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A new paradigm for proppant schedule design Dontsov, E. V.; Peirce, Anthony Feb 27, 2014

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A new paradigm for proppant schedule designE.V. Dontsov and A.P. Peirce?Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, CanadaFebruary 14, 2014AbstractThis study introduces a novel methodology for the design of the proppant pumping schedule for ahydraulic fracture, in which the final proppant distribution along the crack is prescribed. The methodis based on the assumption that the particles have relatively small impact on the fracture propagation,unless they reach the tip region. This makes it possible to relate the proppant velocity to the clearfluid velocity inside the fracture, which is calculated assuming no proppant. Having the history ofthe clear fluid velocity distribution, the prospective proppant motion can be computed. Then, volumebalance is used to relate the final concentration at some point inside the fracture to the correspondinginput concentration at a specific time instant, which helps to avoid solving an inverse problem. Oneexceptional feature of the approach lies in the fact that it is applicable to multiple fracture geometriesand can be implemented using various hydraulic fracturing simulators. To verify the technique, twofracture geometries are considered - Khristianovich-Zheltov-Geertsma-De Klerk (KGD) and pseudo-3D(P3D). It is shown that the developed approach is capable of properly estimating the pumping schedulefor both geometries. In particular, the proppant placement along the fracture at the end of the pumpingperiod, calculated according to the adopted proppant transport model, shows close agreement with thedesign distribution. The comparison with Nolte?s scheduling scheme shows that the latter is not alwaysaccurate, and cannot capture the essential differences between the schedules for the fracture geometriesconsidered.1 IntroductionHydraulic fracturing (HF) is a process in which a viscous fluid injected into a fracture drives crack prop-agation. One of the most common applications is the stimulation of oil and gas wells [1], in which HF isused to break the rock to produce high conductivity channels. To avoid the fracture closing after pumpinghas stopped and the fluid has leaked off, various proppant additives are pumped at some stage. Despitethe fact that many studies have been devoted to proppant transport modelling and investigating the effectsof settling [2, 3, 4], only a few consider the design of a proppant schedule [5, 6, 7]. The appropriate prop-pant schedule is as important as the correct prediction of the fracture footprint, since it directly affects theproppant distribution inside the fracture, and thus influences the conductivity and the production rate.One of the most common approaches in generating the pump schedule is the schedule due to Notle [6].This is a very convenient option, as it provides an analytical formula for the schedule for a given efficiency,total pumping time, and a desired (uniform) concentration inside the fracture at the end of the job. Theapproach is based on the conservation of volume, and the estimation of the total volume of fluid that isleaked during the HF treatment to estimate the pad length. A power law-type schedule is then suggestedand the exponent is calculated based on the proppant volume balance. Not being tied to any fracturegeometry, this scheduling approach is considered applicable to multiple fracture geometries, such as PKN orradial. The universality, seeming reasonable accuracy, and the ease of use are, possibly, the main reasonswhy this scheduling is commonly used, see e.g. [1, 8]. There is an alternative method, developed in [7], whichsuggests using an iterative scheme together with an appropriate proppant transport model to solve an inverse?Email address for correspondence: peirce@math.ubc.ca1problem to generate a pumping schedule. The procedure is to divide the schedule into several intervals andthen adjust the input concentration values iteratively based on the results of the forward problem solutionusing the previous schedule. This iterative algorithm, being effective, lacks the simplicity and may requiresignificant computational resources, especially for more complicated fracture geometries.To address the proppant schedule generation, this study aims to develop an approach, that is moreaccurate than Nolte?s method, but less computationally challenging than the iterative procedure describedabove. In particular, the proppant is assumed to have a minor impact on the fracture propagation, whichallows us to avoid an iterative scheme and lengthy simulations with proppant transport. At the same time,the influence of the fracture geometry and other features that can be built into a HF simulator are takeninto account. One of the biggest advantages of the proposed technique is its applicability to various HFsimulators, which do not need the ability to model the proppant transport itself, see the examples of suchsimulators in [9, 10].The paper is organized as follows: Section 2 outlines the procedure for the method, then Sections 3 and 4illustrate the implementation respectively for the KGD and P3D fracture geometries, and, finally, Section 5provides comparison between different schedules (including Nolte?s schedule) and discusses the applicabilityand possible extensions of the approach.2 Idea behind