Rheological and geometric effects in cementing of irregularly shaped wells

Alondra Renteria1, Amir Maleki1, Ian Frigaard1,2

1 Mechanical Engineering, University of British Columbia, Vancouver, Canada2 Department of Mathematics, University of British Columbia, Vancouver, BC, Canada, V6T 1Z2

ABSTRACTLarge numbers of oil and gas wells, in Canadaand worldwide, allow leakage to surface fromthe reservoir. One common reason is associ-ated with unsuccessful mud removal during theprimary cementing operations, i.e. the drillingmud remains stuck in the narrow annular re-gion. Several factors have to be taken into ac-count in order to design a successful cementjob; namely, the geometry of the well, the rhe-ology of the fluids and the pumping schedule.In this study, we explore how geometric irreg-ularity can combine with fluid rheology to givea wide range of different behaviours.

INTRODUCTIONThis paper looks at both geometric and rheo-logical effects on primary cementing displace-ments, for simplicity focusing on vertical wells.Primary cementing is a process carried out onevery oil and gas well (typically at least twice),in which a steel casing is cemented into a newlydrilled borehole. The cylindrical casing is low-ered into the well creating an annular space be-tween its outside wall and the borehole innerwall. Casings run many hundreds of metresalong the well, penetrating different geologicalstrata. A range of casing diameters are used(larger near the top of the well, smaller in pro-duction zones), but with an annular gap that is2�3cm wide on average. The well is typicallyfull of drilling mud, which is a shear-thinningyield stress fluid, and this must be replacedwith a cement slurry to fill the annular space,where it will harden. The objectives are both tozonally isolate different fluid bearing strata inthe formation and to provide mechanical sup-port to the well. Primary cementing proceeds

by pumping a sequence of fluids down the in-side of the casing to bottom hole, returning up-wards in the annulus from the bottom. Thesefluids are designed (density and rheology) tohelp with the removal of the drilling fluid. Anoverview of the cementing processes is givenby Nelson & Guillot.14

The key geometrical factor in the annulardisplacement is the width of the annular gapand its uniformity. Partly this is influencedby use of centralizers, discussed below, whichmay reduce eccentricity but also constrict lo-cally, and partly by geological factors. Thelatter can related to weak formation or cas-ing/pipe connection locations along the well,where there may be washed out sections of theannulus, i.e. enlargements or washouts. Finally,according to the geomechanical stresses andwellbore orientation the borehole may have anelliptical cross-section, rather than circular.

The challenge of primary cementing is eas-ily understood by anyone who has cementedgarden paving stones into place or grouted largefloor tiles. The annular space in the well is gen-erally eccentric, meaning that to get good cov-erage of cement (full removal of the mud) weneed to remove the drilling mud from an an-nular gap that may be e.g. 5mm wide in thenarrow part of the annulus, pushing the cementslurry into the gap. Not only is the gap po-tentially narrow, but it also extends for many100’s of metres - unlike the floor tile/gardenpaver analogy. This is extremely difficult toachieve and over the years considerable efforthas been expended to model/simulate the pro-cess in order to improve fluid designs. Themodel used in this work is comprehensively de-rived by Bittleston et al.,1 although we in fact

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Figure 1. Uniform eccentric wellboreunwrapped into a Hell-Shaw cell. Schematic

from.15

use a modified version.9 The fluid-fluid dis-placement problem is simplified from the fullNavier-Stokes equations. The derivation usesstandard scaling arguments to simplify the mo-mentum balances and the radial dependency isthen averaged along the thickness of the annu-lus, resulting in a two-dimensional model of thebulk fluid motions in azimuthal (� ) and axial(� ) directions. The narrow annulus formed bythe space between the formation and the casingis conceptually unwrapped, resembling a Hele-Shaw cell with varying gap width H(�) (seeFig.1). The reduced model consists of a seriesof first-order conservation equations, for eachfluid concentrations pumped, and a quasi-linearPoisson-type equation for the stream function,driven by the pump flow rate and by buoyancygradients. Rheological effects enter into thenonlinearity of the stream funcion equation.

