ARMA 16-0150

Well stimulation in tight formations: a dynamic approach

Omidi, O. and Abedi, R.Department of Mechanical, Aerospace & Biomedical Engineering, The University of Tennessee Space Institute, TN, USA

Enayatpour, S.Center for Petroleum and Geosystems Engineering, The University of Texas at Austin, TX, USA

Copyright 2016 ARMA, American Rock Mechanics AssociationThis paper was prepared for presentation at the 50th US Rock Mechanics/Geomechanics Symposium held in Houston, Texas, USA, 26-29 June 2016.This paper was selected for presentation at the symposium by an ARMA Technical Program Committee based on a technical and critical review ofthe paper by a minimum of two technical reviewers. The material, as presented, does not necessarily reflect any position of ARMA, its officers, ormembers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of ARMAis prohibited. Permission to reproduce in print is restricted to an abstract of not more than 200 words; illustrations may not be copied. The abstractmust contain conspicuous acknowledgement of where and by whom the paper was presented.

ABSTRACT: Hydraulic fracturing is widely employed for well stimulation. Different techniques have been utilized inpractice to optimize fracking in the last five decades. However, it has some disadvantages including a lack of controlover the direction of fracture propagation, the high treatment cost along with environmental issues. Producing multiplefractures by dynamic stimulation techniques seems to be more promising in naturally fractured reservoirs, since it is aneffective way for connecting a pre-existing fracture network to a wellbore. In this study, applying high rate loadingswe investigate fracturing in rocks due to explosives and propellants as two common methods for dynamic stimulationof a well. An interfacial damage model implemented in a Spacetime Discontinuous Galerkin finite element framework isutilized to simulate fracturing in rocks. A powerful dynamic mesh adaptivity scheme is implemented to track arbitrarycrack paths and align them with element boundaries. High explosives produce shockwaves causing extreme compressivestresses, which results in crushing and compacting the rock around the wellbore. Propellants can generate a pressurepulse producing a fracturing behavior that loads the rock in tension. The main advantage of this later approach is tocreate multiple fractures and consequently prepare the well for an effective hydraulic fracturing with much lower cost asa re-fracturing solution.Acknowledgments: The authors gratefully acknowledge partial support for this study via the U.S. NationalScience Foundation (NSF), CMMI - Mechanics of Materials and Structures (MoMS) program grant number1538332.

1 INTRODUCTION

Stimulation treatments to enhance hydrocarbon re-covery from shale and other tight formations as un-conventional resources are mainly classified into frac-turing based on hydraulic, thermal and dynamic load-ings. There is no alternative to fracturing itself, sinceit is the only way to artificially fissure the sourcerock to provide sufficient permeability for oil andgas extraction cost-effectively. The technology of hy-draulic fracturing which is also called fracking hasbeen widely employed since 1949. It has been verycommon and popular in oil industry for decades ow-ing to technological advances in practice. Horizon-tal drilling was a complement to this method sincethe late 1980. This technique was developed fur-ther in 1997 by the use of chemicals known as slick-water fracturing in which slickwater is water mixedwith friction-reducing additives. Employing these ad-vances along with the development of multi-well pads

have made gas production from shales technically andeconomically feasible especially in North America inthe last decade.

However, hydraulic fracturing has been a contro-versial topic as an environmental concern during de-velopment of gas shale reservoirs located near popu-lated residential regions. Due to the use of water inthe fracking technique, one of the main worries is itspotential impacts on environment including the dan-ger of possible contamination of groundwater. Thisissue certainly depends on the complexity, scale andfrequency of the hydraulic fracturing required for anarea. The conventional hydraulic fracturing meth-ods usually produce a few number of main hydrauliccracks interacting with natural fissures. Huang et al.in 2011 showed that using water blasting for fractur-ing coal seams can enhance the effectiveness of hy-draulic fracturing by increasing the number and rangeof cracks resulting in an improved permeability [1]. Inthis technique, water pressure blasting is carried out

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by detonating the gel explosive located in a drilledwell. The shock waves generated by the explosioncause a high strain rate in the rock wall surround-ing the hole. The rock breaks and numerous circum-ferential and radial fractures propagate outward. Aconventional hydraulic fracturing is subsequently per-formed as the final stage so that the fissures openby the detonation are further develop and propagate.Reducing the quantity of water required for the hy-draulic fracturing stage, this method is potentiallyapplicable to the formations having low permeability.As noted above, there exist other methods for theformation stimulation that are not water-based. Forinstance, dynamic fracturing by either explosive, elec-tric discharge or propellant and the methods utilizingfluids or chemical materials other than water.