estimation of pumping scheduleThere are several things that have to be prescribed before attempting to obtain the pumping schedule: i) thegeometry of the hydraulic fracture (HF), ii) the size of the fracture at the end of pumping, and iii) the desiredproppant distribution inside the fracture at the end of the job. Here it is assumed that the properties of therock, the fluid and the proppant, as well as the pumping rate are all known and fixed. The first item, i.e. thegeometry, can be interpreted as the type of the HF model that is used for the design, such as KGD, radial,PKN, P3D, or a fully planar HF solver. The second item, i.e. the size of the fracture, is just the length (orradius) of the HF, that is determined during the design process. The last item, which concerns the desiredproppant pattern inside the HF, could be a separate topic for research, since it is unclear what the optimaldistribution of the proppant is that would give the best production rate, see e.g. [11] where the residualfracture opening supported by proppant is analyzed or [12] where the effect of the proppant distributionon the conductivity is studied. Since the investigation of the optimal proppant pattern is beyond the scopeof this study, it is assumed that the desired distribution of the proppant concentration is given, and hencethe schedule should be sufficiently adaptable to be able to accommodate any designed pattern. The mainobjective of this study is the calculation of the pumping schedule for given properties of the materials, HFgeometry, the design length of the HF, and the desired proppant distribution at the end of the fracturingjob.To design the pumping schedule, it is imperative to know where the proppant would go. This can beachieved by estimating the proppant velocity inside HF as a function of time and space. At the same time,assuming that the proppant concentrations are sufficiently small, the proppant velocity can be related tothe fluid velocity, where the latter is calculated assuming no proppant. So, the tentative procedure for thepumping schedule design is the following: i) run an appropriate HF solver without proppant, record timehistories of all velocity components, width and fracture footprint, and find the time required to achieve thedesired fracture size/footprint, ii) find the location of the proppant, injected at time instant ti by ?tracking?its position with time by integrating the actual velocity field, and iii) use conservation of volume to relate theinput concentration (at any given time ti) to the concentration at the end of the pumping. It is important tonote, that this procedure is applicable to any HF solver and can be implemented as a separate module. Also,the solution for the proppant location resembles the Lagrangian approach used in continuum mechanics, andhas a clear advantage over the ?classical? Eulerian approach (i.e. solving the advection equation for proppanttransport) by making it possible to relate the final proppant distribution to the schedule concentration.Another important advantage of the proposed approach is that any desired concentration distribution canbe achieved through the appropriate scheduling, which opens interesting research possibilities for the optimalproppant placement inside the HF.To illustrate the methodology, two fracture geometries are examined: KGD and P3D. The schedules for2xx(t, ti+t) x(t, ti)iQ0t t = titti ti+tittpFigure 1: Schematics of the proppant schedule (top right), the proppant in the borehole at t = ti (top left)and the same proppant at time t in the KGD fracture (bottom picture).both are verified numerically by using a proppant transport model described in [13].3 Pumping schedule for KGD fracturesTo address the proppant scheduling problem in a more straightforward way, first the one-dimensional KGDfracture geometry is analyzed. Given the properties of the rock and the fracture design length le, anappropriate HF simulator (without the proppant) can be used to deduce the total injection time te, as wellas to record the histories of the fracture footprint l(t), width w(x, t), and average fluid velocity V f (x, t). Dueto the nonuniform distribution of the proppant across the fracture width, the average proppant velocity ishigher than that of the slurry. It follows from [13], in the limit of low concentrations, that the ratio betweenthese two corresponding velocities is ? = 1.2. This can be understood in the following way: the proppanttends to concentrate near the centre of a channel, so it flows with nearly the maximum velocity, which isalways higher than the average velocity in the channel. By denoting the position of the particles at timeinstant t, injected at ti, by x(t, ti), the governing equation for the proppant front is?x?t = ?Vf (x, t), x(ti, ti) = 0, ti 6 t 6 te, (1)where the parameter ti enters the problem through the initial condition, so that (1) is actually an ordinarydifferential equation. The solution of (1) allows us to find x(t, ti), i.e. the current location of the proppant,which is injected at any time instant ti. Fig. 1 shows the schematics of the proppant front movement in thefracture. The schedule for the proppant injection concentration is shown in the top right picture, and thered area is proportional to the total amount of proppant pumped during the interval ?t, which is ?iQ0?t,where Q0 is the slurry injection rate. This proppant is schematically shown at t = ti in the top left picture.At this time instant, the particles leave the borehole and enter the fracture. The same proppant (half of itdue to the symmetry), but now at time t > ti, and inside the fracture, is shown in the bottom picture. Tocalculate the ?pad? time, tp, or the time up to which pure fluid is pumped and before the proppant injection(see Fig. 1), one may use the solution for x(t, ti) and requirex(te, tp) = le, (2)i.e. determine tp by requiring that the particles reach the crack tip by the end of pumping. Note that fora more accurate prediction for bigger proppants, it is reasonable to replace le in (2) by a smaller length, l?e,where the latter is calculated based onw(l?e, te) = 2a,30 500 1000 150000.511.5t [s]?