Non-Isothermal Displacements with Step-Profile Time Dependent Injections in Homogeneous Porous Media

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1 Proceedings o the World Congress on Mechanical, Chemical, and Material Engineering (MCM 015) Barcelona, Spain July 0-1, 015 Paper No. 93 Non-Isothermal Displacements with Step-Proile Time Dependent Injections in Homogeneous Porous Media Qingwang Yuan Development Research Department, CNOOC Research Institute, China yuanqw4@cnooc.com.cn Jalel Azaiez The University o Calgary, Department o Chemical and Petroleum Engineering Calgary, Alberta Canada TN 1N4 azaiez@ucalgary.ca Abstract -Miscible non-isothermal low displacements in homogeneous porous media are modeled and analyzed in lows that involve step-proile velocities that alternate between injection and extraction. The viscosity is assumed to vary with both the concentration and the temperature. The low is governed by the continuity equation, Darcy's law and the convection-diusion equations or the concentration and temperature with the assumption o thermal equilibrium. The problem is ormulated and solved numerically using a combination o the highly accurate spectralmethods based on the Hartley s transorm and the inite-dierence technique. Non-linear simulations were carried or a variety o parameters to analyse the eects o the time-dependence o the injection velocity on both the solutal and the thermal ront. It is ound that or the same net low rate, time-dependent injections aect not only the solutal ront but also the thermal rom which can become unstable under time-dependent scenarios when it is known to be stable or lows with a constant injection velocity. The results o this study will be used to improve our understanding o the coupling between heat and mass transer in lows in porous media and to optimize a variety o non-isothermal low displacements. Keywords: Thermo-Viscous ingering, Time-dependent injections, Thermal equilibrium, Homogeneous porous media, Numerical simulations. 1. Introduction Instability at the interace between lowing solutions in a porous medium may takes place as a result o viscosities and/or densities mismatch between the luids. This instability develops in the orm o intruding ingers and is reerred to as viscous ingering or Saman-Taylor instability in case o viscosities mismatch, and as density ingering or Rayleigh-Taylor instability in the case o densities mismatch between the luids. Extensive reviews o such lows have been presented by Homsy (1987) and McCloud and Maher (1995). These ingering phenomena occur in enhanced oil recovery, ixed bed regeneration, groundwater lows, CO sequestration, and soil remediation and iltration, etc. (Hejazi and Azaiez, 010). In most cases, viscous ingering is undesirable since it reduces the sweep eiciency, while in others it can be actually desirable as it promotes mixing. To date, most o the existing studies dealing with this type o instability have been limited to isothermal lows with constant injection velocities. However, in a number o practical applications the displacement velocity is time dependent and involves heat transer. Examples o such applications include a number o enhanced oil recovery (EOR) processes such as the cyclic steam stimulation (CSS) that involves three stages o injection, soaking and production (Mago et al., 005) as well as the CO hu-andpu technique that involves cyclic injection o liquid CO or heavy as well as light oil enhanced recovery (Monger et al., 1991). The success o these processes which can be run as either miscible or immiscible 93-1

2 depends on the importance and duration o each stage o the cycle in conjunction with the eiciency o heat exchange between the dierent components. Recently, Dias et al., (010a; 010b; 01) investigated the possibility o attenuating the ingering instability or immiscible lows in a radial geometry using time-dependent low rates. They reported the optimal low rates or linear low regime and nonlinear low regime. More recently Yuan and Azaiez (014, a-b) examined the eects o step-size dependent lows on the eiciency and stability o reactive displacements in porous media. The authors showed that it is possible to control the amount o chemical product through a judicious choice o the nature and cycle o the time-dependent displacement. However all the studies examining time-dependent injections have so ar been limited to isothermal lows. There are on the other hand a number o studies that have analyzed non-isothermal displacements but under constant injection velocities. These include the studies by Pritchard (009), Mishra et al. (010), Islam and Azaiez (010) and Sajjadi and Azaiez (013, a-b). The present study deals with non-isothermal miscible displacements in rectilinear Hele-Shaw cells under sinusoidal low velocities. The objective o the study is to determine the eects o the requency, amplitude and phase o the velocity on the hydrodynamic instability and to characterize these eects both qualitatively through concentration contours and quantitatively through the sweep eiciency. The objective is to propose guidelines or the choice o the requency, amplitude and phase that allow one to control the low towards either enhancing or attenuating the instability.. Flow Coniguration and Mathematical Model A two-dimensional non-isothermal displacement in which both luids are incompressible, nonreactive and ully miscible is considered. The low takes place in the horizontal direction in a homogeneous medium o constant porosity and permeability k. Fig. 1. Schematic o rectilinear Hele-Shaw cell/two-dimensional porous medium. A luid (Phase I) o viscosity and uniorm temperature 1 T is injected with a time-dependent 1 velocity U(t) to displace a second one (Phase II) o viscosity and uniorm temperature T. Here, the direction o the low is along the x axis and the y axis is parallel to the initial plane o the interace (Figure 1). The length, width and thickness o the medium are denoted by L, W and e respectively. The mathematical model describing the non-isothermal miscible displacement low involves the conservation o mass and momentum and mass-energy balance equations:. v 0 (1) 93-