Different numerical methods have been employedfor the hydraulic fracturing simulations. These nu-merical approaches are conventionally categorizedinto discrete and continuum methods to address frac-turing in such a naturally fissured material. The cate-gory of discrete approaches include peridynamics andparticles methods. Developing a continuum plastic–damage model, Shojaei et al. simulated the hydraulicfracture propagation with the effect of injected porepressure in the developed fracture surfaces [2]. Usinga proposed elastic–plastic damage model for rock anda dynamic solution technique, Busetti et al. studiedthe effect of far-field stresses and pressure distribu-tion in the fracture on the geometric complexity of thefractured regions [3]. Furthermore, as a granular ma-terial, fracturing in tight formations can be modeledby the discrete element method [4, 5]. The extendedfinite element method (XFEM) and the generalized fi-nite element method (GFEM) representing the crackfaces by enriching the solution space with discontinu-ous functions has been widely used for the hydraulicfracture problems [6–8]. Dahi-Taleghani and Olsonused the XFEM in a 2-D implementation and ad-dressed the interaction between the induced hydraulicfracture and natural fractures/weak planes in thesenaturally fractured formations [6].

Including the fluid pressure degrees of freedom inthe formulation, Chen et al. proposed a simultane-ous solution for the fully coupled fluid flow and re-sulting crack propagation by a special XFEM imple-mentation [7]. Besides, Mohammadnejad and Khoeiemployed the XFEM for hydraulic fracture propa-gation by developing a coupled numerical tool [8].Focusing on propagating fractures with complex ge-ometries, like those encountered in early stages ofhydraulic fracturing, Gupta and Duarte proposed aGFEM for the simulation of non-planar 3D hydraulic

fractures [9]. In their proposed GFEM, the represen-tation of fracture surfaces is independent of the vol-ume FE mesh and consequently, complex surface fea-tures like sharp turns and kinks can be accurately cap-tured. Later on, Gupta and Duarte further developedtheir implementation by presenting a coupled hydro-mechanical GFEM formulation for the simulation ofnon-planar three-dimensional hydraulic fractures [10].Gordeliy and Peirce proposed a coupled algorithm us-ing XFEM to solve the elastic crack component of hy-drodynamic equations governing the propagation ofhydraulic fractures in an elastic medium [11]. Bazantand Caner proposed dynamic fracturing based on thekinetic energy of high-rate shearing generated by anunderground explosion to reduce the rock to smallfragments [12]. Using arc pulse electric dischargemethod, Chen et al. computationally and experi-mentally examined how this technique can improvepermeability of concrete [13].

Although the XFEM techniques in which re-meshing is unnecessary have been developed to modelcracks independently of the FEM mesh [14], includingrock mass discontinuities is problematic in XFEM-based simulations. On other hands, the discrete ele-ment methods (DEMs) have been widely used to over-come this limitation of continuum models [15]. InDEM, rock mass discontinuities can be explicitly sim-ulated along with fluid flow in the fracture network.However, since position and geometry of fracturesmust be assumed a priori, initiation and growth mech-anisms cannot be implicitly embedded for a proba-bilistic nucleation in DEM.

Employing the Spacetime Discontinuous Galerkin(SDG) finite element method formulated for elasto-dynamics for our analysis, we discuss an alternativeapproach to simulate dynamic fracturing based on ex-plosive and propellant techniques for stimulation oftight formations. The key feature of the SDG methodis direct discretization of spacetime using unstruc-tured grids that satisfies a special causality condition.This yields a local and asynchronous solution strategywith linear computational cost scaling vs. number ofelements [16]. It also enables arbitrarily high orders ofaccuracy both in space and time per element. Thesefeatures, which are otherwise difficult to achieve withconventional time marching schemes, result in a veryaccurate and efficient numerical method. Of particu-lar importance to hydraulic fracturing, it is the abilityof the method to align inter-element boundaries withuser-specified interfaces in spacetime. This techniquewell captures modes I and II of intact material break-age as well as shearing along newly created and pre-existing fractures.

