and dynamic viscosity; the thermal resistance network consists of convection thermal resistance inside the pipe, thermal conductivity thermal resistance of the pipe wall, thermal resistance of the grouting layer, thermal resistance of the cement ring, and thermal short-circuit thermal resistance between pipes.

[0036] In one possible implementation, the fluid thermophysical property fitting model is a polynomial function of temperature; wherein the density is a cubic polynomial of temperature, the specific heat capacity is a cubic polynomial of temperature, the thermal conductivity is a quadratic polynomial of temperature, and the dynamic viscosity is an Arrhenius exponential function. This fluid thermophysical property model is applicable to liquid water at 0-100°C and can be extended to thermophysical property fitting of other circulating working fluids (such as CO2, organic working fluids, etc.).

[0037] For example, the density It can be represented as: ; Specific heat capacity It can be represented as: The thermal conductivity It can be represented as: The dynamic viscosity It can be represented as: .

[0038] For example, this thermal resistance network can be determined through the following steps. First, the Reynolds number of the fluid inside the pipe is determined using the following formula. And Prandtl : , ;in, The flow velocity inside the pipe is (m / s). The inner diameter of the U-shaped tube is in meters (m).

[0039] Then, the Nusselt number is calculated based on the flow regime. Nu For turbulent flow ( (Using the Dittus-Boelter formula): For laminar flow ( ), take a constant value: For transitional flow ( Linear interpolation is used. .

[0040] (1) Convection heat transfer coefficient inside the pipe : Convection thermal resistance : .

[0041] (2) Thermal resistance of pipe wall (thermal conduction of cylindrical wall) : ;in, The outer diameter of the U-shaped tube (m); The thermal conductivity of the PE pipe is (W / (m·K)).

[0042] (3) Thermal resistance of grouting layer The multi-stage method is used for calculation, taking into account the heat flow deflection effect caused by the difference in thermal conductivity between grouting and soil: , ;in, It is a factor for differences in thermophysical properties; The thermal conductivity of the grouting material is (W / (m·K)). The thermal conductivity of the soil and rock. The borehole radius (m); The outer radius of the U-shaped tube (m); D The distance between the centers of the two pipes is (m).

[0043] (4) Thermal resistance of cement ring Also determined based on the thermal conductivity of the cylinder wall: ;in, The borehole diameter is in meters. The outer diameter of the casing (m); The value represents the thermal conductivity of cement.

[0044] (5) Total thermal resistance inside the borehole The sum of the thermal resistances of each component: .

[0045] (6) Thermal resistance of inter-tube thermal short circuit (heat transfer between two tubes of a U-shaped tube) : .

[0046] In one possible implementation, the piecewise finite-line heat source model determines the response of each segment using a series expansion of an exponential integral function, and approximates the responses of adjacent segments using a finite-line heat source midpoint based on a complementary error function; and handles the heterogeneity of layered soil and rock by using the cross-layer geometric mean thermal conductivity and geometric mean thermal diffusivity.

[0047] For example, for self-segment response ( Using exponential integral functions Temperature response matrix : Among them, the exponential integral function The series expansion is: , ; in, = 0.5772156649 is Euler's constant; The geometric mean thermal diffusivity across layers; It is the geometric mean thermal conductivity across layers.

[0048] For mutual segment response ( Using the midpoint approximation of a finite linear heat source, the temperature response matrix... : ;in, The spatial distance between the midpoints of the two segments; This is the thermal diffusion length; It is a complementary error function; For the length of the section.

[0049] For the far field segment ( Apply exponential decay correction: .

[0050] The borehole wall temperature is obtained by superimposing the original formation temperature and the temperature rise from the linear heat source: ;in, for N Dimensional borehole wall temperature vector (°C); for N Original temperature vector of the strata (°C); G for N×N Response matrix; for NThe heat transfer vector per unit length of each segment (W / m).