3 μ p v () K C C D C C t v. (3) T T D T T t v. (4) In the above equations, v=(u,v) while and stand or the gradient and Laplacian operators in the (x,y) plane. Furthermore, p represents the pressure, the viscosity, k the medium permeability and its porosity. The concentration o the solvent (displacing luid) is denoted by C while the temperature o the solid-luid system which is assumed to be in local thermal equilibrium is T. The corresponding mass and thermal diusion coeicients are reerred to by D C and D T, respectively. The above equations model the propagation o two ronts, one associated with temperature and will be reerred to as the thermal ront while the other corresponds to mass transport and will be called the solutal ront. The parameter that we shall reer to as the thermal lag coeicient can be interpreted as the ratio o the speed o the thermal ront to that o the solutal ront (Bejan and Nield, 006, Pritchard, 009, Islam and Azaiez 010): (5) Cp Cp ( 1 ) Cp s s In the above equation,, s Cp, s, and and the luid phase respectively. Note that 1 Cp are the density and heat capacities o the solid phase with higher values corresponding to smaller dierence between the velocities o concentration and temperature ronts and lower values corresponding to higher rates o dissipation o heat. Throughout this study we will assume that the changes in the luid density and heat capacity are suiciently small such that can be considered constant. The governing equations are made dimensionless as in Islam and Azaiez (005) and Yuan and Azaiez 014, and are expressed in a Lagrangian reerence rame moving with the dimensionless displacement velocity u (t). This results in the ollowing dimensionless equations (using the same notation as or dimensional quantities):. v 0 (6) p μ[ v u (t) i] (7) C v. C (8) t v. ( 1) Le (9) t In the above equations Le DT DC PeC PeT is the Lewis number where Pe C and Pe T are the solutal and thermal Péclet numbers, respectively. Two additional dimensionless groups are also involved, namely the Péclet number Pe UL D and the cell aspect-ratio C A L H that appear in the boundary conditions (Sajjadi and Azaiez 013). To complete the model, a orm or the dependence o the viscosity on concentration and temperature must be speciied. Following Pritchard (009), Islam and Azaiez (010) and Sajjadi and Azaiez (013), exponential dependence is adopted: 93-3

4 μ(c, θ) = exp [β C (1 c) + β T (1 θ)] (10) In this equation c is the natural logarithm o the viscosity ratio 1 / in an isothermal miscible displacement and T is the natural logarithm o the ratio o the viscosity 1T / T in a single luid low with two dierent temperatures at the inlet and outlet boundaries. 3. Numerical Procedure The equations are expressed using a stream-unction vorticity ormulation, where the velocity ield, the stream-unction and the vorticity w (Hejazi and Azaiez 013, Yuan and Azaiez 014, a-b). With this ormulation, the continuity equation is satisied automatically and the governing equations take the orms: C C C (11) t θ θ λ (λ 1) Le( ) t (1) w βc[ ( u (t)) ] βt [ ( u (t)) ] (13) w (14) The partial dierential and algebraic equations are solved using a highly accurate pseudo-spectral method based on the Hartley transorm in conjunction with a semi-implicit predictor-corrector timestepping method along with an operator-splitting algorithm (Islam and Azaiez 005). The code was validated by comparing the concentration contours or constant velocity under isothermal with the study by Islam and Azaiez (005) and or non-isothermal displacements with the results o Islam and Azaiez (010) and Sajjadi and Azaiez (013b). Furthermore, the convergence o the numerical solution was examined by considering cases with dierent spatial resolutions varying rom 18x18 to 51x51 while varying the time step accordingly. Since a resolution o 56x56 resulted in inger structures similar to those obtained with larger number o grid points, it was adopted in all subsequent simulations. 4. Results In this section, the low dynamics will be analyzed in the case o are solved numerically in the case o step-size time dependent injection velocity proiles u (t) with a period T. Such types o velocity proiles correspond to many practical applications where the displacing luid is injected and produced (or soaked) at constant velocities in a certain period o time, or vice versa. Each cycle consists o two stages. The irst with a constant velocity (u (t) = u 1 ) lasts a time (t 1 ) and is ollowed by the second stage where the velocity switches to a new value (u (t) = u ) and lasts a time (t = T t 1 ). The objective is to compare the results with those o a constant injection velocity (u = 1), which requires the average velocity over a period to be also equal to one (u (t) = 1). This implies that the net injected low in both the time-dependent and constant velocity displacements is the same. This requires that t 1 = (u (t) u )T (u 1 u ) = (1 u )T (u 1 u ). Hence, the time-dependent displacement is ully characterized by both u 1, u and the cycle period T. It should be also mentioned that in cases where the displacement velocity changes sign and the process switches rom injection (positive velocity) to extraction (negative velocity) or vice versa, zones with 93-4