The model that we use has been studiedboth analytically and computationally, as wellas being used in various industrial case stud-ies. For uniform wells the dynamics of dis-placement are becoming well understood, asreviewed in the next section. If the wellis irregular however, there is relatively littlestudy. Some of the earliest experimental stud-ies considered the effects of sudden expan-sions on the annular geometry.2, 22 The dan-ger of sudden expansions is to trap drilling

fluid, due to its yield stress. Most relevantscientific studies of the fluid mechanics con-cern single phase flows. Mitsoulis and co-workers11 studied both planar and axisymmet-ric expansion flows with yield stress fluids,showing significant regions of static fluid inthe corner after the expansion. This was stud-ied further in expansion-contraction geome-tries, both experimentally and computation-ally.4, 12, 13 Roustaei & Frigaard18 studied largeamplitude wavy walled channel flows numer-ically, predicting the onset of stationary fluidregions. A more comprehensive study of ge-ometrical variation20 showed that yield stressfluid becomes trapped in sharp corners and inthe small scale features of the washout walls, aswell as filling the deepest parts of the washoutas the depth is increased. For sufficientlylarge yield stress and deep washouts, the ac-tual washout geometry has little effect on theamount of fluid that is mobilized: the flowingfluid “self-selects” its flowing geometry. Con-sidering the effects of (laminar) inertia,19 in-creasing the Reynolds numbers can result ina reduction in flowing area, i.e. contrary tothe industrial intuition that pumping faster isbetter. Very recently17 we have been explor-ing the effects of washout-type irregularity innear-horizontal wells, using a combination ofmodel simulations and lab scale experiments,performed collaboratively.

Here we focus on regular boreholes and ex-plore the effects of varying eccentricity alongthe well. Eccentricity is controlled via theuse of centralizers, which are devices fitted tothe outer wall of the casing, designed to exertnormal forces when in contact with the bore-hole wall. A range of centralizers exist andthere is no standard geometry/mechanical de-sign. These may be fitted every 9 � 40m alongthe well. The effectiveness of centralizationcan be inferred from logging measurementstaken after the cement job. Positioning of cen-tralizers is designed using a range of models;see Gorokhova et al.5 for the state-of-the art. Itmay seem surprising that even in a vertical sec-tion of wellbore the annulus is not fully con-centric: the lowest parts of the casing are in

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Figure 2. Colourmaps showing the progression of two uniform eccentric annular displacements(e = 0.3): a) �y1 = 20Pa, �2 = 1600kg/m3; b) �y1 = 20Pa, �2 = 1700kg/m3. The length of annulus

shown is 500m and time intervals are regularly spaced; W and N denote wide and narrow sides.

compression and the higher parts are in tension.Guillot et al.6 review current practices, explor-ing 2 case studies with vertical well sections ofaround 500m. In both cases the vertical sec-tions, despite frequent spacing of centralisers,show large scale variation in eccentricity (bothpredicted and measured).

We base our study on a vertical well oflength, L = 500m, outer and inner radii, ro =0.1122m & ri = 0.0889m (⇡ 9 & 7 inches).We fix the flow rate to give a mean velocity ofw = 0.333m/s (laminar flows only). The axialcoordinate, � , is measured from the bottom ofthe well. Half of the annulus is modelled, (as-suming symmetry about the wide and narrowsides). The azimuthal coordinate � ranges fromwide side (W: � = 0) to narrow side (N: � = 1).

DISPLACEMENTS IN UNIFORM WELLSTo illustrate the simplest situations we presentresults of 8 simulations: 2 sets of displacedfluid yield stress, 2 density differences and 2uniform eccentricities (e = 0.3, 0.6). The dis-placed fluid 1 (mud) has fixed properties: �1 =1500kg/m3, K1 = 0.1Pa.sn1 , n1 = 0.5. We con-sider 2 yield stresses: �y1 = 10, 20Pa. Thedisplacing fluid 2 (pre-flush or cement slurry)has fixed rheological properties (K2 = 0.04Pa.s,n2 = 1, �y1 = 5Pa) and we consider 2 densities�2 = 1600, 1700kg/m3.

