Let us discuss the two reasons related to reduction and ultimate reversal in temperature rise with time. The primary reason is that our model accounts for fluid heat loss to the overburden and under-burden formations. This heat loss increases with increased fluid temperature. Figure 5 shows the estimated fluid temperature using the rigorous model (solid lines) compared to that estimated assuming no heat loss (everything else remaining the same) to the formation (dashed lines). Note that the maximum temperature after 400 days of flow period is about 10 F higher when heat loss to the formation is not accounted for compared to when it is. Further analyses of ignoring fluid heat loss to the formation are discussed later.
The proposed analytical solution also improves wellbore fluid and heat flow modeling because of more realistic temperature evaluation at sandface. The bottomhole flowing-fluid temperature derived from the analytical model can be coupled with wellbore heat transfer model to allow prediction of flowing-fluid temperature along the wellbore. Accurate flowing-fluid temperature profile along the wellbore is also desirable for well design and production optimization, as well as for pressure transient analysis (Onur and Cinar 2016). An accurate estimation of the reservoir fluid temperature from the analytical formulations can yield a better estimation of well productivity index, which is useful in production optimization and well development planning.
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The advantage of this analytical model over other analytical solutions for reservoir temperature estimation is that heat transfer from/to overburden and under-burden formations \(\dotQ\) is included. While the derivation of the analytical solution neglects property variation, the use of the solution allows for property changes with pressure and temperature. We have shown that \(\dotQ\) is crucial in the estimation of flowing-fluid temperature in a reservoir, especially at long producing times when the reservoir fluid is heated significantly, and the reservoir fluid temperature is very different from that in its surroundings.
An important assumption in our model is that conductive heat flow for our problem is negligible. App and Yoshioka (2013) has shown that the effect of formation thermal conductivity can be represented by the Peclet number. Relating fluid velocity u to production rate q, App and Yoshikawa expressed P e as follows:
Similarly, for our highest production rate of 6200 STB/D, P e = 36.3. Therefore, for the cases considered in this study, omitting thermal conductivity of the formation does not introduce any significant error. We note that for most deepwater assets, economic production rates are expected to be high enough to result in correspondingly high Peclet numbers. As a consequence, the underlying assumptions made while deriving the analytical formulation of this coupled fluid and heat flow problem appear reasonable.
To answer this question, we propose a numerical method to calculate the fluid flow and heat exchange in a fracture controlled reservoir. The models are built using real data from the field. The discrete fracture models (DFMs) are obtained from scanline surveys measured in the field on Las Minas analogue outcrops and processed using the SkaPy script and then extrapolated using the multiple point statistic method as presented in Bruna et al. (2019). The rock properties have been measured in the rock physics laboratory on rocks sampled on the analogue outcrops of Las Minas. The numerical model runs two studies: (i) a static analysis measuring the influence of the stress field on the fracture aperture; (ii) a transient analysis which couples fluid flow in porous media and thermal exchange through the DFMs. Because Acoculco reservoir is composed of limestone, marble and skarn, a comparison is made between these three formations. Therefore, we created three models representing the three formations. Each formation is populated with its own mechanical properties and its own DFM.
Looking at the heat transfer (Fig. 5), a few observations can be made: (i) despite the high fracture density and the permeability trends created by the fractures, the thermal front is circular and, hence, homogeneous. This can be explained by the poor continuity and connectivity of the fractures. (ii) However, even though the fractures (\(\Gamma \)) are not creating a continuous fluid pathway, they are always showing lower temperatures than the matrix (\(\Omega \)). This confirms that the fluid flow propagates quicker in the fractures and, hence, corroborates higher permeability in the fractures. (iii) Considering the implemented conditions of pumping flow rate and the resulting bulk permeability of the reservoir: for this example of the limestone, the injected fluid, pure water at 50 C, needs a 100 m to reach the initial reservoir temperature of 300 C.
When developing subsurface activities such as oil and gas, nuclear waste disposal, CO2 sequestration or, as in this case, enhanced geothermal system, it is fundamental to quantify the role of the fracture system present in the subsurface. Very often, fractures are up-scaled to represent the matrix and the fractures together in a continuum model. In this article, we present a stress-dependent fracture aperture model and, hence, a stress field-dependent fluid flow and heat transfer model, using field data to populate the material properties and the fracture networks. These DFMs are composed of tens of thousands of fractures. The DFMs are separated into fracture sets related to the regional structural context. Each fracture set is characterized with its own aperture value from field measurements, corrected to depth stress conditions. Thanks to these simulations, we can identify the threshold, on the mechanical properties of the fractures, below which deformation takes place. The purpose of this multiple scenarios analysis is to evaluate the risks of the project. Combining multiple formations and multiple well positioning scenarios gives a thorough evaluation of the reservoir performance. This is fundamental in the context of field production risks analysis. Based on our results, the safer scenario for this project would be developed according to the scenario 4, which simulates a well doublet across the different fracture distribution patterns instead of targeting one zone in particular. This is even more important in the case of developing the Marble and the Skarn formations. This method provides, to our knowledge, a more realistic model of the existing and expected fracture network at depth. This would certainly be a major improvement in the development of the EGS technology.
