Abstract In this paper, a numerical model for volumetric prediction of sand production in injector wells is presented. Sanding in injector wells is mainly associated with the backflow and crossflow generated during shutin in addition to the waterhammer pressure pulsing in the wellbore due to fast flow rate changes. Emphasis is given to the geomechanical aspects of sanding such as rock fatigue due to cyclic pressure changes and the concomitant degradation of bonding between the sand grains. This model is robust in capturing the key parameters in the sandstone behavior such as stressdependent elasticity, hardening, softening and dilatancy. Rock degradation is considered to be the necessary condition for sand production which is assumed to obey the erosion mechanics. The model is calibrated and validated using physical model tests carried out under various stresses and fluid flow conditions. The numerical model has been utilized to analyze sanding potential in a cased and perforated injector which will be presented to demonstrate the field application of the proposed concepts.
SPE 156394 A Numerical Model for Predicting the Rate of Sand Production in Injector Wells Azadbakht, S., Jafarpour, M., Rahmati, H., Nouri, A.; University of Alberta; Vaziri, H., BP America Inc.; Chan D., University of Alberta Copyright 2012, Society of Petroleum Engineers This paper was prepared for presentation at the SPE Deepwater Drilling and Completions Conference held in Galveston, Texas, USA, 20–21 June 2012 This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s) Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s) The material does not necessar ily reflect any position of the Society of Petroleum Engineers, its officers, or members Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied The abstract must contain conspicuous acknowledgment of SPE copyright Abstract In this paper, a numerical model for volumetric prediction of sand production in injector wells is presented Sanding in injector wells is mainly associated with the back-flow and cross-flow generated during shut-in in addition to the waterhammer pressure pulsing in the wellbore due to fast flow rate changes Emphasis is given to the geomechanical aspects of sanding such as rock fatigue due to cyclic pressure changes and the concomitant degradation of bonding between the sand grains This model is robust in capturing the key parameters in the sandstone behavior such as stress-dependent elasticity, hardening, softening and dilatancy Rock degradation is considered to be the necessary condition for sand production which is assumed to obey the erosion mechanics The model is calibrated and validated using physical model tests carried out under various stresses and fluid flow conditions The numerical model has been utilized to analyze sanding potential in a cased and perforated injector which will be presented to demonstrate the field application of the proposed concepts Introduction Sand production is a common problem in production and injection wells Extensive research has been carried out in the past couple of decades to identify the key parameters affecting initiation and severity of this phenomenon Detection and management of sand production is more obscure when it comes to injection wells as there is no fluid production and hence no indication of sand initiation and severity Practical problems associated with sand production include erosion of pipelines and surface facilities, reduction in productivity, intervention costs and complexities and other environmental effects These problems cost the oil industry billions of dollars annually (Nouri, 2004) On the other hand, a controllable amount of sand production may omit the need for installing more complex active sand controls involving use of gravel packs which have been used extensively to reduce and avoid sand production from unconsolidated formations (Saucier, 1974) Therefore, understanding the sand production mechanisms and the ability to predict and manage the rate of sand production are beneficial A linked finite difference-finite element (FE-FD) code is used for the sanding assessment of an injection well This model can simulate the impact of injection pressure and shut-in cycles, including the effects of inflow differential pressure (DP) (due to cross-flow or back-flow) and waterhammer (WH) pulses on sanding It also accounts for the in situ strength and its gradual degradation due to stress and pressure changes By simulating the shut-in cycles, the main effects of well operation over the wellbore life can be accounted for The model incorporates the essential physics in water injection operations and accounts for the critical factors with respect to the rock behavior and sanding mechanisms, including the influence of flow rate on sand production Brief Physics of Sand production When a fluid is injected into a reservoir, the following occur depending on the formation consistency: Increase in pressure leads to reduction in effective stress and hence reduction in particleto-particle frictional resistance which particularly impacts the sanding response in unconsolidated or