1. Horizontal Temperature Field Simulation
Currently, articles on the simulation of downhole temperature field during well cementing are in a limited number, and especially those on simulating the temperature field of hot dry rock (HDR) well cementing are comparatively rare. By simulating the horizontal and vertical distribution of the HDR downhole temperature field, we investigated the distributing characteristics and influential factors of HDR downhole temperature field, aiming at guiding operation of HDR well cementing.
1.1. Geometric Model Building
When the depth of hot dry rock (HDR) well is reaching 1000 m, heat-fluid-solid coupling should be considered for HDR numerical simulation, where both stratum thickness and radius are 1 m; the borehole diameter and length are 0.25 m and 1 m, respectively; the relevant wall thickness and length of drill pipe are 0.015 m and 0.9 m, respectively; and the length of drill bit is 0.05 m, as shown in Figure 1.
1.2. Governing Equation
1) Flow equation
Flow in free and porous media is applied, which should satisfy the continuity equation:
where is fluid density, kg/m3; is time, s; is fluid viscosity, Pa∙s; is temperature, K; is volume force, Pa; Total differential Displacement is.
2) Temperature distribution
In the present model, heat transfer within HDR stratum, fluid, drill stem and drill bit in the borehole is considered. Assume that no heat was generated from drill bit, and then the continuity equation of the relevant heat circulation is as follows:
where, is fluid density, kg/m3; is time, s; is temperature, K; is thermal conductivity coefficient, W/(m∙K); is specific heat capacity, J/(kg∙K); is Heat source, W/m3.
3) Convection heat transfer
The governing equation of convection heat transfer between drilling fluid and well wall is given below,
where is thermal conductivity coefficient, [W/m∙K]; is temperature, K; is
Figure 1. Diagram of the geometric model.
normal vector of solid surface; is convective heat transfer coefficient, [W/m2∙k]; is fluid near-surface temperature.
1.3. Boundary and Initial Value
The drilling fluid flows in from the entrance and flows out from the exit as for seepage field. Boundaries and are set as six seepage boundaries, and is set as the seepage wall with a seepage velocity of 8 - 10 m/s. The surfaces of drill pipe and drill bit are set as non-slip wall. The entrance boundary is set as the velocity boundary with a value of 3 m/s, and the velocity of exit boundary is set as zero, with no viscous stress.
As for temperature field, considering heat transfer in the porous media, the values for boundaries a, b, exit and entrance boundaries are all 293.15 K, while those of the borehole wall and boundaries and are 3˚C/100 m, calculated according to geothermal gradient; considering heat transfer in the fluid, the exit and entrance temperatures are also set as 293.15 K; the external temperature concerning convection heat transfer is set the same as borehole wall temperature. The detailed boundary information is shown in Figure 1. The figure was established according to the general structure of hot dry rock drilling.
1.4. Parameter Selection
HDR stratum parameters  -  are used here regarding stratum to be investigated, and polymer drilling fluid is utilized. The materials     of drill pipe and drill bit are alloy and diamond, respectively. Detailed information of the above mentioned parameters is shown in Table 1.
1.5. Mesh Division
Mesh division of the geometric model is shown in Figure 2, through using grid refinement method near the wellbore appropriately, and considering the calculation workload.
Table 1. Model parameters.
Figure 2. Mesh division of the geometric model.
Figure 3. Velocity distribution after 1 h of circulation.
1.6. Analysis of Result
1) Flow field simulation of HDR stratum
Four individual seepage velocities under the well of 10−4 m/s, 10−8 m/s, 10−12 m/s and 0 m/s are used for simulation and calculation, among which the simulation result corresponding to 10−4 m/s is shown in Figure 3. Velocities of the drilling fluid at the bottom of the well related to different seepage velocities are listed in Table 2.
Clearly, the permeability of the well bottom can affect the flow of drilling fluid. The flow velocity at the bottom of the well increases with better well bottom permeability. However, the influence is generally very small, and thus can be ignored.
In Figure 3, different colors refer to different velocity. The red arrow represents the flow line while the black line refers to the seepage of pore water. It can be observed that the drilling fluid velocity is decreasing in the field close to the drill pipe wall and well wall, and the velocity at the drill bit is the highest. Data is the actual discrete, the actual grid may not completely symmetrical.