/?d  CurrentNolte0 20 40 60 80 10000.511.5x [m]?/?d  CurrentNolteDesignFigure 2: Left picture: pumping schedule calculated according to the equation (4) and Nolte?s model withthe efficiency ? = 0.63. Right picture: concentration distribution along the KGD fracture for the currentand Nolte?s schedules.where a is a particle radius. This ensures that there is no proppant in the places where it cannot physicallyfit. Given the desired or design proppant concentration distribution inside the fracture ?d(x), the balanceof mass can be used to obtain12Q0?(ti)?t = w(x(te, ti), te)[x(te, ti)? x(te, ti+?t)]?d(x(te, ti)), (3)where the left side reflects the volume of the proppant injected between ti and ti+?t (the 1/2 factor comesfrom the symmetry), while the right side calculates the same volume at the end of the fracturing job. Bytaking a limit of ?t? 0 in (3), the result can be simplified to?(ti) = ?2w(x(te, ti), te)Q0?x(te, ti)?ti?d(x(te, ti)), ti > tp, (4)which allows us to calculate the proppant schedule for any desired concentration distribution along thefracture. Note that due to the 1D nature of the geometry, Q0 is the injection rate per unit length, i.e.measured in m2/s. Also note that any function ?d(x) can be used in (4), i.e. any proppant distribution canbe achieved without introducing extra complexities.Numerical examples. From the numerical implementation point of view, the task of finding the pump-ing schedule according to (4) requires an appropriate numerical scheme for the solution of (1), numericaldifferentiation in (4), as well as an extensive use of interpolation. The interpolation is used in both (1),since V f (x, t) is computed for a discrete set of x and t, and (4) for the evaluation of w(x(te, ti), te) and?x(te, ti)/?ti (the x values are first interpolated and then numerically differentiated). To achieve high accu-racy and to preclude possible oscillations (that can be caused by spline interpolation), the built-in Matlabfunction ?PCHIP? (Piecewise Cubic Hermite Interpolating Polynomial) is used for the interpolation. Todeal with the numerical solution of (1), the 4th order Runge-Kutta method is used.To verify the proposed approach, an example problem is considered. The set of chosen parameters isE? = 25 GPa for the plane strain modulus, ? = 0.1 Pa?s for the shear viscosity of the fracturing fluid,Q0 =10?3 m2/s for the inlet flux, C ?=5?10?5 m/s1/2 for the Carter?s leak-off coefficient, K1c=1 MPa?m1/2for the fracture toughness and a=0.2 mm for the particle radius. The design length of the fracture is set tole = 100 m, while the target concentration is considered to be uniform and is equal to ?d = 0.2??m. Here?m = 0.585 is the maximum volume concentration that can be achieved [14, 13]. Note that, assuming theproppant mass density of 2300 kg/m3, this concentration can be translated to approximately 2.6 lbs/gal.The HF simulator for KGD fractures, described in [13], is used to calculate the duration of the HF treatmentand to record the history of the average velocity without proppant, which is then used to calculate theschedule. Given the pumping schedule, the same simulator is used for the verification of the design.40 500 1000 150000.511.5t [s]?/?d0 20 40 60 80 10000.511.5x [m]?/?d  Regular concentrationSmall concentrationDesignFigure 3: Pumping schedule and the corresponding concentration distribution along the KGD fracture for?zebra? configuration.Fig. 2 shows the schedule, calculated according to equation (4), and the concentration distribution alongthe fracture at the end of the simulation (black solid lines). In addition, these results are compared toNotle?s scheduling [6] (blue solid lines). Note that Nolte?s model including the correction for the pad lengthis used, see the Appendix for the description of Nolte?s pumping schedule. Despite using the correction,the prediction of Nolte?s model notably underestimates the pad size, which causes premature tip screen-outleading to a fracture length under 80 m (as opposed to the 100 m designed length). At the same time,the current approach shows better performance, as it just slightly distorts the designed 100 m fracturelength, and produces a nearly uniform concentration distribution along the crack. Note that the spikes inthe concentration correspond to plug formation and reflect its size. For the current schedule, the plug juststarted to form and therefore does not significantly affect the fracture propagation. The ?dip? at the proppantdistribution is due to both, numerical diffusion and the effect of coupling between proppant transport andHF propagation, namely the change of the slurry viscosity with concentration. Note that the schedule forthe current approach has a region in which the concentration exceeds ?d, while the resultant concentrationinside the fracture is approximately equal to ?d. This is due to the fact that the ratio between particle andslurry velocities is ?