5 positive mobility ratios will change rom unstable to stable state, or vice versa. This is also true or zones with negative mobility ratios. However or consistency, the zones will be reerred to as stable or unstable based on the low displacement o the constant injection velocity. In all that ollows, the ollowing parameters are ixed as A= and Pe=500, Le=1, and =0.75. For brevity and illustration purposes, the time sequences will not be always presented necessarily at the same time intervals, and only the rames that reveal new and interesting inger structures that help in the discussion, are shown. In all contours, the red and blue colour ields correspond to the displacing and displaced luid, respectively Non-Isothermal Flows with Constant Injection Velocity In this section, dierent scenarios o non-isothermal displacements with unit injection velocity are presented. Figure shows the concentration contours or our dierent scenarios that correspond to positive or negative values o the mobility ratios. In the case o a low where a viscous cold luid displaces a less viscous hot one (β c = 3, β T = 3), the interace is stable. This is expected since in this case both the solutal and thermal ronts are under avourable displacement conditions. The interace diuses with time but no ingers develop (Fig-a). When the thermal ront is unstable while the solutal lone is stable (β c = 3, β T = 3), instabilities grow on the thermal ront and due to the coupling, this results in rather diuse instabilities that are observed in the concentration contours o Fig-b at late times. As discussed by Islam and Azaiez (010) and Sajjadi and Azaiez (013b), this coupling is stronger or larger values o the lag coeicient. (a) (b) (c) (d) Fig.. Concentration proiles or constant injection velocity at dierent times or (a) β c = 3, β T = 3, (b) β c = 3, β T = 3, (c) β c = 3, β T = 3, (d) β c = 3, β T = 3. Figure-c and Figure-d depict the results where the solutal viscosity is unavourable (β c = 3 > 0) and the thermal one is either avourable (β T = 3) or unavourable (β T = 3), respectively. In these cases, ingers develop rapidly resulting in a rapid breakthrough, notably in the case where both the solutal and thermal ronts are unstable (Fig-d). It must also be noted that the number o ingers are larger and their structures are more complex in this last case. 93-5

6 4.. Non-Isothermal Flows with Time-Dependent Injection Velocity The development o the low is now analysed in the case o a step-size time-dependent low displacement. The analysis will ocus on a scenario where the displacement is initiated by an injection ollowed by extraction (u 1 = 3, u = 1). It will also be limited to the case o a less viscous cold luid displacing a more viscous one (β c = 3, β T = 3). As discussed in the previous section, the low leads to an unstable solutal ront but the growth o the ingers is mitigated by the stable thermal ront. Fig.3 show the concentration contours or a constant injection velocity (u = 1), and two time-dependent lows (u 1 = 3, u = 1) with a period T = 00 (case1) and T = 400 (case). (a) (b) (c) Fig. 3. Concentration proiles or constant and time-dependent injection velocities at dierent times and or dierent periods. β c = 3, β T = 3, u 1 = 3, u = 1. (a) Constant velocity, (b) T = 00, (c) T = 400. It is clear that in the early stages o the displacement, the time-dependent lows are more unstable than their constant injection velocity counterpart. Fingers are well developed by t=50 while none have developed in the case o the constant unit velocity. Moreover, up to a time corresponding to hal a period o case1, the structures are identical or both time-dependent injection lows. This is not surprising since up to that time the velocity is the same or both cases. Past this time, the low in case1 moves to an extraction mode where the low reverses and as a consequence the displacement becomes avourable. This low reversal is responsible or an attenuation o the instability and the interace becomes diuse with no 93-6