Figure 2 shows two computed displacementflows in a modest eccentricity well (e = 0.3)with high yield stress mud. Initially the annulusis filled with the displaced fluid (red) represent-ing the mud, then, the displacing fluid (blue) is

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Figure 3. Mud displacement for uniform wells: a) e = 0.3, �y1 = 10Pa, �2 = 1600kg/m3; b)e = 0.3, �y1 = 10Pa, �2 = 1700kg/m3; c) e = 0.6, �y1 = 10Pa, �2 = 1600kg/m3; d) e = 0.6,

�y1 = 10Pa, �2 = 1700kg/m3; e) e = 0.6, �y1 = 20Pa, �2 = 1600kg/m3; f) e = 0.6, �y1 = 20Pa,�2 = 1700kg/m3. Each snapshot is taken near the end of the displacement.

pumped at a constant flow rate from the bot-tom. The snapshots are taken at different timeintervals, regularly spaced throughout the pas-sage of the front along the well. Particularly inFig. 2a (with low density difference), the dis-placing fluid advances mainly in the wide sideof the annulus, leaving behind a uniform mudlayer in the narrow side. Strategies to avoid thistype of unyielded zone in vertical wells havebeen widely studied since the 1960’s3, 7, 8, 10, 21

culminating in rule-based design systems thatcan be improved further with models such asthat used here.16 It is generally accepted thata positive density difference in vertical wells,aids to stabilize the interface preventing the for-mation of a mud channel in the narrow side.On increasing the fluids’ density difference (seeFig. 2b), the displacement front is flat and re-moves the narrow side mud much more ef-fectively. Careful inspection however showsthat the mud removal is not complete: residualmud partially contaminates the displacing fluid.Note too that secondary flows before/after theadvancing front are responsible for dispersingthe preflush ahead of the main front.

For the other 6 displacement flows we plotonly a single snapshot, taken as the displace-ment front nears the top of the annulus; see

Fig. 3. In terms of rheology, the mud’s yieldstress determines the quality of the displace-ment on the narrow side of the annulus to alarge degree, e.g. without a yield stress thestatic mud channel cannot form. For thesmaller yield stress (�y1 = 10Pa) with e = 0.3,the displacements are largely effective; seeFig. 3a & b. In Fig. 3a there remains somemixed fluid on the narrow side, comparable toFig. 2b, illustrating the competition betweenpositive density difference and adverse rheo-logical differences.

When the eccentricity increases (Fig. 3c-f)a bigger density difference is required to ef-fectively displace the mud. Comparing eitherFig. 3c with Fig. 3a, or Fig. 3f with Fig. 2b,we see that we have a mud channel at e = 0.6,but none at e = 0.3. Equally, comparing Fig.2awith Fig. 3e we see the size of mud channelgrows significantly with eccentricity.

Although most of the displacements shownresult in poor/incomplete mud removal, thephysical trends are clear: smaller eccentrictyand yield stress, or larger density difference,all result in better displacements. The con-ditions under which unsteady/steady displace-ments and mud channels can arise are formallyderived in.16

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DISPLACEMENTS IN IRREGULAR WELLSTo study a effects of irregularity comparativelywe have constructed an annular geometry witha 300m long irregular section. The deepest 50mand the top 150m are uniform (see Fig.4). Inbetween we construct a sinusoidal variation ineccentricity with amplitude ±0.2 about fixedvalues e = 0.3 and e = 0.6 for the uniform sec-tions, as illustrated. A total of 10 centralizersare placed between the two uniform sections.The distance between centralizers is 30m. Inan ideal situation, the centralizer would achieve100% standoff (e = 0), but under field circum-stances this is less likely. Here we assume aconstant e = emin over the length of the central-izer (here 40cm).

Figure 4. Eccentricity vs. depth: proposedshape for irregular wells; here e = 0.3 is the

uniform value.

The same 8 displacement flows are run forthe irregular geometries as for the uniform an-nuli (i.e. 2 densities, 2 yield stresses and 2 uni-form eccentricities). Fig. 5 shows the fluid con-centrations near the end of the displacementsfor e = 0.3. Overall, a good displacement isachieved in Fig. 5a, with reasonable removal inthe uniform sections (comparable to Fig. 3a),but larger patches of mixed fluid located at thepoints of maximum eccentricity, in the narrowside of the annulus. On increasing the mudyield stress in Fig. 5b, these zones grow sub-stantially, retaining static mud on the narrowside in patches that clearly follow the eccentric-ity variation. This compares with Fig. 2a for theuniform annulus. The width of static channel inthe uniform section is very close in both simu-lations. For the irregular section, there is moreresidual mud, but the mud channel is brokenperiodically at the centralizer positions, which

could isolate zones better than the uniform mudchannel (although here the removal of the nar-row side mud and zonal isolation is clearly pre-carious).