=1.2, which effectively reduces the mixture concentration by ? when it reaches a steadyflow (i.e., according to the proppant transport model assumptions, when it enters the fracture).To highlight the versatility of the proposed scheduling procedure, Fig. 3 shows the pumping scheduleand resultant proppant distribution for regular (?d = 0.2??m) and small (?d = 0.02??m) concentrationsfor a ?zebra? configuration. This configuration suggests that at the end of the pumping, the proppantshould be concentrated in three equal stripes, placed equidistantly inside the crack. This configuration isinspired by Schlumberger?s HiWAY proppant placement, for which the particles are injected in pulses, thatproduce bridges at the end of pumping. This increases the permeability, as hydrocarbons can flow betweenthe bridges. In practice HiWAY technology suggests numerous pulses, while only three are considered heresince a large number of pulses would require a much finer mesh, which is computationally demanding. Ascan be seen from the results, the calculated schedule indeed leads to the desired proppant placement withinadequate accuracy. As was the case for Fig. 2, the discrepancy for the regular concentration comes mainlyfrom numerical diffusion and the coupling between HF propagation and proppant transport. Since thecoupling is minimal for small concentrations, the blue line (which corresponds to the small concentrationsolution) can be used to estimate the discrepancy caused by the numerical scheme only. Hence, the differencebetween the solutions for regular and small concentrations indicates the effect of coupling. Even though theeffect of coupling leads to visible differences, the overall accuracy of the approach is still satisfactory. It isimportant to note that higher concentrations can lead to bigger differences, which is the limitation of theproposed design approach. At the same time, it is remarkable that the higher viscosity of the slurry (due tothe presence of proppant) does not alter the final fracture length appreciably. This is because the biggestpressure gradients are near the fracture tip, where the width is minimal, while the rest of the fracture is5subject to smaller pressure gradients. When the proppant is introduced, the higher viscosity of the slurryperturbs mostly the small pressure gradients, which are away from the tip, and thus does not significantlyaffect the pressure distribution and fracture propagation. When the proppant eventually reaches the tipregion, it starts to disturb the fracture, but it is already too late since the fracturing job is over once theproppant reaches the crack tip. In other words, even though the particles change the viscosity of a slurry,the time interval during which the proppant can affect the fracture behaviour is small, which makes theconsequences of the coupling effect relatively insignificant.Gravitational settling. One assumption that is implicitly stated is that the fracture is symmetric, whichin particular implies that there is no gravitational settling. As shown in [13], gravitational settling maybreak the symmetry as the proppant sinks towards one side of the fracture and causes early tip screen-out.It is possible to account for gravitational settling in the schedule, but in this case the design should beperformed for the downward fracture (i.e. finish pumping when the proppant reaches the downward movingcrack tip). In the opposite case, the downward fracture could be screened, which can affect the other portionof the fracture and cause inapplicability of the scheduling approach. When gravitational settling is present,equation (1) should be replaced with?x?t = ?Vf (x, t) + V s, x(ti, ti) = 0, ti 6 t 6 te, (5)where V s is the settling velocity, which can be estimated for small particle concentration asV s = 2a29?f (?p??f )g,where a is a particle radius (it is assumed that all particles are spherical and equal in size), ?p??f is thedifference between particle and fluid mass densities, ?f is the fluid viscosity, while g is the gravitationalacceleration. As with (1), equation (5) can be solved numerically and the solution can be used to calculatethe schedule according to (4).4 Pumping schedule for P3D fracturesThe design of a pumping schedule for the P3D geometry [15] is conceptually similar to that for the KGDmodel, but there are several notable differences. One of them comes from the fact that a line source isused in the P3D model, as opposed to a point source. At the same time, another difference is related tothe two-dimensional nature of the proppant flow, and the presence of the vertical velocity component. Forthe purpose of schedule calculation, it is assumed that the gravitational settling is negligible, and hence thevertical component of the velocity is zero along x axis (z = 0), see Fig. 4. In this case, the proppant thatis pumped in the vicinity of z= 0 remains close to the x axis during the fracture growth. This fact allowsus to ?track? the proppant along the x axis and not to consider its vertical migration. The schematics ofthe P3D fracture together with the ?proppant tracking box? are shown in the Fig. 4. The function x(t, ti)has the identical meaning as for the KGD fracture, it calculates the position of the proppant at time t, thatwas injected at time ti. As will be shown later, the height