NUMERICAL SOLUTION OF THE START-UP OF WELL DRILLING FLUID FLOWS

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1 Proceedgs of ENCIT Copyright by ABCM 3 th Brailian Congress of Thermal Sciences and Engeerg December 5-,, Uberlandia, MG, Brail NUMERICAL SOLUTION OF THE START-UP OF WELL DRILLING FLUID FLOWS Gabriel Merhy de Oliveira, gabrielm@utfpr.edu.br Cear Otaviano Ribeiro Negrão, negrao@utfpr.edu.br Admilson Teixeira Franco, admilson@utfpr.edu.br UTFPR - Federal University of Technology - Parana Av. Sete de Setembro, 365, CEP Curitiba-PR-Brail Andre Leibsohn Marts, aleibsohn@petrobras.com.br Roni Abensur Gandelman, roniag@petrobras.com.br TEP/CENPES - PETROBRAS S/A Cidade Universitária, Q.7, Ilha do Fundão, CEP.94-95, Rio de Janeiro-RJ Abstract. The drillg fluid is designed to build up a gel-like structure, when at rest, order to avoid cuttgs to drop at the bore bottom and therefore to prevent the bit obstruction. As consequence, high pressures, which can be larger than the formation pressure and can damage the well, are needed to break up the gel when circulation resumes. Due to its thixotropic effect, the gel viscosity remas high for a while after the circulation restarts. The gelation may have significant importance, specially, deep waters where high pressures and low temperatures take place. The current work presents a compressible transient flow model of the start-up flow of drillg fluids, order to predict borehole pressures. The model comprises one-dimensional conservation equations of mass and momentum and one state equation for the calculation of the fluid density as a function of the pressure. The considered geometry is a concentric annular pipe of length L. Its ternal diameter is D and external one, D. For a circular pipe, the ternal diameter is made equal to ero. The ma difference from previous model was the type of boundary condition: Constant flow rate at the pipe let rather than the constant pressure. Both Newtonian and non-newtonian Bgham fluid flows are considered. The governg equations are discretied by the Fite Volume Method usg the fully implicit formulation and the first-order upwd scheme. The resultg non-lear algebraic equations are iteratively solved. The model results were corroborated with an analytical solution for Newtonian flows. Case studies are conducted to evaluate the effect of fluid flow properties, well geometry and flow rate on borehole pressures. For Bgham fluid flow one can observe that large pressures (compared with Newtonian fluid flow) are observed when constant flow rate are put as boundary condition. Pressure peaks caused by the acoustic wave propagation can be more tense low compressible fluid flow, low viscosity fluid and smaller length pipes. The model can be applied to predict the time-dependent pressure field and pressure peaks durg the restart of drillg fluid operations. Keywords: start-up flow, drillg fluid, Bgham fluid, compressible flow, transient flow. INTRODUCTION In oil well drillg, a fluid is pumped through a drill pipe. The fluid flows to the bottom of the hole and returns by the annular space between the drill pipe and the well wall, carryg the cuttgs formed while drillg. Durg drillg stoppages, the fluid gellifies order to avoid cuttgs to fall down the annular space obstructg the bit. The flow restart requires higher than the usual operational pressures to break the gel down. High pressure peaks may damage the well structure. Therefore, the study of the wave pressure propagation durg the flow restart is essential to predict the pressure peaks. The mathematical modelg can be quite useful to understand the phenomenon and to predict pressure peaks durg the flow restart of a gellified material. Although some works were dedicated to study the problem most of them were applied to the start-up of waxy crude oils at low temperature. For stance, Sestak et al. (987), Cawkwell and Charles (987), Chang et al. (999) and Davidson et al. (4) considered that the waxy crude oil filled completely the pipe and them it is displaced by a non-gellified oil. Chang s et al. (999) e Sestak s et al. (987) works disregarded the ertial effects the momentum equation and shear forces are based on the fully developed flow. The time variation of flow occurs only due to the time-change of the rheology properties. Davidson et al. (4) cluded the compressibility effect the Chang s et al. (999) model but did not considered the ertia terms the momentum conservation equation. Although Cawkwell and Charles (987) have disregarded the advective terms the momentum conservation equation they considered the fluid flow as transient and compressible. Vay et al. (6) presented a D transient model to simulate the start-up of a Bgham fluid. In a second work, Vay et al. (7) developed a D model and compared the results to their D model with good agreement. Nevertheless, Vay et al. (7) showed the solution of the one-dimensional model is much faster their D counterpart. In order to reduce the computational time, the same authors (Wachs et al., 9) developed a.5d model which they merged the D and the D model. Besides, they evaluated the compressibility and thixotropy the fluid flow restart. The current work presents a mathematical model for the start-up of drillg fluids horiontal pipes. The transient flow is considered isothermal, one-dimensional and compressible. Differently from the above works, both the transient