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The other features this SDG can possibly ad-dress are fracture branching and coalescence, mate-rial heterogeneity along with rock mass discontinu-ities. Adaptive meshing allows propagation of cracksin the domain based on crack-path trajectories thatare obtained as a part of the solution according to acrack growth criterion. Thus, this model does not suf-fer the mesh-dependent effects encountered in mostother numerical fracture models. There are severaldistinct differences between our approach and theXFEM/GFEM methods. First by dynamically ad-justing the mesh in spacetime, we align the elementboundaries with predicted crack paths; unlike theXFEM methods no special discontinuity functions arerequired. Second, complicated crack pattern scenar-ios such as microcracking and crack bifurcation donot pose additional challenges in our approach, asagain no enrichment functions are needed to modelthem within the elements. Finally, since the elementsare not enriched with crack field functions, no spe-cial quadrature rules are required for their integra-tion [17].

2 FORMULATION

2.1 Overview of the SDG method

Discontinuous Galerkin (DG) techniques are a specialclass of finite element methods in which discontinu-ous basis functions are used for elements. Continuitybetween elements is enforced a priori in continuous fi-nite element methods (CFEMs), while this conditionis weakly imposed via the jumps between elementsin DG methods. Although they have more averagedegrees of freedom per element, there are two dis-tinct advantages over CFEMs in a transient analy-sis. First, in DG methods mass matrix has a blockdiagonal form resulting in a linear solution cost scal-ing versus number of elements. This is achievablein CFEMs only by the so-called mass lumping whichcan adversely affect the convergence rate in the prob-lems where higher orders of accuracy are importantand desirable. Second, due to their discrete solutionspace and treatment of fluxes between elements, DGmethods typically perform much better for problemstending to either preserve discontinuities and shocksin initial and boundary conditions or create them ifnonlinearities are present. Consequently, they do notsuffer from global nonphysical oscillations, a featurecommonly observed with CFEMs.

2.2 Adaptive schemes for dynamic frac-turing in rocks

2.2.1 Mesh adaptive operations

The most practical time discretization procedures em-ploy either implicit or explicit operators to advancethe solution in time. Herein, instead of using a timeintegration scheme, as illustrated in 1(a) for a sim-ple 1d×time causal mesh we directly discretize thespacetime using unstructured grids that satisfy a spe-cial causality constraint enabling us to utilize localsolutions for small collections of connected elements.High temporal orders of accuracy and adjusting themper element are distinct features of direct discretiza-tion of spacetime with finite elements. Both aspectsare particularly challenging within the framework ofmethods that employ a time marching scheme, dis-tinct from spatial finite element, to advance the so-lution in time. Moreover, the asynchronous struc-ture of the SDG method has very important impli-cations for multiscale simulations and parallel com-puting. Explicit time marching methods are oftenmore efficient but suffer from very restrictive timesteps for spatial meshes with vastly different elementsizes. While hybrid implicit-explicit (IMEX) and lo-cal time stepping (LTS) alleviate this problem to someextend, the time advance for distinct spatial elementsizes are completely decoupled in the SDG method re-sulting in a very efficient scheme for multiscale grids.Besides, in parallel computing, usual time marching(Figure 1(b)) results in global synchronization whichcan be a limitation in parallel efficiency. The asyn-chronous structure of the SDG method directly ad-dresses this issue. The causal SDG meshes enableasynchronous, element-by-element solutions with lin-ear complexity [16,18].