of the box, ?z(t), varies with time due to thepresence of the vertical velocity component. Since the particles can easily be traced along the x coordinate,it is natural to establish the target concentration along the x axis as well. As for the KGD fracture, thesimulations without proppant are first performed, and for given problem parameters and design length le,the pumping time te is calculated, and the histories of the appropriate quantities are recorded. Knowingthe history of the x component of the average fluid velocity at z=0, V fx , the motion of the particles can bedescribed by solving?x?t = ?Vfx (x, t), x(ti, ti) = 0, ti 6 t 6 te, (6)which is identical to (1). Also, the ?pad? time, tp, is calculated in a similar fashion asx(te, tp) = le,6xzHFracture footprintStress barriersz(t)x(t, ti)x(t, ti+t)l(t)Figure 4: Schematics of the P3D fracture with the proppant ?tracking? region.where the final length of the fracture le can be replaced by l?e for bigger proppants. Here l?e is determinedfromw(l?e, te) = 2a,where w(x, t) is the fracture width along x axis for z = 0. With reference to Fig. 4, the volume of theproppant at the injection point and at the end of the fracturing can be equated to find?(ti)V fx (0, ti)w(0, ti) ?z(ti) ?t =[x(te, ti)? x(te, ti+?t)]w(x(te, ti), te)?z(te)?d(x(te, ti)), (7)where ?d(x) is the design concentration distribution along the x axis, V fx (0, ti) is the average fluid velocityat the inlet (and z= 0), while w(0, ti) is the corresponding width of the fracture at that point. By notingthat?z(te)?z(ti)= ?? teti?V fz (x(t, ti), t)?z(ti)dt, (8)where ?V fz is the difference between the vertical components of the fluid velocity at the top and bottom ofthe ?proppant tracking box?, equation (7) can be simplified to?(ti) = ?w(x(te, ti), te)w(0, ti)V fx (0, ti)?x(te, ti)?ti(1 + ?? teti?V fz (x(t, ti), t)?z dt)?d(x(te, ti)). (9)Note that ?V fz /?z is evaluated at z=0 and its history has to be precomputed in addition to the history ofthe horizontal velocity component V fx . It is important to understand that ? should not enter on the left sidein (7), since the ratio between the proppant and slurry fluxes is ?(ti), while the slurry flux is proportional toV fx (0, ti). At the same time, since the proppant?s vertical velocity (as opposed to the fluid?s) is responsible forthe vertical ?box? growth, ? appears in (8). As with the expression for the KGD fracture (4), the relation (9)can be used to design a proppant schedule for any target concentration profile along the x axis ?d(x), whichadds versatility to the approach.Numerical examples. To illustrate these developments for P3D fractures and to check the validity of (9),several numerical examples are considered. The parameters used for the calculations are E? = 25 GPa forthe plane strain modulus, ?=0.1 Pa?s for the shear viscosity of the fracturing fluid, Q0 =10?2 m3/s for thetotal inlet flux, H = 25 m for the reservoir layer, ?? = 2.5 MPa for the magnitude of the stress barriersC ? = 5 ? 10?5 m/s1/2 for the Carter?s leak-off coefficient, K1c = 1 MPa?m1/2 for the fracture toughness,a=0.2 mm for the particle radius, g = 9.8 m/s2 for the gravitational acceleration and ?p? ?f = 1300 kg/m37for the difference between particle and fluid mass densities, see Fig. 4 and [15, 13] for more details. Note thatthe gravitational settling is formally included in the simulations, but, since a relatively small particle size isconsidered, there is almost no distortion in the symmetry due to settling. As noted in [13], the dimensionlessparameter that determines the settling extent isGs =16??a2gQ0E?3(te?tp)3??4H4 ,where te is the total pumping time, while tp is the time at which the proppant is first injected, see the top rightpicture in Fig. 1. The parameter Gs reflects the ratio between proppant travel time and the settling time,so when Gs  1, then settling occurs before the end of the pumping, while if Gs  1, then, practically, thegravity does not alter the particle distribution. For the set of parameters under consideration, Gs = 0.035,which indeed suppresses the effect of settling. As with the KGD fracture geometry, the design length of thefracture is set to le = 100 m, while the target concentration is considered to be uniform and is equal to?d=0.2??m, where ?m=0.585. The HF simulator for P3D fractures, described in [13], is used to calculatethe duration of the HF treatment and to record the history of the average x component of the velocity andthe derivative of the vertical velocity component (assuming no proppant), which are then used to calculatethe schedule using (9). Numerical techniques, that were used for dealing with the KGD fracture schedulingin Section 3, are utilized for the numerical solution of (6) and for the interpolation, which are both nessecaryfor the evaluation of (9). Given the pumping schedule, the same HF simulator for the P3D fracture, thistime with proppant, is used for verification purposes.To evaluate the accuracy of the proposed scheduling, the top left picture in Fig. 5 compares the schedulethat is calculated according to (9) with that suggested by Nolte [6]. The differences are similar to those foundfor the KGD model, see Fig. 2, namely, Nolte?s approach underestimates both the time of the first proppantinjection, tp, and the maximum concentration near the end of pumping. The consequences are similar as well,i.e. Nolte?s schedule leads to premature tip screen-out (which in turn leads to a shorter fracture