[0051] In some embodiments, the above-described iterative solution of the friction-through temperature distribution of the downcomer and riser in each time step based on the borehole wall temperature and fluid energy balance equation may include the following steps.

[0052] For the downcomer (inlet pipe, from the surface to the bottom of the well), the energy balance equation is: ;in, Mass flow rate (kg / s); First item For heat dissipation from the fluid to the formation; the second item This refers to the thermal short-circuit loss from the downcomer to the upcomer.

[0053] For the riser pipe (the outlet pipe, from the bottom of the well to the surface), the energy balance equation is: Among them, the first item For the heating of the formation to the fluid, the second term This is a thermal short-circuit heating method for the downcomer to the upcomer.

[0054] bottom of U-shaped tube ( The boundary condition is that the temperature is continuous (ignoring heat loss at the elbow): For the downcomer, a first-order upwind differential scheme is used from... arrive Solve sequentially; for the riser pipe, from arrive Solve the problem sequentially.

[0055] In one possible implementation, the under-relaxation iterative update includes the following steps: solving for the downcomer temperature distribution; applying a temperature continuity condition at the bottom; solving for the riser temperature distribution; updating the heat transfer per unit length of each segment based on the current fluid temperature; updating the borehole wall temperature by calling the segmented finite line heat source model; and using under-relaxation techniques to update the borehole wall temperature to improve iterative stability. This under-relaxation iterative solution strategy ensures numerical stability during periods of rapid borehole wall temperature change in the strong transient phase (the initial few hours to days of operation).

[0056] In one possible implementation, the relaxation factor of the under-relaxation iteration ranges from 0.5 to 0.9; the convergence tolerance in the convergence criterion ranges from [missing value]. to .

[0057] For example, the above process may include the following steps: Updating using under-relaxation techniques. T b To improve iteration stability: ;in, =0.7 is the relaxation factor; superscript and This represents the iteration step. The convergence criterion is: ;in, The wellhead water temperature, To achieve convergence tolerance.

[0058] In one possible implementation, the method is implemented using MATLAB or Python programming; the method includes a graphical user interface for parameter input, real-time calculation progress display, result visualization, and data export.

[0059] The following is combined Figures 2 to 6 The workflow of the technical solution of this application will be further described in detail with specific embodiments.

[0060] Example 1: This example uses the conversion of a typical abandoned oil well into a single U-tube geothermal well as an example. The wellbore structure of this single U-tube geothermal well can be referenced. Figure 2 As shown, (a) is the longitudinal structure of the wellbore, and (b) is the AA cross section of the wellbore.

[0061] Step 1: Input parameter settings. Set the geometric parameter to: well depth. = 2500 m, borehole diameter = 215.9mm, outer diameter of the casing = 177.8 mm, casing wall thickness = 9.19 mm, outer diameter of U-tube = 32 mm, U-shaped tube wall thickness =3 mm, pipe center distance = 65 mm, U-tube type is single U-shaped.

[0062] Physical property parameters are set as follows: thermal conductivity of soil and rock = 2.5 W / (m·K), specific heat capacity of rock and soil = 1000 J / (kg·K), density of rock and soil = 2650 kg / m³, thermal conductivity of cement = 1.0 W / (m·K), thermal conductivity of PE pipe =0.42 W / (m·K), thermal conductivity of steel casing = 45 W / (m·K), thermal conductivity of grouting material = 0.6 W / (m·K).

[0063] The calculation parameters are set as follows: inlet water temperature = 10℃, circulation flow rate Q= 3 m³ / h, surface temperature = 16℃, geothermal gradient = 3.0℃ / 100m, simulation time step = 360 days, time step = 1 / 24 day (i.e., 3600 seconds), number of segments N = 100, convergence tolerance = 1×10 -4 .