The SDG method’s features discussed above are ofparticular importance to dynamic fracturing in rocks.First, high gradient mechanical solutions around mov-ing crack tips require high order and stable numericalmethods. Second, to resolve fracture process zonewith high fidelity and maintain an efficient solutionscheme, a very small ratio of the elements around thecrack tip to the largest elements in the domain resultsin highly multiscale grids. Thus, a spatially and tem-porally high order numerical method such as the SDGmethod that can efficiently simulate multigrid meshesis of immense importance. Another level of complex-ity arises from moving wave fronts and evolving crackpaths in dynamic fracture problems. The interestedreaders are referred to [17,19] for the general analysisof the SDG method, and [20,21] for adaptive featurespertained to elastodynamics and fracture mechanics.

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Figure 1: SDG Solution scheme on causal spacetimemesh in 1d×time (top). Global coupling in non-causalmesh (bottom). Reproduced from [18]

2.2.2 Crack path tracking strategy

There are different crack path tracking strategiesavailable in finite element methods including contin-uum buck damage models, discrete cracking by eitherfixed or adaptive meshes and the XFEM-based sim-ulations. Continuum damage models do not capturephysical fracture faces and cracks are represented onlyas smeared regions in the bulk with a localized path.Furthermore, utilizing a fixed discretization, crackpropagation is restricted to existing element bound-aries. Obviously, handling mixed mode loadings inwhich the crack path is not predictable is challengingand the predicted crack path is not reliable in simu-lations using fixed meshes.

On the other hand, adaptive meshing scheme is asuitable approach for modeling crack propagation be-cause element boundaries are aligned with predictedcrack paths. However, simulating crack growth us-ing the classical FEM is quite difficult because thetopology of the domain changes continuously. On theother hand, allowing to simulate arbitrary disconti-nuity with a fixed mesh, the XFEM method follows acrack path within the elements and the domain doesnot need to be re-meshed as the crack propagates.Although the XFEM alleviates the problem of mod-eling arbitrary cracks and discontinuities of the finiteelement mesh, incorporation of nonlinear mechanismssuch as damage and plasticity at the crack interfaceis still a challenge in the available XFEM implemen-tations. Furthermore, different crack pattern topolo-gies, such as microcracking and bifurcation, demand

the derivation and inclusion of additional enrichmentfunctions. Our approach to model complex fracturepatterns fall into adaptive meshing category discussedabove.

2.2.3 Probabilistic nucleation

Stochastic distribution of natural fractures and pre-existing crack-like defects plays a critical role in frac-ture process of rocks. Cracks initiated from these de-fects are often accelerated through increasing stressconcentrations induced by initial fracture growth andlocal inhomogeneity. Energy absorption and stressrelease due to these local features of fracturing pro-cess result in a very non-uniform failure pattern atmesoscale. Since deterministic continuum modelstreat the material as perfectly homogeneous, theyare capable of predicting simultaneous failures at re-gions of high stress, which is not physical. To ad-dress this issue at the continuum level, a probabilis-tic nucleation model can be devised where cracks nu-cleate from pre-existing fractures that are randomlydistributed in rocks.

As discussed in the previous section, the SDGadaptive meshing scheme aligns the boundaries ofspacetime elements with crack-path trajectories hav-ing arbitrary position and orientation. The cracksnucleation is based on a probabilistic model trying tosimulate the nucleation process from randomly dis-tributed pre-existing fractures which can be observedexperimentally. Assuming a random distribution ofmaterial defects throughout the domain, the situationof individual defects varies according to the proba-bility distribution. Let s̃j

N (x) denote the nucleationstrength of a defect j in the mesoscopic neighborhoodof a point x in the simulated area as illustrated in Fig-ure 2. Let us assume that the number of defects inN(x) is n = ρNA while ρN is the density of defectsand A denotes the spatial measure of N(x).

V

x1

x2

xn

n = ρNA

Figure 2: A possible distribution of defects as sam-pling points around a vertex

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Assuming that the material’s resistance to nucle-ation at any defect is isotropic, nucleation of a newfracture surface occurs at defect j when the maximumeffective traction over all possible surface orientationsat the defect, denoted by s∗, exceeds that defect’snucleation strength; i.e. when s∗ > s̃j

N . Therefore,having several sampling points in a specified regionsurrounding a vertex, we need to examine this crite-rion with s∗ > s̃min

N in which s̃minN is the minimum

of nucleation strength among all possible samplingpoints as:

s̃minN = Min{s̃1

N , s̃2N , s̃

3N , ..., s̃

nN} (1)

The effective traction over the surface with angleθ shown in Figure 3 is defined as the following:

seff (θ) =√< sn(θ) >2 +β2 (st(θ))2 (2)

where β is the shear stress factor controlling modemixity; sn and st are the normal and tangential com-ponents of traction acting on the interface in bondedmode. The positive-part operator ensures that onlytensile stresses drive the damage evolution.