length) andsmaller concentration near the inlet, see Fig. 5. With regard to the accuracy of the current approach, thereis also a ?dip? near the fracture tip and a small plug starts to form thereafter. Despite the fact that P3Dand KGD consider different types of fractures, the reasons for the ?dip? are similar, namely, the couplingbetween proppant transport and HF propagation and numerical diffusion. Note that the term with thevertical velocity derivative in (9) plays an important role, and its absence can lead to observable inaccuracyof the final concentration, while the tp stays unaffected. Of course, the degree of influence depends on theproblem parameters, and in particular on the fracture growth in the vertical direction and the specificityof the line source implementation in the HF simulator (i.e. the variation of source intensity versus z atx=0). In addition, the developed proppant plug on the bottom right picture in Fig. 5 has a strange shape,namely, the particles are concentrated near the top and the bottom of the fracture, leaving the central partunderpropped. Unfortunately, this is due to an inaccuracy in the P3D model [15, 13], in which a uniformpressure along every vertical cross-section is assumed, which in turn leads to unrestricted motion of theslurry in the vertical direction. The fact that the slurry is transported to the tip region mainly throughthe central part of the fracture, and the leak-off occurs uniformly along the height, together lead to strongoff-central vertical velocities in the tip region, which bring the proppant away from the centre to the sidesof the fracture. This feature is more pronounced for the smaller particle size as considered in this paper,while bigger particles form a plug some distance away from the crack tip, and are influenced to a much lesserextent, see [13]. Note that even though Fig. 5 indicates smaller sensitivity of the proppant placement tothe schedule type, than Fig. 2, one should always keep in mind that those are the examples for one set ofparameters, and some variations are possible for different problem parameters.To show the capabilities of the proposed scheduling paradigm, Fig. 6 shows the pumping schedule andthe results of the simulations for regular (?d = 0.2??m) and small (?d = 0.02??m) concentrations for P3Dfractures with ?zebra? distributions of proppant. As for the KGD geometry, ?zebra? configuration is definedby requiring a specified proppant placement in three equal equidistant stripes along the x axis by the endof the simulation. Due to the planar nature of the P3D model, the stripes form an interesting shape, whichreflects the velocity pattern inside the fracture. The comparison between the solutions for small and regularconcentrations allows us to estimate the effect of coupling between proppant transport and HF propagation.This coupling is more pronounced than for the KGD geometery (see Fig. 3) and leads to some notable visual8Figure 5: Top left: comparison between the current and Nolte?s (efficiency ? = 0.39) pumping schedulesfor P3D fracture. Top right: comparison between the normalized proppant concentration along the x axisfor the current and Nolte?s schedules with the design concentration distribution. Bottom left: footprint ofthe P3D fracture for the current schedule with the colour indicating the normalized proppant concentration?/?d, so that the colour associated with 1 corresponds to the desired concentration specified in the design.Bottom right: footprint of the P3D fracture for Nolte?s schedule with the colour indicating the normalizedproppant concentration ?/?d.9Figure 6: Top left: pumping schedule for ?zebra? proppant distributions for the P3D fracture. Top right:comparison between normalized proppant concentration along the x axis for regular and small concentrationswith the design concentration distribution. Bottom left: footprint of the P3D fracture for regular concen-tration with the colour indicating the normalized proppant concentration ?/?d. Bottom right: footprint ofthe P3D fracture for a small proppant concentration with the colour indicating the normalized proppantconcentration ?/?d.100 0.2 0.4 0.6 0.8 100.511.5t/tmax?/?dNolteKGDP3DKGD withstress barriers? = 0.1? = 0.5? = 0.9Figure 7: Comparison between different schedules for different values of the efficiency ?.distortions of the proppant pattern, see the bottom pictures in Fig. 6. Also note that both bottom picturesare not perfectly symmetric, which is due to the presence of small gravitational settling.5 CommentsComparison between different schedules. While Figs. 2 and 5 show the comparison between Nolte?sschedule and the predictions based on the equations (4) and (9), it is instructive to make a comparison fora broader range of efficiencies ?. To accomplish this goal, Fig. 7 shows the comparison between differentschedules for ? = 0.9, 0.5, and 0.1. Nolte?s schedules are compared to the corresponding KGD and P3Dschedules, as well as to the schedule for a KGD fracture, which is affected by the presence of symmetricstress barriers. The stress barriers are placed 60 m from the inlet and have a magnitude ??