[0064] Step 2: Geometric parameter calculation. Based on the input parameters, the inner diameter of the casing is calculated. = 159.42 mm, inner radius of the casing = 79.71 mm, drilling radius = 107.95 mm, cement ring thickness = 19.05 mm, inner diameter of U-tube =26 mm, inner radius of U-shaped tube = 13 mm, outer radius of U-tube = 16 mm, single-pipe flow area = 5.309×10 - 4 m². Discrete depth coordinates The length from 0 to 2500 m is evenly divided into 100 segments, each segment having a length of... = 25.25 m.

[0065] Step 3: This embodiment uses uniform soil and rock parameters and does not load data of layered physical properties.

[0066] Step 4: Time discretization. Total simulation time = 360 × 86400 = 31,104,000 seconds, which is discretized into 62 logarithmically spaced time steps. The early time step density is higher to capture fast transient processes, and the time step spacing is increased in the later time steps to improve computational efficiency.

[0067] Step 5: Initialization. Bottom hole temperature: = 16 + 3.0×2500 / 100 = 91℃. Initially, all temperature fields were set to the original geothermal temperature of the strata.

[0068] Step 6: Iterative solution. First time step ( t It takes about 8-12 iterations to converge within 3600 s; as the simulation time progresses, the borehole wall temperature tends to stabilize, and the number of iterations in the later time steps is reduced to 3-5.

[0069] Step 7: Result Output. The final outlet water temperature after 360 days of simulation. = 32.92℃, temperature rise = 22.92℃, total heat exchange = 79.2 kW, heat transfer intensity per unit length = 31.7 W / m.

[0070] The relevant parameters and main results of Example 1 are shown in Table 1 below.

[0071] Table 1:

[0072] Reference Figure 3 The diagram illustrates the water temperature distribution along the wellbore after 360 days. It shows that the temperature of the downcomer (inlet pipe) gradually increases with depth, rising from 10°C at the wellhead to approximately 34°C at the bottom, absorbing heat from the formation along its path. The temperature of the riser (outlet pipe) slowly decreases from 34°C at the bottom to approximately 32.9°C at the wellhead, continuing to absorb heat from the formation during this period, but some heat is transferred to adjacent downcomers through thermal short-circuiting. The borehole wall temperature lies between the downcomer and riser temperatures, indicating that the borehole wall acts as an intermediate medium for heat exchange, simultaneously transferring heat to both the downcomer and riser.

[0073] Reference Figure 4 The data illustrates the evolution of the outlet water temperature and temperature rise during 360 days of operation. In the early stage (first 30 days), the outlet water temperature decreased rapidly, from an initial temperature of about 35°C to about 34°C, reflecting the rapid cooling of the rock and soil near the borehole. In the middle stage (30-180 days), the rate of temperature decrease slowed down, entering a transitional phase. In the later stage (180-360 days), the outlet water temperature tended to reach a quasi-steady state, with an outlet water temperature of about 32.9°C at 360 days, representing a temperature rise of about 22.9°C.

[0074] Reference Figure 5 and Figure 6 The data shows the changes in heat exchange (total heat exchange) and heat exchange intensity per unit length over time, respectively. The trends are consistent with the changes in outlet water temperature, both exhibiting a rapid decline in the early stages followed by stabilization. After 360 days, the heat exchange is approximately 79.2 kW, and the heat exchange intensity is approximately 31.7 W / m. This allows for calculations under a single operating condition to be completed within seconds, enabling rapid scanning and optimization design of the aforementioned multi-dimensional parameter space.

[0075] Example 2: Comparative analysis of different flow conditions.

[0076] In this embodiment, keeping other parameters the same as in Embodiment 1, the circulation flow rate was set to 1, 2, 3, 5, and 8 m³ / h, respectively. The heat exchange performance under different flow rates was compared through actual calculation by the program, and the calculation results are shown in Table 2 below. =2500m, =3.0℃ / 100m, =10℃, 360 days.