disaggregated materials Under high injection pressure and/or waterhammer pressure pulsing, particularly if exceeding the overburden stress, sand may reach a fluidized state This condition may exacerbate sanding right after shut-in depending on the injection magnitude and shutin rate In weakly consolidated formations, injection and shut-in cycles particularly if combined with waterhammer pulses may result in the destruction of cementation and turn the material into an unconsolidated sand mass with consequences as discussed above In competent and cemented formations, high injection may result in development of fractures but no major sanding is expected as a result of this phenomenon Every cycle of shut-in and start-up will promote sand face fatigue (gradual weakening of cementation due to strain cycles) and this may eventually breakdown all cementation The extent and acceleration of degradation depend on the magnitude of injection, shut-in rate (impacting magnitude of cross-and back-flow and waterhammer intensity) and strength properties of the formation In water injectors, the injection pressure and water are likely to destroy the multi-grain structure and hence create a high potential and volume of sanding Numerical Model Description A numerical tool that links a finite difference (FD) code with a finite element (FE) code is used for this study The distinguishing features of this model that are of critical importance to the proposed study include: The drilling phase is simulated in a fully coupled manner to capture the critical processes that happen during drilling more accurately The injection/shut-in cycles are performed using sequential coupling In that, the FD code performs the mechanical calculations and the FE code does the fluid flow calculations This is done to take advantage of fast fluid flow calculation of the FE code The model allows for rock strength degradation with changes in stress and strain associated with the injection cycles and the waterhammer pressure pulses The model is capable of computing sand production The model accounts for the very rapid inflows following shut-ins and hence can capture the effects of the shut-in rate It can track changes in hydro-mechanical properties (e.g., stiffness, permeability) as a result of sanding and changes in effective stress with injection Two main components of this numerical model are the constitutive model and sanding criterion which are briefly described below SPE 156394 Constitutive model The importance of precise and descriptive modeling of constitutive behavior of rocks can not be overemphasized This part plays a vital role in any sand production simulation As shown schematically in Fig 1, results of laboratory experiments indicate that granular materials usually demonstrate strain softening at Low Effective Confining Stress (LECS) and strain hardening at the state of High Effective Confining Stress (HECS) (Vermeer & de Borst, 1984; Sulem et al., 1999) These facts are taken into account in formulation of the yield surface which expands (strain hardening) or contracts (strain softening) as a function of the hardening parameter which will be introduced later Fig 1: Different stress-strain regimes at various confining stresses (Vaziri et al., 2007) Elasto-palstic constitutive models have shown to model sandstone behavior with adequate accuracy In this paper the same approach as Sulem et al (1999) and Nouri et al (2009) is undertaken in which a bilinear Mohr-Coulomb (MC) model is calibrated using laboratory tests This model involves the calibration of elastic properties, initial and peak yield surfaces, friction hardening, cohesion softening and mobilized dilation angle The last three parameters are expressed as a function of the hardening parameter that is itself a function of principal plastic strains For the sake of brevity, the details of the constitutive modeling are not presented here but the interested reader can refer to the above references Sanding criterion The sanding criterion used in this study is based on erosion mechanics (Detournay, 2006) In this logic, it is assumed that sanding will start when both of the following conditions are met: a) All cohesion (which represents cementation) is lost; that is, real cohesion degrades to zero, and b) The totally disaggregated or cohesionless sand particles are broken away from sand mass and carried into the perforation/wellbore by the action of hydrodynamic forces (erosion process) This process causes the porosity of the elements to increase (as a result of sand removal) until it reaches the critical porosity, i.e., the porosity at SPE 156394 which the rock matrix collapses (Rahmati et al, 2011) There are almost no field cases involving injectors where the sanding events have been recorded as they occurred This makes validation or even calibration of any modeling effort difficult Having said that, the following measures are undertaken into considerations to deal with the general uncertainties: • Maximize value from laboratory tests performed on formation rock samples The rigor involved in back-analyzing properties that capture the rock behavior is shown later