2 Proceedgs of ENCIT Copyright by ABCM 3 th Brailian Congress of Thermal Sciences and Engeerg December 5-,, Uberlandia, MG, Brail and advective terms are cluded the equations. Besides, the friction factor concept for fully developed flows for both Newtonian and Bgham fluid is applied to the computation of viscous effects. The governg equations are discretied by the Fite Volume Method with the fully implicit and first-order upwd schemes. The set of discretied equations are solved iteratively.. MATHEMATICAL MODEL. System Geometry and Hypothesis Although the drillg fluid flows through the drill pipe and the annular space (see Fig. a), the geometry is simplified to a horiontal annular space. The length, ternal and external diameters are, respectively, L, D and D, as shown Fig. b. To simulate a pipele flow, the ternal diameter can be taken as ero. a) b) P and Q( t ) t P out Coluna de Perfuração P(, t ) R R Espaço Anular Cascalhos Poço Broca c) V (, t ) P P()and t Q() t Q t L P(, t ) P out R R Fluido de Perfuração V(, t ) L Figure. (a) Scheme of the wellbore, drill pipe and drill bit. Problem doma at: (b) t and (c) t.. Governg Equations Considerg the fluid flow as one-dimensional, as shown Fig., the governg equations of mass and momentum can be written, respectively, as: ( V ) t ( V) ( VV) P f V t D h () () where, V and P are the average values of density, velocity and pressure the cross sectional area, respectively, f is the Fanng friction factor, which is evaluated accordg to the fluid flow, fluid properties and geometry (pipe or annular). t is the time and is the axial position. Dh is the hydraulic diameter, which is defed as D D for the annular space and as D for the pipe ( D ). Once the flow is compressible, density is function of pressure and the flow is isothermal. Besides, the pipe is considered completely rigid. By defition, the fluid compressibility,,can be written as (Anderson, 99): P T (3) Assumg the compressibility is constant for the pressure range analyed, Eq. (3) can be rearranged and tegrated from a reference state to any other: P P ln (4) where and P are, respectively, the density and pressure for the reference state. Accordg to Anderson (99), the fluid compressibility, the density and the pressure wave speed are correlated by the equation:

3 Proceedgs of ENCIT Copyright by ABCM 3 th Brailian Congress of Thermal Sciences and Engeerg December 5-,, Uberlandia, MG, Brail (5) a where a is the pressure wave speed for an elastic pipe..3 Constitutive Equation and Friction Factor The drillg fluid is considered to be a Bgham plastic: (6) where is the shear stress, is the shear stra rate, is the yield stress and is the fluid plastic viscosity. This constitutive equation is implicitly taken to account the calculation of the friction factor. The friction factor depends on the kd of fluid, its properties and flow geometry. In the current work, the flow is assumed lamar. For the non-newtonian Bgham flow a circular pipe, the Fanng friction factor is employed (Chang et al., 999): f T 6 He He Re 6Re 3 Re 4 t, t, 3 7 t, t, ft t, (7) where Re t, VD h is the Reynolds number and He t, D h is the Hedstrom number. The subscript T means the friction factor is computed for a circular pipe. The t and subscripts of the Reynolds and Hedstrom numbers means they are vary with time and the axial position, respectively. Eq. (7) is reduced to the Newtonian friction factor as the Hedstrom number (or the yield stress) approaches ero. The followg Fanng friction factor is adopted to the annular pipe (Fontenot and Clark, 974): f A 4 He, t He t, e Re, t 8Re t, Re t, 3 (8) where the dex A means annular space and is the conductance of the Bgham fluid. For a ero yield stress,, and Eq. (8) is reduced to the lamar flow friction factor of a Newtonian fluid a annular space. Accordg to Fontenot and Clark (974), the fluid flow can be considered to be lamar if Re. t,.4 Dimensionless Governg Equations The axial position, the time and density are normalied accordg to the followg:, L t a t and L (9) The dimensionless velocity is defed as a function of the pipe let velocity ( the pressure gradient ( V, t P V, t, P, t V ) of a Newtonian steady-state flow: is a geometric parameter of the annular space which is defed as: V ) and the pressure as a function of P t, 3LV and P () P D h D D D D D D ln D D () For a circular pipe, is equal to., and for a narrow annular space (.5 D D ), its value is approximately.5. In the current work, all annular spaces are considered narrow.