Similarly, an existing fracture surface will extendfrom its edges (endpoints in two dimensions), in adirection determined by the maximizing surface ori-entation, when s∗ > s̃j

N for a defect j in the neigh-borhood of the edge.

Vθ

Psn

st

sMeff

s̃

Figure 3: The circumferential distribution of effectivestress and crack propagation criterion

When nucleation or extension is examined in ourimplementation, we generate a number of defect testsproportional to the volume of the element or regionunder consideration. For each defect test, we gen-erate a randomized nucleation strength according toan assumed probability distribution. If any test indi-cates a positive result, then we implement nucleation

or extension through the adaptive meshing capabili-ties of our spacetime code. The probability distribu-tion is designed to ensure that nucleation and exten-sion occur at effective traction levels somewhat lowerthan the damage threshold traction, s. This ensuresthat the new fracture surface is in place, with dam-age initialized to zero, when the threshold traction isattained [17].

2.3 Interfacial damage-contact model

A two-scale, delay-damage cohesive model for crack-ing and contact problems is employed herein to cap-ture dynamic fracturing in rocks where as a highlyheterogeneous material, crack nucleation and growthfrom weak points is the primary mechanism of frac-ture. As shown in Figure 4, a macroscopic damageparameter D describes the area fraction of debondedinterface in the mesoscopic neighborhood N(x) of apoint x on a cohesive interface. The damage param-eter vanishes where the interface is undamaged, andD = 1 where the interface is, locally, completely dam-aged, i.e., debonded. A time-delay equation governsthe macroscopic evolution of D in response to the lo-cal stress state, while the standard Riemann solutionfor a material interface holds in fully bonded meso-scopic zones and the contact model developed in theprevious chapter governs the interfacial response inmesoscopic, debonded regions. Rather than refer to atraction-separation relation, we use a simple averag-ing of the mesoscopic model to determine the macro-scopic cohesive response.

rg

1 D

1

1

x

Bonded

Debonded

Contact

Separation

Stick

Slip

Figure 4: Schematic of contact-mode hierarchy andcorresponding fractions

Figure 5 illustrates an active, fully developed frac-ture process zone in the employed interfacial-damagecohesive model. The cohesive surface tip (CST) is theleading edge of the cohesive process zone where inter-facial damage begins to accumulate from D = 0 untilcomplete damage, D = 1, is attained at the trailingedge of the process zone.

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Figure 5: Cohesive process zone and the crack prop-agation mechanism

The crack propagation criterion is tested everytime a patch is pitched over an active CST vertex,such as V in Figure 3. Vertex V may be the tip ofan advancing crack as illustrated in the figure or bea newly nucleated CST. The weighted averages of Dand (1−D) is used to define the macroscopic targetRiemann stress at each point as:

σ∗ = (1−D) σ∗B +D σ∗

D (3)

where σ∗B and σ∗

D are the bonded and debondedstress vectors, respectively. Any point on an in-terface can take three different statuses of contactin debonded mode (i.e., separation, contact–stickor contact–slip). Figure 4 shows the hierarchy ofmesoscale contact modes in the neighborhood of amacroscopic point in the contact set. At the firstlevel of subdivision, the debonded part of the inter-face is partitioned into regions that are either in con-tact mode or in separation mode, with respective areafractions η ∈ [0, 1] and 1−η, relative to the debondedarea. The connectivities of these regions may be ar-bitrarily complex. However, the connectivities do notinfluence the macroscopic target stress and velocityvalues generated by our simple averaging scheme, sowe are not concerned with them here. However, wedo treat η as a continuous variable to support regu-larization of the separation–contact transition. Thesecond level of subdivision partitions the contact re-gion into contact–stick and contact–slip regions, withrespective area fractions γ ∈ {0, 1} and 1− γ relativeto the area of the contact region. We treat γ as abinary variable because we have no need to regularizethe stick–slip transition. See [18] for identifying themesoscale parameters, η and γ.