along with validation • Use rigorous numerical analyses In this case, we have used a complex numerical model along with a fine mesh and complex procedures to capture very short duration events, such as pressure waves, relatively short events such as shut-in and injection build up and longer term operations during steady injection Numerical Model Calibration Calibration of the numerical model involves calibration of both constitutive model and the sanding criterion Constitutive Model Calibration The bilinear MC with combined hardening/softening model was calibrated using a series of uniaxial and triaxial tests Fig shows this model schematically Fig 2-a shows the hardening behavior wherein line (0) stands for the initial yield surface Once a stress state reaches line (0), plastic deformation begins Further loading increases the friction coefficient or the slope of the line up to the peak yield surface (line 1) This is shown by the upward arrows from line (0) to line (1) Up to this point, the tension cut-off is approximately constant both for the low and high effective confining stresses ( and ) Additional deformation after the peak results in the softening of the material and shrinkage of the yield surface This is demonstrated in Fig 2-b by the downward arrows from line (1) to line (2) During softening, tension cut-off shrinks to the residual value ( ), and it is equal to zero for fully degraded sandstone, as depicted in Fig 2-b However, the friction coefficient remains constant That is, the line is lowered to the residual state with the same slope as that of the peak Line (2) is the new yield surface during softening when the residual tension cut-off gradually decreases to zero leading to the development of shear bands (Jafarpour et al., 2012) Tension cut-off can be related to the mobilized cohesion, C, by the following relationship: q C / tan where (1) is the friction angle of the rock In this work, the hardening parameter is Equivalent Plastic Strain (EPS), which is defined by the following (Vermeer and de Borst, 1984): 2 ö2 æ1 1 EPS = ç ( De1ps - Demps ) + ( Demps ) + ( De3ps - Demps ) ÷ è2 ø 2 where emps (2) e1ps e3ps (3) e jps , j 1,3 are the principal plastic shear strain increments The MC envelope as shown in Fig varies as a function of (EPS) and hence can simulate the strength degradation Fig shows the elastic properties of a pay sandstone layer with UCS of 1,250 psi as a function of the confining stresses Shear and bulk moduli increase with increase in confining stress which is due to the closing of pre-existing micro-cracks As plastic deformations start, friction and dilation are mobilized as a function of EPS until they reach the peak values after which they remain constant Fig shows the mobilized friction and dilation angles for the same sandstone Fig 2: a) Hardening and b) softening of the bilinear MohrCoulomb model (Sulem et al., 1999) SPE 156394 Fig 3: Bulk and shear moduli as a function of confining stresses Fig 6: Comparison of numerical and experimental stressstrain response for a triaxial test Sanding Model Calibration The main parameters in the sanding model are critical porosity, critical flow rate and erosion rate coefficient These parameters are calibrated using laboratory data obtained by testing perforated rock samples The rate of the produced sand mass is proportional to the specific flow rate (Detournay et al., 2006): Fig 4: Mobilized friction and dilation angles as a function of EPS After reaching the peak stress state, the rock enters the softening stage, which is demonstrated through cohesion degradation as shown in Fig ( Fig shows the comparison between the triaxial test measurements and the numerical results for the sandstone at a certain confining pressure As seen, the constitutive model predicts the rock behavior with reasonable accuracy ) (4) where is the specific mass flux, is the specific discharge normal to the boundary, is the critical value of specific discharge, λ is the erosion rate coefficient, is the rock porosity and is the grain density Papamichos et al (2001) showed that using a constant erosion coefficient may result in physically unrealistic behavior To improve prediction of sanding rate, they suggested an erosion coefficient λ as a function of EPS as follows: { Fig 5: Mobilized cohesion during hardening and softening )( ( ) (5) where EPSres stands for residual equivalent plastic strain and is calculated at residual strength state Using the Law of Conservation of Mass, the rate of the change of porosity is related to the rate of generated sand mass by the equation (Detournay et al., 2006): (6) where is the boundary surface area of the element and is the volume of the element Sand is produced at a rate given by Eq until the porosity of the element reaches a critical value As the porosity increases the material becomes less competent This phenomenon is represented in the model SPE 156394 by degrading the bulk and shear modulus with the increasing porosity as follows: (7) (8) pressure and time axes in Fig and Fig are not to scale In case of a PSD, the shut-in rate is in such a way that minimizes the resulting waterhammer