In this study, we developed a transient fully coupled model for wellbore CO2/water flow, which considers the complicated mass and heat transfer mechanisms in different flow patterns and the dynamical coupling between wellbore and reservoir. Subsequently, the proposed model is applied to analyze the multiphase flow process during a drilled CO2 kick.
The mass and heat transfer characteristics in the two-phase flow are significantly governed by the flow patterns. In this study, the model developed by Hasan and Kabir [24] is used to flow pattern identification, as shown in Table 1.
The determination of the mass transfer coefficient, which is related to the fluid properties (such as density, viscosity, and diffusivity), flow velocity, and annulus size, is challenging. Considering the laminar flow and turbulent flow conditions, the expression presented by Cussler [25] is employed.where is the pipe length, m; is the kinematic viscosity, m2/s; and is the gas diffusivity coefficient, m2/s.
In drillingengineering, the research methods of wellbore temperaturefield are mainly divided into the analytical method and numericalmethod. Holmes14 used the steady-statelinear heat transfer approximation to replace the heat transfer betweenthe annulus fluid and the formation and derived an analytical modelto calculate the steady-state heat transfer between the fluid in thedrill string and the fluid in the annulus. The calculation resultsshow that the maximum temperature of the fluid in the annulus doesnot appear at the well bottom. Based on this research result, Kabir15 assumes that the heat exchange process betweenthe fluid in the drill string and the fluid in the annulus belongsto steady-state heat transfer, and the heat exchange between the annulusfluid and the near wellbore area belongs to pseudo-steady-state heattransfer. Combined with the energy conservation law, a one-dimensionaltransient heat transfer model based on dimensionless time has beenestablished. Based on the above model and considering the differentconstruction environment, mode, and other factors, researchers haveestablished different analytical models to analyze the temperaturedistribution law of wellbore and formation and the heat transfer efficiencybetween fluid and near wellbore area in the construction process ofcementing16 and oil and gas production.17,18
The numerical method is based on the one-dimensional numericalmodel established by Raymond19 to calculatethe temperature distribution of annulus fluid in vertical wells understeady and pseudo-steady conditions. Keller20 added the inner heat source generated by the drill string rotationand the friction resistance of drilling fluid during drilling to theRaymond model, established the two-dimensional transient heat transfermodel, and considered that these inner heat sources had a significantimpact on the energy balance during drilling fluid circulation. In1998, J. Romero21 first considered theinfluence of circulation time, displacement, and seawater temperatureon wellbore temperature field during deep-water drilling and developedthe corresponding calculation program. In 2013, Mou22 divided the drilling fluid in the wellbore into severalgrids in the radial direction, established the wellbore temperaturefield model considering the radial temperature gradient of the drillingfluid, discretized and solved the mathematical model using the finitedifference method, and considered that the axial heat conduction ofthe drilling fluid had a weak influence on the wellbore temperaturedistribution. In 2014, Li23 pondered theheat source generated during horizontal well drilling, establisheda numerical model calculating the horizontal well temperature field,and analyzed the heat source distribution in the process. In 2016,Li24 considered the influence of bottomhole assembly and casing program on wellbore temperature distributionand established a wellbore-formation transient heat transfer modelsuitable for the process of drilling fluid circulation and well shut-inunder overflow conditions. In 2019, Yang25 established a transient heat transfer model suitable for deep-watermultigradient drilling based on the conservation equations of mass,momentum, and energy, aiming at the problem of variable mass heattransfer caused by the hollow glass spheres entering the annulus fromthe drill string during deep-water multigradient drilling, and analyzedthe wellbore temperature distribution law in deep-water multigradientdrilling through this model. Compared with the numerical method, theanalytical method has the characteristics of high computational efficiencyand unconditional convergence, but it is difficult to obtain the analyticalsolution of the nonlinear partial differential equations representingthe wellbore temperature field and the influence of the thermophysicalparameters of the wellbore fluid, heat sources, and different casingprogram on the wellbore temperature distribution cannot be fully considered.As a result, the analytical model is difficult to effectively completethe research on the wellbore temperature distribution during deep-waterdrilling and drilling into deep reservoirs. This also makes more andmore scholars prefer to use numerical methods to study the transientheat transfer process between wellbore and formation. 2ff7e9595c
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