4 Proceedgs of ENCIT Copyright by ABCM 3 th Brailian Congress of Thermal Sciences and Engeerg December 5-,, Uberlandia, MG, Brail By employg the dimensionless variables of Eqs. (9) and (), Eqs. () to (3) can be re-written as: / 3 ( V ) () Re t / 3 ( V) ( VV) 3 P f V Re t Re (3) d dp (4) where Re (the Reynolds number), (the dimensionless compressibility), (the aspect ratio) and the Bgham number are the characteristic parameters of the problem: ovdh Re, 3 V P Re a, Dh Dh and B (5) L 8 V.5 Initial and Boundary Conditions As an itial condition, the drillg fluid is admitted to rest with the pipe and therefore, the velocity field is null ( V(, t ) ). Initially, the fluid density is uniform with the whole ( ). Therefore, accordg to Eq. (4), the itial pressure filed is also uniform and equal to ero Pt (, ). To restart the fluid flow of the gellified fluid, a pump pressuries another fluid to the pipe. Therefore, the gellified fluid starts to be displaced. It is assumed that the pump provides a constant volumetric flow at the pipe let or a constant average velocity ( V(, t ) ). At the pipe outlet, the density is assumed and consequently, the pressure, as constant ( (, t ) and P (, t) 3. SOLUTION OF THE EQUATIONS 3. Discretiation of the Equations ). The governg equations of the problem are discretied by employg the Fite Volume Method, as proposed by Patankar (98). The pressure and velocity grids are staggered, as the pressure is evaluated at the fite volume faces and the velocity at the center of the volumes. Fig. shows a grid scheme for an annular space doma. The doma is divided to N fite volumes of length. In Fig., the pressure position is dicated by a small i and the velocity position by a capital I. With this grid, i ranges from to N and I, from to N. Note that these are average values of the variables across the sectional area. The fully implicit and the first order up-wd schemes are employed as terpolation functions time and space respectively. i I i i I i I i i N I N i N P V V I P i i P N N VN PN N 3. Solution Method Figure. View of the employed grid. The non-lear discretied equations are solved iteratively at each time-step. The whole pipe is divided to N fite volumes of length, as shown Fig.. In the current work, a 5 volume grid is employed. Initially, the pressure and velocity fields are estimated and the governg equations are solved iteratively until convergence. After convergence, the solution evolves to the next time-step and the computation is repeated. This time evolution takes places until the steady-state is reached.