7 sharp ingers. At the same time, case is still in the injection mode and the ingers are able to develop and extend urther in the low. The situation is reversed past the hal-period o case, where the low now shits to extraction and the ingers are strongly damped in case while they are reinvigorated in case1. Actually, as the long stage o extraction, case ends up with a very wide diused interace that subsequently is not propitious or the growth o strong ingers. In this low scenario, case1 ends up being the most unstable displacement and results in the earliest breakthrough time, while case is the least unstable one and exhibits an interace that is less unstable than that o the constant injection low. (a) (b) (c) Fig. 4. Temperature proiles or constant and time-dependent injection velocities at dierent times and or dierent periods. β c = 3, β T = 3, u 1 = 3, u = 1. (a) Constant velocity, (b) T = 00, (c) T = 400. The eects o the time-dependent low on the corresponding thermal ront are depicted in Figure 4. It is clear that in the constant injection low, the thermal ront is basically stable and the small wavy interace is attributed to the coupling with the solutal ront. This coupling is however short and the ronts separate as the thermal ront recedes due to heat losses to the porous medium while the solutal ront advances with strong ingers. This changes when the injection velocity is time-dependent and the coupling is now stronger and more lasting leading to shaper and more developed thermos-ingers. This is due to two actors. The irst one is a natural result o the more unstable solutal ront that is able to grow at the early stages o the displacement when the two ronts are still close. This allows the thermal ront to 93-7

8 develop instabilities (see Fig.4b-c at t=50 and t=100) when that o the constant injection is still stable. Even though these thermos-ingers are not well developed they become the precursors o the ingers that develop later even when the low reverses into an extraction. The second actor is associated with the initial stronger injection velocity (u 1 = 3) that delays the eects o heat diusion into the porous medium and help maintain a sharper thermal ront where ingers can grow easier. It is interesting that in spite o the low reversal, the thermos ingers were able to maintain their growth and were not supressed even at late times. 5. Conclusion In this study, the eects o time-dependent step-proile injection velocities on the development o ingering instabilities in homogeneous porous media were examined. The analysis ocuses on nonisothermal miscible displacements where both heat and mas are transported by the low. The development and progress o the instabilities are illustrated through iso-suraces o both the concentration and the temperature o the luids. The dynamics o the low were analysed or an injection-extraction process and compared with those arising rom a constant injection velocity that results in the same average low rate. It is ound that the time-dependent velocity aects the hydrodynamic instability and in particular it has an important impact on the thermal ront due to the coupling between the solutal and the thermal ronts. The results o the study will be a cornerstone or understanding and optimizing processes that involve timedependent injections o hot/cold luids in porous media. Acknowledgements Financial support rom the China Scholarship Council (CSC) and the Natural Sciences and Engineering Research Council o Canada (NSERC) are grateully acknowledged. The authors would like also to acknowledge WestGrid or providing computational resources. Reerences Bejan, A., & Nield, D. (006). Convection in Porous Media, 3 rd Edition. Springer. Dias, E. O., Parisio, F., & Miranda, J. A. (010a). Suppression O Viscous Fluid Fingering: A Piecewise- Constant Injection Process. Phys. Rev. E., 8, Dias, E. O., & Miranda, J. A. (010b). Control O Radial Fingering Patterns: A Weakly Nonlinear Approach. Phys. Rev. E., 81, Dias, E. O., Alvarez-Lacalle, E., & Carvalho, M. S. (01). Minimization O Viscous Fluid Fingering: A Variational Scheme For Optimal Flow Rates. Phys. Rev. Lett., 109, Hejazi, H., & Azaiez, J. (013). Nonlinear Simulation O Transverse Flow Interactions With Chemically Driven Convective Mixing In Porous Media. Water Res. Res., 49, 8, Homsy, G.M. (1987). Viscous Fingering In Porous Media. Ann. Rev. Fluid Mech., 19, Islam, N., & Azaiez, J. (005). Fully Implicit Finite Dierence-Pseudo Spectral Method For The Simulation O High Mobility-Ratio Miscible Displacements. Int. J. Num. Meth. Fluid., 47, Islam, N., & Azaiez, J. (010). Miscible Thermo-Viscous Fingering Instability In Porous Media: Part : Numerical Simulations. Transp. Porous Media, 84, Mago, A., Barruet, M., & Nogueira, M. (005). Assessing The Impact O Oil Viscosity Mixing Rules In Cyclic Steam Stimulation O Extra-Heavy Oils. Proc.. o the SPE Annual Technical Conerence and Exhibition, SPE McCloud, K.V., & Maher, J.V. (1995). Experimental Perturbations to Saman-Taylor Flow. Physics Reports, 60, Mishra, M., Trevelyan, P. M., Almarcha, C., & De Wit, A. (010). Inluence O Double Diusive Eects On Miscible Viscous Fingering. Physical Review Letters, 105, Monger, T., Ramos, J., & Thomas, J. (1991). Light Oil Recovery From Cyclic CO Injection: Inluence O Low Pressures Impure CO, And Reservoir Gas. SPE J., 6, 5-3. Pritchard, D. (009). The Linear Stability O Double-Diusive Miscible Rectilinear Displacements In A Hele-Shaw Cell. Eur. J. o Mechanics, B/Fluids, 8,

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