pressure pulses whereas during a UPSD the shut down periods are rapid enough to create considerable WH pulses Some fieldscale transient analysis data were used to estimate the WH pressure pulses for the well under study After the element reaches the critical porosity, it is kept in the mesh at residual stiffness properties to represent infill materials Further details about sanding model calibration can be found in Rahmati et al (2011) Finite difference mesh and boundary conditions Fig shows a close-up of the FE mesh near the wellbore Two types of injection shut-downs are expected: planned shut-down (PSD) and unplanned (or emergency) shutdown (UPSD) Fig shows the schematics of an injection cycle with planned shut down Fig 9: Schematic of UPSD with cross-flow Depending on reservoir heterogeneity, fluid injection can create different pressure gradients in different layers due to differences in permeability, porosity, compressibility, etc Upon injection shut down, fluid may flow from high pressure layers with low permeability to layers having a lower pressure and usually higher permeability; a process which is known as interlayer cross-flow Another type of cross-flow is the flow of fluid from the high-pressure layer to the low-pressure layer through the wellbore This is called intra-well cross-flow Fig 10 shows schematically a possible scenario for intrawell cross-flow Upper layer has lower permeability and will retain the injection pressure, which upon shutdown will become the driving force to squeeze the injection fluid into the lower layer with higher permeability Fig 7: Close up of the FD mesh Fig 10: Schematic of intra-well cross-flow Fig 8: Schematic of PSD with cross-flow Fig shows a schematic of a cycle with unplanned shut down The time intervals for various sections of the injection/shut down cycles are selected in a way to assure establishment of steady state flow conditions The Depending on differential pressures and magnitude of fluid flow, cross-flow can have a serious impact on sanding behavior of a well Cross-flow is incorporated in the injection/shut down cycles by applying a drawdown (DD) after the shut-down period Also, cases are examined without the cross-flow effect, i.e., zero DD after the shut down 6 Basic Simulation Steps First, in-situ stresses are initiated and the model is solved for equilibrium Then, reservoir pressure is reduced to simulate the production induced depletion in the reservoir After depletion, multiple injection and shut-in cycles are applied to the perforation cavity The model computes changes in stress, strain and any associated degradation in strength, which along with seepage forces may result in sand failure and production SPE 156394 very early and picks up quickly in this case Although water weakening effect might seem inevitable in injector wells; proper measures can be taken to ensure the compatibility of the injected water and the formation rock to minimize the impact of this factor Water weakening effect Experimental observations indicate that water contact can have a high impact on rock strength and hence on sanding potential of a well (Santarelli et al., 2000; Han and Dusseault, 2002) Data from literature are used to correlate the UCS of samples with non-native water saturation to the UCS of dry samples This correlation is then used to reduce rock strength parameters accordingly to account for water weakening effect in the numerical simulations Results The numerical model has been used to perform some sensitivity analysis to assess the effect of various parameters on sanding Figures 11-13 show the results of sensitivity analysis for one of the rocks in this study In all the cases, only one parameter has been changed at a time and everything else is kept the same In all these plots, the horizontal axis shows the time after the start of injection operations Fig 11 shows the effect of perforation size on sanding response In case of a larger perforation size, sanding starts earlier and has a considerably higher magnitude This points out the importance of proper perforation size selection Generally, the smaller the perforation size, the less the risk and severity of sanding but care should be taken not to compromise the well productivity in this process A more in-depth description of the effect of this factor on sanding is yet to be investigated Fig 12: Effect of water induced weakening on sanding Fig 13 demonstrates the effect of cross-flow on sanding Cross-flow appears to have the same effect as perforation size in terms of pattern and magnitude This exemplifies the impact cross-flow can have on sanding response of wellbores Fig 13: Effect of cross-flow on sanding The effect of rock strength on sanding behavior is shown in Fig 14 Three different rock types were used in this part having different UCS values and the result are plotted for the first, second and third year after the start of injection operations Such a parametric study can help the production engineers in selecting a rock strength cut-off below which the formation shouldn’t be perforated Fig 11: Effect of perforation size on sanding Fig 12 shows the effect of