5 Proceedgs of ENCIT Copyright by ABCM 3 th Brailian Congress of Thermal Sciences and Engeerg December 5-,, Uberlandia, MG, Brail As the fluid flow is transient and compressible, the equations are hyperbolic. To avoid stabilities and numerical errors, the literature (Wylie et al., 993, Fortuna, ) suggests that the time and space grids be related by the pressure wave speed accordg to: t CFL Independently to discretiation scheme used (implicit or explicit), a good practice to reduce numerical diffusion and/or dispersion is to keep CFL. (Feriger and Peric, 996). As the pressure and velocity grids are a half volume staggered, the best results were observed for CFL.5. Therefore, CFL.5 is employed for all cases shown below. 4. RESULTS AND DISCUSSION 4. Model Validation In order to validate the model, the results of an analytical solution are compared with the current model results. As Eqs. (), (3) and (4) do not have a known analytical solution, they are simplified for low compressible flow case. The hypothesis are: i) the non-lear advective terms, V e V V, are disregarded mass and momentum conservation equations, respectively; ii) the density is substituted to the mass conservation equation by employg the equation of state (Eq. (4)); iii) the mass conservation equation is substituted to the momentum conservation equation (Eq. (3)) and iv) the fluid is considered Newtonian ( or B ). The mass and momentum conservation equations are derived respect to and t, respectively, and they are subtracted from each other: (6) V V V t t (7) where is a constant that represents the viscous effect ( 3 Re ). This second-order lear equation depends only on the velocity and equivalent equation can be obtaed for pressure: P P a P t t For the constant flow rate boundary condition at the pipe let, the solution of equations Eqs. (7) and (8) can be written as: (8) t V, t e s j cosjt sjt j j j (9) t e P, t cos s cos j j jt jt () j j 4 j where j j and 4. j j For an annular pipe with Re 5,., B,. and.5, is equal to.98. Figs. 3 and 4 compare the current model and analytical solution results. Fig. 3 presents the time change of pressure three positions along the pipe; at the let ( ), at the middle (.5 ) and near pipe outlet (.9 ) and Fig. 4 shows the velocity change at.,.5 and at the pipe outlet ( ). As already said, a 5 fite volume grid was employed the comparison. As can be seen, the results are quite close. Grid refg can approximate the results even more. However, the computational time creases exponentially. Therefore, the 5 grid volumes are used for all simulation shown below. One can see that both pressure and velocity show oscillation with time. These oscillations are due to pressure propagation that reflects both pipe ends. The pressure behavior Fig. 3 can be understood by analyg the pressure change with the axial position several time stants (see Fig. 5). In Fig. 5a are shown the pressure field at five dimensionless times (.,.5,.5,.75 and.). Note that before t. the pressure wave did not propagate through the whole pipe. For t.5, the pressure wave traveled through half the pipe length. In other words, pump pressure is only felt on half of the pipe and the other half is kept at ero pressure.

6 Proceedgs of ENCIT Copyright by ABCM 3 th Brailian Congress of Thermal Sciences and Engeerg December 5-,, Uberlandia, MG, Brail Analytical Solution Figure 3. Time change of pressure. Re 5,., B,. e.5. Analytical Solution Figure 4. Time change of velocity. Re 5,., B,. e.5. In Fig. 5b are shown the pressure profiles between time t. and t 3.. At the pipe end and time t., the pressure wave reflects and starts to propagates on reverse direction; from the pipe outlet to the let. The propagation on this direction takes place until t., as the pressure reaches the let and reflects aga. The reflection occurs contuously before beg completely dissipation by the viscous force. Fig. 5c illustrates the pressure dissipation between 3. and 4. and also the pressure filed for t., which is almost the steady-state. In the steady-state, the pressure changes learly with the pipe length. a) b) c) Figure 5. Pressure change with pipe length for several dimensionless times. Re 5,,, B,, and, 5. One can see Fig. 3 that the let pressure rises contuously to its maximum value ( P.8 ) at approximately t.. At this time, the pressure wave reflects at the let and the pressure falls down immediately. The pressure then contues to decrease as the pressure wave propagates back to the outlet and forward to the let, when the pressure jumps aga at t 4.. This cycle takes place until the pressure wave is completely dissipated and the let pressure stabilies at.. For.9, the pressure peak length is shorter than its let counterpart. Nevertheless, the its first peak is 7.3 times higher than the steady-state value ( PRP. ). Fig. 4 shows that the fluid flow any position starts when the pressure wave reaches such position. Besides, the wave reflection alters significantly the fluid velocity, as the outlet velocity is about.4 times higher than the let value at time t Sensitivity Analysis This section presents the fluence of the ma parameters the fluid flow, namely, Reynolds number, dimensionless compressibility and Bgham number.