=2.5 MPa. Toachieve the desired value of the efficiency for both KGD and P3D models, the Carter?s leak-off coefficientis adjusted. The comparison shows significant variability among different models. As mentioned previously,Nolte?s model underestimates tp - the time instant at which the proppant is introduced, and the maximumconcentration near the end of pumping. But what was not clear so far, is the big difference between theschedules for the KGD and P3D geometries, which clearly indicates that there is no universal schedule thatcan work for all fracture geometries. Even with the same fracture type (KGD), the introduction of the stressbarriers affects the schedule to some extent. This also supports the fact that a universal schedule can not begenerated. One peculiar feature that can be seen from Fig. 7 is the hierarchy between the schedules, namelyNolte?s schedule suggests the earliest proppant injection, followed by the corresponding KGD and, finally,P3D schedules. The difference between Nolte?s and the KGD schedules can be related to the ratio betweenparticle and slurry average velocities, ?=1.2, which allows the proppant to reach the crack tip faster. At thesame time, since the P3D model has another dimension, it effectively introduces another factor, which is theratio between the peak proppant velocity and the average proppant velocity with respect to the vertical, i.e.the z direction. This can be seen on the bottom pictures in Fig. 6. The proppant at z=0 is already at the tip,but, on average, the boundary of the corresponding first ?zone? of the proppant is some distance away formthe tip. Note that the ratio between the peak proppant velocity and average slurry velocity is important,since the first ?moves? proppant forward, while the second is responsible for the fracture growth. This ratiois higher for P3D than for KGD fractures, which allows proppant to reach the crack tip notably faster for aP3D geometry. While the differences between the schedules are prominent for high efficiencies, they becomeless pronounced for smaller efficiencies, see Fig. 7. Regarding the implementation, Nolte?s schedule is theeasiest to deal with since analytical formulas are used, at the same time, once executed, the current approachmakes it possible to calculate a reasonably accurate schedule on a computer in mere seconds for KGD andtens of seconds for P3D fractures.11Assumptions and limitations. Despite the fact that the scheduling has been verified numerically, itis essential to understand all the assumptions behind the model. First, it is assumed that the presenceof proppant does not disturb fracture propagation. This is a critical ingredient for developing a pumpingschedule, but, at the same time, it limits the applicability to relatively small concentrations, for whichthe viscosity of the slurry is not significantly perturbed by particles. However, for the design proppantconcentration ?d = 0.2??m, which is used for the verification, the change in the apparent viscosity isapproximately 20%, see the proppant transport model in [13]. Despite this notable change in the viscosity,Figs. 2 and 5 show good agreement between the designed proppant placement and the one that is calculatedusing the generated schedule. As discussed before, this is due to the fact that the particles spend little time atthe near-tip region, and since this part of the fracture is primarily responsible for the propagation, the higherviscosity of the slurry does not alter the fracture footprint much. However, due to the nonlinear variationof the slurry viscosity versus particle concentration, higher design concentrations could introduce biggerdiscrepancies, so it is essential to verify the schedule via numerical simulation. In addition to the reducedaccuracy for higher concentrations, the proposed schedule cannot be used for tip screen-out applications, sincethe proppant plug near the crack tip significantly changes the fracture behaviour relative to the correspondingfracture with pure fluid.The calculation of a pumping schedule, that is proposed in this paper, is always consistent with somehydraulic fracturing model, such as KGD or P3D model. For this reason, it is clear that the correctness ofthe HF model plays a crucial role in scheduling. This factor should not be underestimated, as, according toFig. 7, there might be a significant difference among various fracture models. In addition, the scheduling istailored to the specific proppant transport model [13], which, in particular, provides the value for ?, see (1).The validity and the applicability range of this model are also essential for scheduling.Possible extensions. One of the biggest advantages of the proposed scheduling procedure lies in itsapplicability to multiple hydraulic fracturing models. As an example, a HF simulator with a more accurateleak-off model can be used for the design. Another possibility is to use a HF simulator that accounts forturbulent motion near the wellbore. In the latter case, it might be necessary to introduce ?