2) HDR temperature field simulation
In Figure 4, the continuous and discontinuous color legend refers to the geothermal gradient and stratum isotherm, respectively. From analysis of Figure 4, it can be found that during the circulation of drilling fluid, when the temperature of returning drilling fluid under certain well depth is lower than the stratum temperature, the temperature of the upward moving fluid will be raised via continuously absorbing the heat from the well wall rock, thus resulting in temperature decrease of well wall rock due to heat loss. With the circulation of drilling fluid in the well arriving at a stable state, the heat exchange between drilling fluid in the well and well wall rock reaches a balance state gradually, thus the temperature difference between them also decreases gradually. When the time is long enough for drilling fluid keeping stable, the temperature of the drilling fluid in the well will gradually approaches the earth temperature. Therefore, as a function of time, the stratum temperature near the well wall, is a problem concerning transient heat transfer.
In Figure 4, the color legends refer to the geothermal gradient and stratum isotherm separately.
2. Longitudinal Temperature Field Simulation
2.1. Longitudinal Simulation of Well Structure (Example)
As for HDR geothermal well, the well surface and bottom temperatures are 20˚C and
Table 2. Flow velocity corresponding to different seepage velocities at the bottom of the well.
Figure 4. Stratum temperature distribution and geothermal gradient (Drilling fluid circulation after 60 min).
220˚C respectively, and the well depth is 2000 m. Well structure is shown in Figure 5.
2.2. Continuous Circulation Simulation of Temperature Distribution before Well Cementing
Simulation results of temperature distribution of circulating drilling fluid at 0 h, 1 h, 5 h, 10 h, 24 h, 48 h and 96 h are presented in Figure 6. It can be observed that the temperature of upper stratum increases with increasing circulation time, but that of lower stratum decreases. Seen from Figure 7, it can be found that the temperature variation of upper part is smaller than that of lower part. During 96 hours’ circulation simulation, the temperature variation at selected time of upper stratum is within 10˚C, while that of lower stratum exceeds 20˚C.
2.3. Discontinuous Circulation Simulation of Temperature Distribution before Well Cementing
Figure 8 shows temperature variation at seven different times during 96 hours’ circulation simulation, but the difference between that shown in Figure 7 is that individual temperature is obtained on condition that the circulation is stopped at certain selected
Figure 5. Configuration of well structure for longitudinal temperature simulation.
Figure 6. Comparison of annular space temperature of circulating drilling fluid at different time.
Figure 7. Comparison of annular space, sleeve and earth temperature after 96h circulation of drilling fluid.
Figure 8. Recovery temperature at different time.
time. The temperature at different stopping time of upper part decreases while that of lower part increases. Comparison of temperature distribution at different stopping time reveals that temperature variation of upper and lower parts are 5˚C and 16˚C, respectively, and temperature difference of upper annular space is far smaller than that of lower part. Meanwhile, regarding comparison between Figure 6 and Figure 8, it can be found that temperature change of discontinuous case is less than that of continuous case.
2.4. Temperature Variation during Injection of Cement Slurry
After being stable for 96 h, the ahead fluid and cement slurry will be injected successively, and then the relevant temperature variation of the annular space is shown in Table 3.
Table 3. Annular space temperature variation during injection of cement slurry.
It can be observed from Table 3 and Figure 9, the temperature of the lower stratum is high but that of the upper stratum is low, and the temperature variation of the upper and lower stratum is larger, with a value about 40˚C. The cement slurry in lower position will solidify at first to avoid the bridge plug, because in general, the higher the temperature, the faster the cement solidification. Consequently, this whole process can help improve cementing quality. The temperature of subsequent cement slurry will be increased due to convective heat transfer from solidified cement in the lower position, thus its solidification can be quickened, without changing the condition that the cement temperature of lower layer is higher than that of the upper. During waiting on the cement setting stage, the subsequent annular space temperature will return to the stratum temperature gradually.
When conditions are ripe, we can test how the simulations correspond to the real experimental data.
1) The seepage at the bottom of the well can accelerate the velocity of flow in the wellbore, but the impact is negligible.
2) The temperature of upper stratum increases with the increase of circulation time, but the temperature variation of lower stratum has an opposite trend.
3) At stable circulation state, the temperature of upper part decreases, while that of lower part increases. The temperature variation of upper annular space is far lower than that of lower part in stable state. The temperature variation in stable state is less than
Figure 9. Temperature distribution during injection of cement slurry.
that in cyclic state.
4) During injection of cement slurry, the temperature variation of upper and lower stratum is greater. During waiting on the cement setting period, the subsequent annular space temperature will return to stratum temperature gradually. Generally, the control agent of cement slurry solidification is very sensitive to temperature, so the influence of stratum temperature variation on cement slurry solidification should be considered during circulation of drilling fluid and injection of cement slurry, aiming at ensuring construction safety and increasing well cementing quality.
This research was supported by NSFC project “Study on the evolution mechanism of composite materials with high temperature resistance, high pressure resistance, and low elastic modulus”, No. 51474192; and “Basic research funding of central universities”, No. 2652015067.