7 Proceedgs of ENCIT Copyright by ABCM 3 th Brailian Congress of Thermal Sciences and Engeerg December 5-,, Uberlandia, MG, Brail 4.. The Effect of the Reynolds Number The fluence of the Reynolds number the time change of let pressure ( ) can be seen Fig. 6. As the ertial effects decreases comparison to the viscous effect (reduction of Re ), the pressure wave undergoes a larger dissipation. The dissipation is related to the necessary time to reach the steady-state; the smaller the Reynolds number the larger the time to reach the steady-state. However, for higher Reynolds number, the pressure wave has enough energy to reflect several times before beg completely dissipated, creasg the time to reach the steady-state. Although Fig. 6 do not show the steady-state for Re, the steady-state is Reynolds number dependent ( PRP.4). The steady-state pressure is higher than. because the fluid is a Bgham plastic ( B.). In section 3..3, this result is better explaed. Fig. 7 shows the velocity changes with time for three different positions,.,.5 and, and Re. The outlet velocity starts to change only at t.8 because of pressure wave dissipation. The velocity.5 changes its time evolution at t.3. This faster crease of velocity takes place after the pressure wave gog forward and back to the middle of the pipe. This effect also accelerates the fluid at., as shown Fig Re= Re= Re= Re= Figure 6. Effect of Reynolds number on the time change of pressure at.., B.,. and Figure 7. Time change of V at.,.5 and. Re,., B.,. and The Effect of the Dimensionless Compressibility Fig. 8 shows the let pressure change as a function of the dimensionless compressibility. One can see the pressure wave dissipation creases with the dimensionless compressibility. Pressure peaks are only observed for smaller compressibility. Besides, to keep the same flow rate for a higher compressible fluid, a higher let pressure is necessary. For., the let steady-state pressure is.4, and for., this value is approximately 6. higher ( PRP.56). Despite the different Reynolds numbers, the curve for. of Fig. 8 and that for R e of Fig. 6 are identical. The analysis of Eqs. () and (3) shows the dimensionless governg parameters of the flow are:, 3 Re and B. Therefore, the results cocide because all three parameters of these two cases are the same. One can conclude that the effect of creasg of the aspect ratio is equivalent to the crease of the Reynolds number. Fig. 9 shows that the pressure wave reaches the pipe outlet at the expected time ( t. ) for the less compressible case and for the highest compressibility, this time creases to about t 5.. This delay is related to the pressure wave dissipation as the compressibility creases. Besides, the steady-state average velocity at the pipe outlet, as the let pressure, creases with the fluid compressibility. The difference between the imposed let velocity, V(, t)., and the steady-state outlet velocity is due to the conservation of mass. The high pressure at the pipe let makes the let density smaller than the outlet counterpart. In order to keep the same mass flow rate, the average outlet velocity must be higher than the let one. For the most compressible case, the outlet velocity is about.7 times higher than the let one ( VRP ( ).7 ).

8 Proceedgs of ENCIT Copyright by ABCM 3 th Brailian Congress of Thermal Sciences and Engeerg December 5-,, Uberlandia, MG, Brail 4 3 =. =. =..5.5 =. =. =. 5 5 Figure 8. Effect of the time change of P at. Re, B.,. and Figure 9. Effect of the time change of V at. Re, B.,. and The Effect of the Bgham Number The Bgham number is directly related to the yield stress of the fluid. A fluid with a high yield stress requires a high pressure to start-up. The effect of the Bgham number on the flow start-up is now analyed. Fig. shows that the steady-state pressure for a low compressibility, Newtonian ( B=) fluid stabilies at.. To provide the same flow rate, the steady-state pressure creases with the Bgham number. For B=5., the let steady-state pressure is 7.3 larger than the Newtonian fluid pressure. As shown Fig., the pressure oscillation are reduced with the crease of the Bgham number. In comparison with other Bgham numbers, the pressure wave takes a