(x) in (1), asthe proppant distribution along the width of the fracture for turbulent motion won?t resemble its laminaranalog, and thus the proppant to slurry average velocity ratio could change.Since the accuracy of the scheduling approach deteriorates for higher proppant concentrations, it might beuseful to adjust the average velocity history iteratively. In other words, given an initial guess for a schedule,one may run the appropriate HF simulator with proppant and record the history of the average velocity.Then, this velocity history can be used to recalculate a schedule. This process can be repeated until theresults converge. Note that the proppant velocity history can be recorded right away, which eliminates theuse of ? in (1). The current design approach therefore represents the first step in such an iterative process.The iterative approach, although effective, is computationally demanding, and sacrifices the efficacy of theoriginal non-iterative methodology.6 SummaryThis paper introduces a universal approach for designing a proppant schedule, which complements a givenhydraulic fracture simulator. The main idea is based on the assumption that the proppant particles do notaffect the fracture propagation until they reach the tip region. This makes it possible to precompute thehistory of the velocity distribution assuming no proppant, which is then used to evaluate the prospectiveproppant movement. Once the prospective movement is calculated, volume balance is used to relate thedesired proppant concentration at a given point in space (concentration does not have to be uniform) to acorresponding input concentration at a certain time instant. In this way, the schedule is obtained withoutsolving an inverse problem. The scheme is illustrated for two fracture geometries, namely, KGD and P3D.It is shown that Nolte?s schedule suggests earlier proppant injection, which leads to a premature tip screen-out, while the current approach produces more accurate results and does not alter the desired final fracturelength. In addition, the effect of coupling between proppant transport and hydraulic fracture propagation isstudied. As expected, smaller proppant concentrations lead to more accurate results, while the agreement12for higher concentrations is still adequate. The comparison between the schedules for different fracturegeometries together with Notle?s schedule for different efficiencies shows that knowledge of the efficiencyalone is not sufficient to predict the schedule. In particular, a noticeable difference between all of the modelsis observed. This demonstrates that there is no universal schedule that is applicable to multiple fracturegeometries, while, on the other hand, the proposed technique is able to calculate the pumping schedule inan accurate and fast manner.AcknowkedgementsThe support of the British Columbia Oil and Gas Commission and the NSERC discovery grants program isgratefully acknowledged.References[1] M.J. Economides and K.G. Nolte, editors. Reservoir Stimulation. John Wiley & Sons, Chichester, UK,3rd edition, 2000.[2] A.A. Daneshy. Numerical solution of sand transport in hydraulic fracturing. J. Pet. Tech., pages132?140, 1978.[3] A.T. Mobbs and P.S. Hammond. Computer simulations of proppant transport in a hydraulic fracture.SPE Prod. Facil., pages 112?121, 2001.[4] E.M. Shokir and A.A. Al-Quraishi. Experimental and numerical investigation of proppant placement inhydraulic fractures. In SPE Latin American and Caribbean Petroleum Engineering Conference, 2007.[5] H.R. Crawford. Proppant scheduling and calculation of fluid lost during fracturing. In SPE AnnualTechnical Conference and Exhibition, 1983.[6] K.G. Nolte. Determination of proppant and fluid schedules from fracturing-pressure decline. SPE Prod.Eng., pages 255?265, 1986.[7] H. Gu and J. Desroches. New pump schedule generator for hydraulic fracturing treatment design. InSPE Latin American and Caribbean Petroleum Engineering Conference, 2003.[8] M. M. Rahman and M. K. Rahman. A review of hydraulic fracture models and development of animproved pseudo-3D model for stimulating tight oil/gas sand. Energy Sources, Part A, 32:1416?1436,2010.[9] J. Adachi, E. Siebrits, A. Peirce, and J. Desroches. Computer simulation of hydraulic fractures. Int. J.Rock Mech. Min. Sci., 44:739?757, 2007.[10] A. Peirce and E. Detournay. An implicit level set method for modeling hydraulically driven fractures.Comput. Methods Appl. Mech. Engrg., 197:2858?2885, 2008.[11] L.B. Neto and A. Kotousov. On the residual opening of hydraulic fractures. Int. J. Fract., 181:127?137,2013.[12] C.L. Cipolla, E. P. Lolon, M.J. Mayerhofer, and N.R. Warpinski. The effect of proppant distributionand un-propped fracture conductivity on well performance in unconventional gas reservoirs. In SPEHydraulic Fracturing Technology Conference, 2009.[13] E.V. Dontsov and A.P. Peirce. Proppant transport in hydraulic fracturing: Crack tip screen-out inKGD and P3D models. J. Fluid Mech. (submitted), 2014.[14] F. Boyer, E. Guazzelli, and O. Pouliquen. Unifying suspension and granular rheology. Phys. Rev. Lett.,107:188301, 2011.13[15] J. I. Adachi, E. Detournay, and A. P. Peirce. An analysis of classical pseudo-3D model for hydraulicfracture with equilibrium height growth across stress barriers. Int. J. of Rock Mech. and Min. Sci.,47:625?639, 2010.7 Appendix: Nolte?s pumping scheduleThis appendix summarizes Nolte?s pumping schedule that is used in this paper for comparison purposes. Asfollows from [6], the proppant concentration, which should lead to uniform distribution of proppant at theend of the fracturing job, can be written as?(t) = ?d( t? tpte ? tp)1???fd/?, t > tp,where ?d is the design concentration, te is the total injection time of the fluid and slurry, tp =((1??)2+fd)teis the time at which proppant is introduced, ? denotes the efficiency calculated as the ratio between the volumeof the fracture and the total volume that is injected, and fd = 0.05 is a correction factor. As discussed in [6],this correction factor makes it possible to match the data obtained through a numerical simulations. Notethat it is implicitly assumed that ?(t) = 0 for t < tp.14

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