[0077] Table 2:

[0078] It can be seen that the outlet water temperature does not change monotonically with the flow rate, but rather exhibits a non-monotonic characteristic of first rising and then falling. At a flow rate of 1 m³ / h (laminar-transitional flow, Re≈10800), the outlet water temperature is 31.7℃; when the flow rate increases to 2 m³ / h (entering vigorous turbulent flow, Re≈21600), the outlet water temperature rises to a maximum of 35.1℃; thereafter, as the flow rate continues to increase to 8 m³ / h, the outlet water temperature monotonically decreases to 22.6℃. Regarding the total heat exchange, it continuously increases from 25.0 kW at a flow rate Q=1 to 116.5 kW at Q=8 (an increase of 4.7 times). This indicates that in practical engineering, a comprehensive trade-off optimization between heat exchange power and outlet water temperature is needed based on specific heat demand. Thus, because a single solution only takes a few seconds, a rapid and detailed scan of the full range of flow parameters can be completed.

[0079] Example 3: Comparative analysis of different U-tube types.

[0080] In this embodiment, the heat exchange performance of single-U and double-U downhole heat exchangers is compared under the same total flow rate, i.e., 6.0 m³ / h. The double-U scheme uses the equivalent piping method to distribute the total flow rate to two parallel pipes (3.0 m³ / h each), with other parameters the same as in Embodiment 1. The actual calculation results are shown in Table 3 below. =2500m, =3.0℃ / 100m, 360 days.

[0081] Table 3:

[0082] It can be seen that, under the same total flow rate (6.0 m³ / h), the double U-tube scheme is superior to the single U-tube scheme in all aspects. The outlet water temperature increased from 25.7℃ in the single U-tube to 32.9℃ in the double U-tube (an increase of 7.2℃), and the total heat exchange increased from 108.2 kW to 158.5 kW (an increase of 46.5%). The main reasons are: (1) The double U-tube distributes the total flow rate to two parallel pipes, and the flow velocity in each pipe decreases from 3.14 m / s in the single U-tube to 1.57 m / s, doubling the residence time of the fluid in the wellbore and making the heat exchange more sufficient; (2) The heat exchange area of ​​the double U-tube is doubled; (3) The reduced flow velocity in the pipe reduces the flow resistance and the pump power consumption. However, the double U-tube also faces the problems of increased thermal interference between the two pairs of pipes and limited casing space. In this way, the performance comparison of different U-tube types was achieved under a unified solution framework through the equivalent pipe method, providing an efficient quantitative analysis tool for the selection of downhole heat exchangers.

[0083] The various embodiments in this specification are described in a progressive manner. For the same or similar parts between the various embodiments, please refer to each other. Each embodiment focuses on describing the differences from other embodiments.

[0084] The above embodiments are only used to illustrate the technical solutions of this application, and are not intended to limit this application. Although this application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features therein. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of this application.

Claims

1. An analytical solution method for determining the heat exchange performance of abandoned oil wells converted into geothermal wells, characterized in that, The method includes: Obtain the geometric parameters, physical property parameters, and calculation parameters of the target oil and gas well, and determine the derived geometric quantities of the borehole and U-tube based on the geometric parameters; The thermal diffusivity of multiple segments in the target oil and gas well is determined based on the physical property parameters; the total simulation duration is determined based on the simulation time step in the calculation parameters, and the total simulation duration is discretized into multiple time steps; Based on the calculation parameters, each temperature field is initialized using the linear geothermal gradient formula. Based on the current local fluid temperature, temperature parameters are determined using a temperature-dependent fluid thermal property fitting model; and a thermal resistance network is determined based on the heat transfer physical process of the buried pipe and the current local fluid temperature. A piecewise finite linear heat source model is used to construct the temperature response matrix of the borehole wall of the target oil and gas well, and the borehole wall temperature is obtained. Based on the borehole wall temperature and the fluid energy balance equation, the friction temperature distribution along the downpipe and riser is iteratively solved in each time step. The borehole wall temperature is updated using under-relaxation iteration, and the change in wellhead water temperature is used as the convergence criterion. If convergence is determined, the water temperature, total heat transfer, heat transfer intensity per unit length, and temperature distribution along the borehole at each time point are output.