longer time to reach the pipe end for B= 5.. Besides, the pressure peak at.9 is relatively small comparison to the steady state pressure B= B=.5 B=. B= B= B=.5 B=. B= Figure. Effect of B the time variation of P at. Re,.,. and Figure. Effect of B the time variation of P at.9. Re,.,. and CONCLUSIONS This work presents a mathematical model to simulate the flow start-up of drillg fluids. The viscous effect is taken to account by employg the friction factor concept. The governg equations (conservation of mass, momentum and the equation of state) are normalied and discretied by the Fite Volume Method, implicit and up-wd scheme. The system of non-lear algebraic equations is solved iteratively at each time-step. The numerical results are compared to analytical solution results for Newtonian fluids and a good agreement is obtaed. A sensitivity analysis of Reynolds and Bgham numbers, dimensionless compressibility and aspect ratio is conducted and the conclusions are: i) The smaller the Reynolds number the higher the pressure wave dissipation and consequently, the smaller the pressure peaks. The reason for that is the crease of the viscous effect comparison to the ertia effects; ii) Low compressibility fluids present more oscillations. As the compressibility creases the oscillations reduce or even disappear;

9 Proceedgs of ENCIT Copyright by ABCM 3 th Brailian Congress of Thermal Sciences and Engeerg December 5-,, Uberlandia, MG, Brail iii) The aspect ratio is provides the same effect as the Reynolds number. In other words, for a large pipe length comparison the diameter (small aspect ratio), there is a large region to dissipate the pressure energy, contributg to a smooth pressure propagation. The opposite is also true for a large aspect ratio. iv) The higher the Bgham number the larger pressure magnitude and the smaller the pressure oscillation. The reason is the larger apparent viscosity as the Bgham number creases. 6. ACKNOWLEDGEMENTS The authors would like to thanks PETROBRAS S/A, ANP (Brailian National Oil Agency) and CNPq (The Brailian National Council for Scientific and Technological Development) for their fancial support to this work. 7. REFERENCES Anderson, J.D.,99, Modern Compressible Flow, Ed. McGraw-Hill, N. York, U. States, 65 p. Cawkwell, M.G. and Charles, M.E.,987, An Improved Model for Start-up of Pipeles Contag Gelled Crude Oil, J. of Pipeles, Vol. 7, pp Chang, C., Rønngsen, H.P. and Nguyen, Q.D.,999, Isothermal Start-up of Pipele Transportg Waxy Crude Oil, J. of Non-Newtonian F. Mechanics, Vol. 87, pp Davidson, M.R., Nguyen, Q.D., Chang,C. and Ronngsten, H.P.,4, A Model for Restart of a Pipele with Compressible Gelled Waxy Crude Oil, J. of Non-Newtonian F. Mechanics, Vol. 3, No. -3, pp Feriger, J.H. and Peric, M., 996, Computational Methods for Fluid Dynamics, Ed. Sprger-Verlag, Berl, German, 364 p. Fontenot, J.E. and Clark, R.K.,974, An Improved Method for Calculatg Swab and Surge Pressures and Circulatg Pressures a Drillg Well, Soc. of Petroleum Eng. J. (SPE 45), Vol. 4, No 5, pp Fortuna, A. O,, Técnicas Computacionais para Dâmica dos Fluidos. Conceitos Básicos e Aplicações, Ed. da Universidade de São Paulo, S. Paulo, Brail, 46 p. Patankar, S.V., 98, Numerical Heat Transfer and Fluid Flow, Ed. Taylor & Francis, Washgton, U. States, 97 p. Sestak, J., Cawkwell, M.G., Charles, M.E. and Houskas, M.,987, Start-up of Gelled Crude Oil Pipeles, J. of Pipeles, Vol. 6, pp Vay, G., Wachs, A. and Agassant, J.F.,6, Numerical Simulation of Weakly Compressible Bgham Flows: The Restart of Pipele Flows of Waxy Crude Oils, J. of Non-Newtonian F. Mechanics, Vol. 36, No. -3, pp Vay, G., Wachs, A. and Frigaard, I., 7, Start-up Transients and Efficient Computation of Isothermal Waxy Crude Oil Flows, J. of Non-Newtonian Fluid Mechanics, Vol. 43, No -3, pp Wachs, A., Vay, G., and Frigaard, I. 9, A.5D Numerical Model for the Start Up of Weakly Compressible Flow of a Viscoplastic and Thixotropic Fluid Pipeles. J. of Non-Newtonian F. Mechanics,Vol. 59, pp Wylie, E. B., Streeter, V. L., and Suo, L., 993, Fluid Transients Systems, Ed. P. Hall, N. Jersey, U. States, 463 p. 8. RESPONSIBILITY NOTICE The authors are the only responsible for the prted material cluded this paper.