water induced weakening on sanding This parameter appears to have the highest impact on sanding response of the rock Sanding starts SPE 156394 Assuring the compatibility of the injected water with formation rock to reduce the water weakening effect Optimizing the perforation size as far as it doesn’t compromise well productivity Avoiding perforation of weak layers or using sand control devices in such layers Acknowledgment We thank BP for the permission to publish this paper The financial support provided by NSERC is also acknowledged Fig 14: Sanding response as a function of rock UCS Discussion and Conclusions To provide an insight into sanding behavior of injector wells and the effect of different parameters that contribute to sanding, a comprehensive numerical model is proposed Using geomechanics principles and basic physics of sand production, a set of criteria were incorporated into this model for describing conditions required for sanding initiation and propagation This numerical model can take into account the effects of the following parameters on sanding behavior of injectors: Rock strength Different rock types were used in this work having different UCS values The outcome can help the production engineers in selecting a rock strength cutoff below which the formation shouldn’t be perforated Water-weakening effect In water injectors, formation rock may lose some of its strength due to chemical and/or mechanical effects of water contact This study shows that this factor had a significant impact on sanding severity Cross-flow Effect In heterogeneous reservoirs, crossflow is a very likely phenomenon after the injection shutdown Results of this work show that this factor can have a considerable impact on sanding behavior of wellbores One should note that in this work, only one parameter is changed at a time to study the corresponding effect on sanding The fact is that in a real case scenario, all this factors are present and the overall sanding response of the formation is determined by taking into account the effect of all the individual parameters Results of this study shows that considering the following factors can be helpful in reducing the risk and severity of sanding in injector wells: Reducing the number of unplanned shut-ins in order to reduce the waterhammer events In case of inevitable unplanned shut-ins, it is ideal to make the shut-in time (time required to shut down the pumps and/or close the wellhead) as long as possible to reduce the severity of waterhammer pulses Nomenclature C&P = Cased and Perforated DD= Drawdown DP= Differential pressure EPS = Equivalent Plastic Strain EVS = Effective Vertical Stress HECS = High Effective Confining Stress LECS = Low Effective Confining Stress MC = Mohr-Coulomb PSD = Planned Shut-down P = mean stress q = tension cut-off = initial yield tension cut-off = peak tension cut-off = residual tension cut-off = tension cut-off at HECS = tension cut-off at LECS T= square root of the second invariant of the deviatoric stress UCS = Unconfined Compressive Strength USD = Unplanned Shut-down WH = Waterhammer e jps , j 1,3 = Principal plastic shear strain increments = friction coefficient H = friction coefficient at HECS L = friction coefficient at LECS = Friction angle = grain density References Detournay, C., Tan, C., Wu, B 2006 Modeling the mechanism and rate of sand production using FLAC 4th international FLAC symposium on numerical modeling in geomechanics, paper: 08-10 8 Han, G and Dusseault, M.B 2002 A Quantitative Analysis of Mechanisms for Water-Related Sand Production, paper SPE 73737 presented at the 2002 SPE Int Symposium and Exhibition on Formation Damage Control Held in Lafayette, LA, Feb 20-21 Jafarpour, M., Rahmati, H., Azadbakht, S., Nouri, A, Chan, D., Vaziri, H (in press) Determination of Mobilized Properties of Degrading Sandstone Journal of Soils and Foundations Nouri, A 2004 A Comprehensive Approach to Modeling and Eliminating Sanding Problems During Oil Production Ph.D dissertation, Dalhousie University, Halifax, Nova Scotia Nouri, A., Kuru, E., Vaziri, H 2009 Elastoplastic modeling of sand production using fracture energy regularization method Journal of Canadian Petroleum Technology, Vol 48, No.4, pp 64-71 Papamichos, E., Vardoulakis, I., Tronvoll, J., Skjaerstein, A 2001 Volumetric sand production model and experiment International journal for numerical and analytical methods in geomechanics, Vol 25, No 8, pp 789-808 Rahmati, H., Nouri, A., Vaziri, H., Chan, D (in press) Validation of predicted cumulative sand and sand rate against physical model test, JCPT Santarelli, F.J., Skomedal, E., Markestad, P., Berge, H.I and Nasvig, H 2000 Sand Production on Water Injectors: How Bad Can It Get? SPE Drill & Completion, 15, no.2, 132 Saucier, R.J., 1974 Considerations in Gravel Pack Design, Journal of Petroleum Technology, Vol 26, No.2, pp 205-212 Vaziri, H., Nouri, A., Hovem, K., Wang, X 2008 Computation of Sand Production in Water Injectors SPE Production & Operations Vol 23, No 4, pp 518-524 Vermeer, P.A., De Borst, R., 1984: Non-Associated Plasticity for Soils, Concrete and Rock, Heron 29, pp 1–62 SPE 156394