Rapid contact transitions and sharp moving frontsmake dynamic contact modeling very challenging.Numerical solutions typically suffer from nonphysi-cal oscillations or under/overshoots at these transi-

tions. Artificial diffusion, although may reduce thesefeatures, excessively smoothens the solution in gen-eral. The set of contact property solutions employedpreserve the characteristic structure of elastodynam-ics and enable us to capture these sharp transitionswith almost no numerical artifacts. In addition, thesesolutions were integrated with the interfacial dam-age model to study combined contact and fractureproblems and smooth transitions of various contactmodes occur. For example, the direction of the sliptransition is ambiguous at stick-slip transition and itcalls for regularization in almost all numerical meth-ods. It has been demonstrated that this transition isin fact continuous and no regularization is requiredwhile this set of solutions is used. The only separa-tion to contact transition physically induces shocksand a regularization scheme with tunable maximumpenetration was implemented to treat this issue. De-tails of contact model’s features and formulation havebeen provided in [18]

3 NUMERICAL APPLICATIONS

In the stimulation techniques in which static loadingsare applied, the cracks generated is proportional tothe energy transferred to the volume of material be-ing broken. However, dynamic loadings apply a largeamount of energy to a small volume of material. Inthis situation, a large area of cracks will be created.As the loading wave spreads inside the material, it willcreate fragmentations, thereby connecting the naturaland induced fractures. Utilizing explosives as a stim-ulation technique in tight formations to increase pro-duction is a very old technique, which is also widelyemployed in mining and known as well shooting. Be-sides, the burn of a propellant in a well is a rapid oxi-dation reaction causing the release of gaseous energy.Figure 6 shows three typical pressure-time profiles ofexplosive, propellant and the conventional hydraulicfracturing events in which the loading rates differ inthe scales noted in the figure. As observed, a propel-lant event is therefore intermediate in comparison tothe other two methods in its energy release rate.

Two loading rates are considered herein for theapplication part to demonstrate the performance ofour dynamic approach for tight formation stimula-tions. As the first problem, a well is pressurized in avery high loading rate to indicate an explosive load-ing. In the next examination, we simulate a perfo-rated wellbore subjected to an intermediate loadingrate. Therefore, the second problem followed in thissection is devoted to propellant fracturing from theoriented perforations.

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Figure 6: Pressure-time profiles of three stimulationapproaches

The fluid pressure along fracture faces, specificallynear the wellbore can be treated as a nearly constantdue to the high in-situ stresses acting as effective con-fining pressures in the deep reservoirs and the lowfluid viscosity. Utilizing analytical solutions for thisproblem, references [11, 22] showed that consideringthis behavior is reasonable. Allowing that, it is alsopractical to deduce that the fluid and fracture frontsoccur simultaneously and are coincident, which meansthe fluid lag is negligible. It should be noted that thesize of the fluid lag is inversely proportional to σ3

0, inwhich σ0 is the far field confining stress [23]. In thefollowing, the maximum and minimum compressivestresses are denoted as σH and σh, respectively.

3.1 Explosive fracturing

A pressurized wellbore in a domain, which is sub-jected to far field confining stresses as bi-axial trac-tions, is considered in the following examination. Theproblem sketch is illustrated in Figure 7. The diam-eter of the wellbore is 0.30 m and is subjected to theconfining in-situ stresses as σh = σH = 3.6375 MPa.The material properties are: Young’s modulus E = 20GPa, Poisson’s ratio v = 0.2 and the tensile strengthof 2 MPa. The fluid pressure is assumed to increase intime as a dynamic loading from a stabilizing constantpressure being applied as a static load. A maximumpressure of 38.8 MPa based on the profile shown inFigure 6 with the total duration of 0.4 ms is applied

to the sidewalls of the well. Since, there is no explicitnucleation point for fractures to be initiated from,the probabilistic nucleation method discussed in theprevious section is used here.