As seen in the uniform cases, increasing thedensity difference results in a better displace-ment. Fig. 5c & d show the displacement un-der same conditions as in Fig. 5 a & b but withthe larger density difference. The mud is nowremoved more effectively, resulting in no mudchannel, but still there are patches of contami-nated displacing fluid.

The higher eccentricity irregular wellbore isevaluated in Fig. 6 for the same parameters pre-sented in Fig. 5c-f. The eccentricity now variesfrom emin = 0.4 to emax = 0.8. In this case,there is residual mud in every case (Fig. 6 a-d). Again it appears that the mud channels arewider than for the uniform annuli, but poten-tially may achieve partial isolation Notice thatincreasing the yield stress (Fig.6b) makes thedisplacement of the mud, even at the central-izer’s position (emin), quite challenging. In thisparticular scenario, when the centralizer doesnot position the casing to give eccentricity e <emin = 0.4, the resulting mud channel is aboutthe same width as that produced in the uniformcase without centralization (Fig.3e). Again, onincreasing the density difference in Fig.6c & d,the displacement is improved and the residualmud is significantly reduced.

A HELICAL DISPLACEMENTSo far we have studied the effect of an irregu-lar sinusoidal eccentricity along the well. Thelevel of eccentricity changes at different depthswhile the position of the narrow side is keptfixed. Particularly in a vertical well, the az-imuthal position of the narrow and wide side ofthe casing will not be fixed. Just for a prelimi-nary exploration of this type of effect we havemodified our uniform geometry such that theposition of the wide side rotates 4 times aroundthe wellbore over the 500m length. Thus, wehave a helical eccentric pathway along the well.We now also perform the computations over afull annulus using periodicity conditions in � .

Due to space limitations only a single exam-

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Figure 5. Mud displacement in the irregular annulus with e = 0.3±0.2: a) �y1 = 10Pa,�2 = 1600kg/m3; b) �y1 = 20Pa, �2 = 1600kg/m3; c) �y1 = 10Pa, �2 = 1700kg/m3; d) �y1 = 20Pa,

�2 = 1700kg/m3.

Figure 6. Mud displacement in the irregular annulus with e = 0.6±0.2: a) �y1 = 10Pa,�2 = 1600kg/m3; b) �y1 = 20Pa, �2 = 1600kg/m3; c) �y1 = 10Pa, �2 = 1700kg/m3; d) �y1 = 20Pa,

�2 = 1700kg/m3.

ple is shown, as an appetizer to the complexi-ties that will arise in a more complete study tofollow. Figure 7 shows a laminar displacementof two Newtonian fluids with identical viscos-ity (0.01 Pa.s) and density (1100 kg/m3) alongthe helical channel (with e = 0.3). As the flu-ids are identical here, this is simply a dispersionexample. The helical motion of the fluids is ev-ident and the revolving eccentricity appears to

result in secondary flows that improve the dis-placement over that expected in a uniform an-nulus. It can be expected that more complexrheolgies and the introduction of yield stresswill lead to unyielded zones and more difficultdisplacements.

CONCLUSIONSWe have given an overview of recent studieson fluid-fluid displacement in a long thin annu-

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Figure 7. Mud displacement for irregular well with helical geometry: �1 = �2 = 1100 kg/m3,K1 = K2 = 0.01, n1 = n2 = 1, �y1 = �y2 = 0 Pa.

lus, covering uniform and irregular geometries.In nominally vertical wells, increasing the den-sity difference in eccentric wells aids the dis-placement in both, uniform and irregular ge-ometries. However, in presence of a sufficientlyhigh yield stress, neither the use of centralizersnor moderate density difference can prevent thedevelopment of a mud channel. The wellboreirregularity leads to a corresponding patterningof narrow side mud channels, which may inmarginal cases give a degree of zonal isolationnot present in uniform annuli.

ACKNOWLEDGEMENTSWe thank the following organizations for fund-ing that has contributed to this research:NSERC, Schlumberger, BCOGRIS, and theMexican National Council for Science andTechnology (SENER-CONACYT) for finan-cial support (AR).

REFERENCES1. S.H. Bittleston, J. Ferguson, and I.A.

Frigaard. Mud removal and cement placementduring primary cementing of an oil well lami-nar non-newtonian displacements in an eccen-tric hele-shaw cel. J. Eng. Math., 43:229–253,2002.