2. The method according to claim 1, characterized in that, The method further includes: Obtain the layered physical property parameters from the layered data file; The determination of the thermal diffusivity of multiple segments in the target oil and gas well based on the aforementioned physical property parameters includes: The layered physical property parameters are mapped to multiple segments by linear interpolation, and the thermal diffusivity of each segment is determined.

3. The method according to claim 1, characterized in that, The geometric parameters are well depth, borehole diameter, casing outer diameter, casing wall thickness, U-tube outer diameter, U-tube wall thickness, tube center distance, and U-tube type; the U-tube type is a single U-tube, double U-tube, triple U-tube, or quadruple U-tube downhole heat exchanger. The physical properties are thermal conductivity of soil and rock, specific heat capacity of soil and rock, density of soil and rock, thermal conductivity of cement, thermal conductivity of PE pipe and thermal conductivity of steel; the calculation parameters are inlet water temperature, circulation flow rate, surface temperature, geothermal gradient, simulation time step, time step, number of segments and convergence tolerance. The derived geometric quantities are casing inner diameter, casing inner radius, borehole radius, cement sheath thickness, U-tube inner diameter, U-tube inner radius, U-tube outer radius, single-tube flow area, discrete depth coordinates, and segment length.

4. The method according to claim 3, characterized in that, The method further includes: When the U-tube type is a double U-tube, triple U-tube, or quadruple U-tube downhole heat exchanger, the total flow rate of the target oil and gas well is evenly distributed to multiple equivalent single U-tubes using the equivalent pipe method. The target heat exchange performance of each of the multiple equivalent single U-tubes is solved, and the total heat exchange performance is obtained by superimposing the multiple target heat exchange performances. The outlet water temperature is determined based on the mixing temperature of the multiple equivalent single U-tubes.

5. The method according to claim 1, characterized in that, The temperature parameters are density, specific heat capacity, thermal conductivity, and dynamic viscosity; the thermal resistance network consists of the convection thermal resistance inside the pipe, the thermal conductivity thermal resistance of the pipe wall, the thermal resistance of the grouting layer, the thermal resistance of the cement ring, and the thermal short-circuit thermal resistance between pipes.

6. The method according to claim 5, characterized in that, The fluid thermal property fitting model is a polynomial function of temperature; wherein, the density is a cubic polynomial of temperature, the specific heat capacity is a cubic polynomial of temperature, the thermal conductivity is a quadratic polynomial of temperature, and the dynamic viscosity is an Arrhenius exponential function.

7. The analytical solution method according to claim 2, characterized in that, The piecewise finite line heat source model determines the response of each segment using a series expansion of an exponential integral function, and approximates the response of each segment using a finite line heat source midpoint based on a complementary error function; furthermore, it addresses the non-uniformity of layered soil and rock by using cross-layer geometric mean thermal conductivity and geometric mean thermal diffusivity.

8. The method according to claim 1, characterized in that, The under-relaxation iterative update includes the following steps: solving the temperature distribution in the downcomer; applying a temperature continuity condition at the bottom; solving the temperature distribution in the riser; updating the heat transfer per unit length of each segment based on the current fluid temperature; updating the borehole wall temperature by calling the segmented finite line heat source model; and using under-relaxation techniques to update the borehole wall temperature to improve iterative stability.

9. The method according to claim 8, characterized in that, The relaxation factor in the under-relaxation iteration ranges from 0.5 to 0.9; the convergence tolerance in the convergence criterion ranges from [missing value]. to .

10. The method according to any one of claims 1 to 9, characterized in that, The method is implemented using MATLAB or Python programming; the method includes a graphical user interface, which is used to implement parameter input, real-time calculation progress display, result visualization, and data export.

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