Figure 7: Problem sketch for the explosive fracturingapplication

Evolution of damage which is mostly in shearmode is shown in Figure 8. Furthermore, fracturevisualizations of the solution are illustrated in Fig-ure 9 for the corresponding time frames shown inthe damage evolution figures. When the explosiveis detonated in practice, an extremely high and com-pressive pressure pulse, or shock wave is generated,which far exceeds the tensile strength of the forma-tion rock. The high pressures of the detonation causethe rock to yield and compact. After the stress wavepasses, the rock unloads elastically, leaving an en-larged, deformed wellbore, a zone of compacted rockand a region of greater compressive stress. Herein,the cracks occurring are almost all in shear damagemode. As reported from experiments in practice, thiscrushed region may reduce permeability and produc-tivity around the wellbore.

3.2 Propellant fracturing

Before performing hydraulic fracturing in practice,wellbores usually are cased and then perforated to iso-late the well from undesirable regions and to considersome operational considerations in the field alongwith stability concerns. These perforations gener-ated during the process of a well completion play therole of a transmission channel between the wellboreand the reservoir. In fact, a perforation may serve asan initial fracture to help with crack nucleation and

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(a) time ≈ 0.033 ms (b) time ≈ 0.062 ms (c) time ≈ 0.077 ms

(d) time ≈ 0.098 ms (e) time ≈ 0.148 ms (f) time ≈ 0.207 msFigure 8: Evolution of damage in solution of a well stimulation by explosive fracturing technique

(a) time ≈ 0.033 ms (b) time ≈ 0.062 ms (c) time ≈ 0.077 ms

(d) time ≈ 0.098 ms (e) time ≈ 0.148 ms (f) time ≈ 0.207 msFigure 9: Visualization of fracture evolution in solution of a well stimulation by explosive fracturing technique

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slightly force propagation direction to perform an ef-ficient treatment. Therefore, perforations are impor-tant in the complex fracture geometries around well-bore. The success of stimulation treatment throughperforations depends on several parameters includingits length, diameter along with permeability of therock around the perforation. The enhanced perme-ability of the rock around the wellbore controls recov-ery flow through a perforation. By shortly reviewingeffective parameters, which are important for a perfo-ration design, the function of perforation in hydraulicfracturing is discussed in the following. Perforationphasing, which is the angle between the two successiveperforations, is another important parameter affect-ing production rate and needs to be carefully assessedat its design stage. Common perforation phasing an-gles are 60o, 90o, 120o and 180o.

Herein, as sketched in Figure 10 one applicationwith phasing angles of 90o is considered. All param-eters are same as the previous example, except therate and duration of the loading. In this applica-tion, a maximum pressure of 19.4 MPa during a totaltime of 2 ms is applied to the sidewalls of the welland inside the perforation surfaces. Unlike the ex-plosive stimulation example in which there was notany nucleation point to initiate cracking, here thereare four crack tips and we can expect that all frac-turing start from these tips. The results of damageevolution and fracture opening visualizations for sev-eral time frames have been plotted in Figures 11 and12, respectively. The damages occurred in this sim-ulation are mostly in tensile mode where we expectto have fracture opening in the visualizations of thesolution shown in Figure 12.

Figure 10: Problem sketch for the propellant fractur-ing application in a well with four perforations

As observed, a propellant stimulation rapidly ex-ceeds the fracture pressure of the rock and maintainthe pressure but do not crush the rock as we no-ticed in the explosive stimulation application. As aresult, propellants pressurize and break down all per-forations along a significant fracturing and penetratefarther in comparison with explosive fracturing. Theadvantages of propellant stimulation approach overthe conventional hydraulic fracturing is that multiplefractures are created, consequently the entire consid-ered zone is stimulated. Besides, there is no need toinject fluids, which makes it a better choice to pro-tect the environment. Of course, it is not a com-plete replacement for hydraulic fracturing and whenhydraulic fracturing is not practical economically, itcan be an initial treatment prior to fracking.

