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Login To See SPE Member Price USD 110Addressing both steady and unsteady-state fluid flow and related heat-transfer problems, the Second Edition of Fluid Flow and Heat Transfer in Wellbores strikes the perfect balance between theory and practice to aid understanding. Three new chapters on application of theory have been added and include probing pressure traverse in various wellbore multiphase fluid-flow situations, estimating flow rates from temperature data, translating off-bottom transient-pressure data to that at the datum depth, and a detailed discussion around newly discovered wellbore safety and integrity issues. Fundamental aspects of drilling, fluid circulation, and production operations form the foundation of this update of the 2002 original publication.FREE WITH PURCHASESelected Papers Supplement* which contains 5 papers written by the author team-Rashid Hasan and Shah Kabir.- A Simplified Model for Oil/Water Flow in Vertical and Deviated Wellbores- A Robust Steady-State Model for Flowing-Fluid Temperature in Complex Wells- Analytic Wellbore Temperature Model for Transient Gas-Well Testing- Sustaining Production by Managing Annular-Pressure Buildup- Interpretation of Cleanup Data in Gas-Well Testing From Derived Rates*This Selected Papers Supplement will be delivered digitally to each purchaser after payment has been processed.

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 dissolution of sour gas in the wellbore two-phase flow is a diffusion process governed by concentration difference. Therefore, the gas dissolution rate is a function of gas concentration and mass transfer coefficient:where is the gas concentration, mol/m3; is the gas concentration at saturation, mol/m3; is the molecular mass of CO2, kg/mol; is the interphase mass transfer coefficient between CO2 and liquid phase, m/s; and is the contact area of CO2 and liquid phase, m2. In this study, the contact area in different flow patterns is estimated using the model proposed by Sun et al. [23].

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.

Furthermore, the thermophysical properties of CO2 vary greatly accompanied with the complicated mass and heat transfer processes. The detailed calculation models for thermophysical parameters of CO2 are presented in Appendix A.

The integrated model is solved using the simulation method proposed by Sun et al. [23], which employs a fully implicit scheme, constant space steps, and varying time steps. The overall simulation process is consisting of three layers of iterations. At first, the phase velocities and fractions at different space blocks are calculated based on a drift-flux model. Subsequently, the pressure field is estimated using a predictor-corrector shooting technique. With updating the fluid properties and multiphase flow parameters, the temperature distribution in the drilling pipe and annulus are iteratively simulated until a desired convergence tolerance is achieved.

The formation fluids can enter the wellbore driven by pressure underbalance, if the bottom hole pressure at open hole section is less than the pore pressure during drilling. The understanding of phase transition and kick migration is important for early kick detection and wellbore control procedure. Using the proposed model, we simulate and analyze the flow behaviors of CO2 and water-based fluid in the scenario of a drilled CO2 kick. The basic parameters for kick simulation [19] are shown in Table 3.

The gas influx rate at the open hole section is mainly governed by the wellbore pressure distribution, which consists of hydrostatic pressure and friction pressure. As the gas kick enters the wellbore, the flow velocity and friction pressure of fluid mixture will increase abruptly, which can lead to the sudden increment of bottom hole pressure. Subsequently, the decrease of hydrostatic pressure plays a dominant role, and the bubble pressure decreases gradually.

Figure 6 shows the distributions of free gas along the wellbore during methane and CO2 kicks. For a given depth, the void fraction of methane is significantly larger than that of CO2. Because the gas influx rate for methane kick is larger under the same pressure underbalance, since the viscosity and percolation resistance of methane in reservoir are much smaller than that of CO2. Affected by the large gas influx rate, the velocity of the fluid mixture is larger and gas front rises faster during the methane kick.

Another important application of temperature logs is the identification of fluid loss or feed zones from temperature data obtained under hydraulic testing conditions (Okandan 2012; Steingrimsson 2013). Examples of using temperature measurements in boreholes are multifold. Pehme et al. (2010) identified hydraulically active fractures in dolomite and sandstone aquifers; Klepikova et al. (2011) estimated local transmissivities and hydraulic head differences; Nian et al. (2015) predicted flow rates in oil and gas production wells. These authors stressed the satisfactory accuracy of temperature-derived flow velocities compared to direct flow measurement. In recent years, fiber-optic distributed temperate sensing (DTS), which is a robust means of acquiring continuous temperature profiles instantaneously along the length of the cable (Großwig et al. 1996), has also been extensively used to improve the accuracy of flow rate profiling and the detection of fracture zones (Read et al. 2013; Coleman et al. 2015; Read et al. 2015; Bense et al. 2016).

Figure 1 shows the schematic of typical wellbore flow and heat transfer scenarios. The cold drill fluid is considered to be either injected both in through the drill pipe and the annulus (coflow); or injected in the drill pipe and circulated back to the surface (counterflow). The simulator assumes the wellbore to be treated either as a one-dimensional or a two-dimensional structure depending on the problem being studied. When a two-dimensional wellbore structure is considered, the wellbore components, such as the fluid inside the drill pipe, the drill pipe wall, the annulus, and the casings, are treated as different regions (region 1, 2, 3, 4, respectively) in which the temperatures (T1, T2, T3, and T4) need to be solved as individual variables (Fig. 1). These variables are linked through the interfacial heat transfer relationships between the fluid and the solid. The injection fluid was assumed to be pure liquid water. Fluid properties such as density, viscosity, and heat capacity were calculated according to the IAPWS-IF97 formulation (Cooper and Dooley 2007). The fundamental assumptions of the models considered in this work are: the geometries of the wellbore and formation are cylindrical, the fluid is incompressible, fluid flow is in the axial direction only, the rock formation is impermeable, there is no radial temperature gradient within the fluid when the wellbore is considered to be a two-dimensional structure, thermal dissipation and expansion effects are negligible.

Schematic of the heat exchange model between the wellbore and the formation. Governing equations are solved in four regions: the fluid inside the drill pipe (region 1), the drill pipe wall (region 2), the annulus (region 3), casing-cement-formation (Region 4). The solid arrow pointing downwards and the dashed arrow pointing upwards in the annulus refer to coflow and counterflow scenarios in the wellbore, respectively

where ρf is the fluid density, Cp, f is the fluid specific heat capacity, vz and vr are the axial and radial flow velocities, respectively, λf is the thermal conductivity.

As mentioned above, the thermal exchange between different wellbore regions is modeled via thermal transfer relations at their interfaces (Table 1, BC2). The heat transfer coefficient, h, is the proportionality constant between the heat flux and the thermodynamic driving force for the heat flow (i.e., the temperature difference between adjacent wellbore components, ΔT). In this work, the heat transfer coefficients under forced convection and shut-in condition are correlated and calculated using different approaches.

So far in most theoretical and simulation studies, pure conductive heat flow in a static water column is assumed when estimating temperature recovery during borehole shut-in (Shen and Beck 1986; García et al. 1998; Espinosa-Paredes et al. 2001; Yang et al. 2015). The heat transfer coefficient in the borehole fluid is then approximated by:

However, several studies have reported the existence of another key factor in the heat transfer, which is free convection caused by density differences arising from vertical temperature gradients (Diment 1967; Gretener 1967; Pfister and Rybach 1995; Berthold and Börner 2008; Eppelbaum and Kutasov 2011; Klepikova et al. 2018). The critical parameters for the free convection process can be indicated by the following equation (Diment and Urban 1983): 2b1af7f3a8