2. C.R. Clark and G.L. Carter. Mud displace-ment with cement slurries. Society of PetroleumEngineers Conference Paper, SPE 4090, 1973.3. M. Couturier, D. Guillot, D. Hendriks, and

Callet F. Design rules and associated spacerproperties for optimal mud removal in eccentricannuli. Society of Petroleum Engineers Confer-ence Paper, SPE 21594, 1990.4. P.R. de Souza Mendes, M.F. Naccache,

P.R. Varges, and F.H. Marchesini. Flowof viscoplastic liquids through axisymmet-ric expansions-contractions. J. Eng. Math.,142:207–217, 2007.5. L. Gorokhova, A. Parry, and N. Flamant.

Comparing soft-string and stiff-string meth-ods to compute casing centralization,. Societyof Petroleum Engineers, Drilling and Comple-tions, SPE 163424, 2014.6. D.J. Guillot, B.G. Froelich, E. Caceres, and

R. Verbakel. Are current casing centralizationcalculations really conservative? Society ofPetroleum Engineers. doi:10.2118/112725-MS,2008.7. A. Jamot. Déplacement de la boue

par le latier de ciment dans l’espace annu-laire tubage-paroi d’un puits,. Revue As-soc. Franc. Techn. Petr., 224:27–37, 1974.

ANNUAL TRANSACTIONS OF THE NORDIC RHEOLOGY SOCIETY, VOL. 26, 2018

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8. C.F. Lockyear, D.F. Ryan, and M.M. Gun-ningham. Cement channeling: how to predictand prevent. SPE Drilling Engineering. SPEpaper 19865., 5(03):201–208, 1990.

9. A. Maleki and I.A. Frigaard. Primary ce-menting of oil and gas wells in turbulent andmixed regimes. J. Eng. Math., 107:201–230,2017.10. R.H. McLean, C.W. Manry, and W.W.Whitaker. Displacement mechanics in primarycementing. Journal of Petroleum Engineering.SPE paper 1488., 1967.11. E. Mitsoulis and R.R. Huilgol. Entry flowsof bingham plastics in expansions. J. Non-Newt. Fluid Mech., 122:45–54, 2004.12. M.F. Naccache and R.S. Barbosa. Creep-ing flow of viscoplastic materials through aplanar expansion followed by a contraction.Mech. Res. Comm., 34:423–431, 2007.13. B. Nassar, P.R. de Souza Mendes, andM.F. Naccache. Flow of elasto-viscoplasticliquids through an axisymmetric expansion-contraction. J. Non-Newt. Fluid Mech.,166:386–394, 2011.14. E.B. Nelson and D. Guillot. Well cement-ing. Schlumberger, second edition, 2006.15. S. Pelipenko and I.A. Frigaard. Two-dimensional computational simulation ofeccentric annular cementing displacements.J. Eng. Math., 64:557–583, 2004.16. S. Pelipenko and I.A. Frigaard. Visco-plastic fluid displacements in near-vertical nar-row eccentric annuli: prediction of travelling-wave solutions and interfacial instability. J.Fluid Mech., 520:343–377, 2004.17. A. Renteria, A. Maleki, I.A. Frigaard,B. Lund, A. Taghipour, and J.D. Ytrehus. Ef-fects of irregularity on displacement flows inprimary cementing of highly deviated wells.J. Petr. Sci. Engng: submitted May 2018, un-der review., 2018.18. A. Roustaei and I.A. Frigaard. The oc-currence of fouling layers in the flow of ayield stress fluid along a wavy-walled channel.J. Non-Newt. Fluid Mech., 198:109–124, 2013.

19. A. Roustaei and I.A. Frigaard. Resid-ual drilling mud during conditioning of unevenboreholes in primary cementing. part 2: Steadylaminar inertial flows. J. Non-Newt. FluidMech., 226:1–15, 2015.20. A. Roustaei, A. Gosselin, and I.A.Frigaard. Residual drilling mud during condi-tioning of uneven boreholes in primary cement-ing. part 1: Rheology and geometry effects innon-inertial flows. J. Non-Newt. Fluid Mech.,220:87–98, 2015.21. C.W. Sauer. Mud displacement during ce-menting: state of the art. Journal of PetroleumTechnology. SPE paper 14197., 39(09):1–091,1987.22. J.J.M. Zuiderwijk. Mud displacement inprimary cementation. Society of Petroleum En-gineers Conference Paper, SPE 4830, 1974.

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