4 CONCLUSIONSOne of the most important applications of hydraulicfracturing nowadays is to improve the recovery of un-conventional hydrocarbon reservoirs. Having an ap-propriate fracture propagation model in rocks is acrucial issue for a hydraulic fracture design. Manyapproaches have been developed to efficiently per-form crack growth simulations, which are mostlybased on either efficient remeshing techniques or theXFEM/GFEM employing fixed meshes, but theseare mainly limited to the linear elastic fracture me-chanics (LEFM) framework. In this paper, an in-terfacial damage model implemented in a SpacetimeDiscontinuous Galerkin (SDG) framework is utilizedto simulate nucleation and then propagation of hy-draulically induced fractures in an oil reservoir. TheSDG method offers many advantages over conven-tional and extended/generalized finite element meth-ods including dynamic adaptive meshing, interfacetracking, and element-wise conservation. To facili-tate crack propagation in any arbitrary direction weuse the SDG’s powerful adaptive meshing capabilitiesto align cracks with inter-element boundaries; Unlikethe XFEMmethods no special discontinuity functionsare required.

Hydraulic fracturing has been widely employedfor well stimulation in the last five decades. Dif-ferent techniques, equipment, fracturing fluids andproppants have been utilized in practice to optimizethe fracturing process. However, it has some disad-vantages including a lack of control over the direc-tion of fracture propagation, the high treatment costalong with some challenging environmental issues.Dynamic stimulation techniques generating multi-ple fractures in a wellbore to enhance gas recovery

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(a) time ≈ 0.102 ms (b) time ≈ 0.510 ms (c) time ≈ 0.715 ms

(d) time ≈ 0.863 ms (e) time ≈ 1.037 ms (f) time ≈ 1.165 msFigure 11: Evolution of damage in solution of a well stimulation by propellant fracturing technique

(a) time ≈ 0.102 ms (b) time ≈ 0.510 ms (c) time ≈ 0.715 ms

(d) time ≈ 0.863 ms (e) time ≈ 1.037 ms (f) time ≈ 1.165 msFigure 12: Visualization of fracture evolution in solution of a well stimulation by propellant fracturing technique

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are nowadays attracting more attention in oil indus-try. Producing multiple fractures seems to be morepromising in naturally fractured reservoirs, since it isan effective way for connecting a pre-existing fracturenetwork to a wellbore. Applying high rate loadingsby explosives and propellants are the two commonmethods for dynamic stimulation of a well.

The technique using explosives, which is alsocalled well shooting in the literature, has been foundto have damaging effects on the well since it causescrushing and plastic flow near the borehole which canpossibly result in a severely locked formation ratherthan the multiple fracture pattern desired for en-hancing permeability. Whereas the pressure loadingrate is too slow, a single extensive hydraulic fractureis produced. On the other hand, dynamic stimula-tions through propellants utilizing intermediate load-ing rate in comparison with slow rates in hydraulicfracturing and fast rates in explosive fracturing cangenerate multiple fractures without excessive crush-ing or residual damage around the borehole.

Although hydraulic fracturing has been employedfor several decades in oil industry, a thorough un-derstanding of the interaction between induced hy-draulic fractures and pre-existing natural fractures isstill challenging. Our approach is also applicable tohydraulic fracturing where an induced major crackpropagates and intersects natural fractures which inturn are hydraulically loaded and extended to inter-sect other fissures resulting in a complicated fracturenetwork. Furthermore, incorporation of macro-microcrack interactions can explain discrepancies for track-ing efficiency between real productivity and compu-tational estimations. The future work focuses on uti-lizing the developed interfacial damage model in cap-turing the interactions between hydraulically inducedfractures and natural fractures.

Water usage is reduced or completely eliminatedin the stimulation methods using dynamic loadings,however they usually suffer from some potential dis-advantages including higher costs, riskier due to us-ing a dangerous substance and limited possibility tooperate at depth. Although these new techniquescould help address the environmental concerns, thehydraulic fracturing is still the preferred approach bythe oil industry. Therefore, a combination of thesetechniques might be more effective. The main ad-vantage of propellant fracturing is to create multiplecracks and consequently prepare the well for an effec-tive hydraulic fracturing with much lower cost as are-fracturing solution.

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