International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Computational simulation of hydraulic fracturing nonlinear dynamics using Gaussian processes surrogates Zio, Souleymane; Rochinha, Fernando A. Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Computational Simulation of Hydraulic Fracturing NonlinearDynamics using Gaussian Processes SurrogatesSouleymane. ZioPhd Student, Dept. of Mechanical Engineering , Federal Univ. of, Rio de janeiro, BrazilFernando A. RochinhaProfessor, Dept. of Mechanical Engineering , Federal Univ. of, Rio de janeiro, BrazilABSTRACT: High-Fidelity physics based computational models enables the design and optimization ofcomplex engineered processes. Moreover, important and strategic decisions might be taken relying onthose computational models predictions. Therefore, there is a need for improving their robustness andreliability. Therefore, understanding the impacts on the predictions due to unavoidable input and modelstructures uncertainties, often referred to as Uncertainty Quantification (UQ), has become a major issue.A key aspect in this context is the demand of a significant computational effort involving many-queriesof a computer code. That might be lessen by the use of reduced order models or any form of surrogates.Here, we employ Gaussian Processes (GPs) as a surrogate (often referred to as emulators) for a computercode devoted to Hydraulic Fracturing simulation.1. INTRODUCTIONHydraulic fracturing (HF) is a widely used engi-neering process in the petroleum industry for im-proving the productivity of the oil and gas reser-voirs. The fracturing fluid is pumped into thewell at an elevated pressure, this pressure must begreater than the minimum in-situ stress to createa fracture in the reservoir rock. The mathemati-cal model of this process relies on complex non-linear and free boundary problems leading to com-plex computer models. To resolve this free bound-ary and multi-scale problem, the implicit level setis combined with the plane asymptotic solutions atthe tip of the fracture (Peirce, 2014). The solution atthe tip depends on the different physical processesnamely, storage viscosity, storage-toughness, leak-off viscosity and leak-off-toughness. Each of theseprocesses has different values of the asymptotic so-lution at the tip of fracture which depends also ofthe reservoirs rock properties. Typically, the mod-eling of a HF involves scenarios that present un-certainties in the Geomechanics properties, like,for instance, but not limited to, Elasticity Modu-lus and confining stresses. In literature, severalmethods are used to take into account uncertaintyover the parameters in the numerical simulation.Methods such as Monte Carlo (MC), general Poly-nomial Chaos(gPC) (Ghanem and Spanos, 1991),Multi-element generalized Polynomial Chaos(ME-gPC)(Xiaoliang Wan, 2005), stochastic colloca-tion (Zio and Rochinha, 2012), Adaptive stochasticcollocation (ASCM) present some difficulties re-lated to computational demand, convergence, andto calculate some statistics information. One ofthe common aspects of these methods is the de-mand of a high computing effort involving many-requests from a computer code. That might belessen by the use of reduced order models or anyform of surrogates(often referred to as emulators)(,Mingjie Chen). Here, we employ a nonstationaryMultivariate Gaussian process as surrogate(MGPs)(Bilionis and Zabaras, 2008) for a computer codedevoted to Hydraulic Fracturing simulation. TheMGPs constructs in this work can be efficient toapproximate the nonlinear and nonsmooth behav-iors of HF response in heterogeneous, discontinu-ous reservoir rock.112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20152. HYDRAULIC FRACTURE MODELINGModeling the evolution of a hydraulic fracturein a geologic formation involves distinct physicalprocesses: deformation of the surrounding elasticmedium, the fracture propagation, the flow of aviscous fluid within the fracture, and the leak-offof the fracturing fluid. In order to achieve a bal-ance between accuracy and easiness of the simu-lation, a number of reasonable simplified models,which are readily justifiable in the present context,are combined to describe the multiphysics scenario.The fracture grows in limit equilibrium, follow-ing the linear elastic fracture mechanics , within alayered formation portraying homogeneous layers(spatially uniform values of the toughness Kic of theElastic Modulus E and of the Poisson ratio ν). As-suming a plane strain state, the elastic problem, andconsequently the flow problem, is reduced to a onedimensional spatial setting (Adachi and Detournay,2008) with coordinate x coincident with the frac-ture propagation direction and occupying a domainthat evolves with the time t described by the interval`(t)= [llow(t), lup(t)]. The fracture is propagated bythe injection, through the well, of an incompress-ible Newtonian fluid with dynamic viscosity µ . Asthe flow takes place in the confined interior volumeof the fracture delimited by its aperture w(x, t) ,lubrication equation is assumed to model the flowbehavior characterized by the pressure p f (x, t) andfluid flux q(x, t). The whole process is controlledby the injection rate Q0 and the physical and ge-ological conditions imposed by the rock formation,namely: the confining stress σ , perpendicular to thefracture aperture direction, and the four material pa-rameters E ′, µ ′, K′, and C′ defined as: E ′ = E1−ν2 ;µ ′ = 12µ; K′ = 4( 2pi )12 Kic; C′ = 2Cl . Here, E ′ is theplane strain modulus, µ ′ the alternate viscosity, C′the leak-off coefficient and K′ the rock toughness.The governing equations are presented belowin a dimensionless form resulting from a scalingthat allows to unveil the different fracture evolutionregimes, which will not be discussed here due tothe lack of space, but are pivotal in the design ofthe numerical approach proposed in (Peirce and De-tourney, 2008). This scaling introduces character-istics length, time, pressure, and fracture opening:`∗, t∗,w∗, p∗, to be determined in line with the afore-mentioned propagation regimes. Therefore, thequantities of interested are expressed as: x = `∗χ,t = t∗τ , w = w∗Ω, p f = p∗Π f , σ0 = p∗Σ0Φ(χ),Q = Q0Ψ(τ). The quantities χ,τ,Ω,Π f representthe dimensionless spatial coordinate, time, fractureaperture and fluid pressure. Moreover, Φ(χ) is thespatial distribution of the confining stress.Elasticity EquationΠ=Π f (χ,τ,ϖ)−Σ0(ϖ)Φ(χ,ϖ) =−GE4pi∫ `up(τ,ϖ)−`low(τ,ϖ)Ω(χ ′,τ,ϖ)(χ−χ ′) dχ′, (1)where Π(χ,τ,ϖ) stands for the nondimensionalnet pressure (resulting from the difference be-tween fluid pressure and the geological confiningstresses).Lubrication Equation:∂Ω∂τ +GcH(τ− τ0(χ))√(τ− τ0(χ))=Gm∂∂χ [Ω3∂Π f∂χ ]+GvΨδ0(χ), (2)where H is the Heaviside function and δ0 is theDirac function centered at the injection points.Boundary and Propagation Conditions:limξ→0Ω3∂Π f∂ξ = 0,Ω(`low,lup, ., .) = 0, (3)limξ→0Ωξ 1/2= Gk, (4)where ξ = `(τ, .)− χ is a local coordinate repre-senting the distance from a fracture interior pointto the fracture tip. The first condition correspondsto no flux through the tips; the second implies thatthe fracture aperture vanishes, and the third one re-flects the aperture asymptotic behavior correspond-ing to the linear fracture mechanics. The dimen-sionless quantities G j above, whose values deter-212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015mine the propagation regimes, are defined as fol-lows: GE =p∗`∗E ′w∗, Gm =µ ′Q0w3∗p∗, Gv =Q0t∗w∗`2∗, Gc =C′`2∗Q0t1/2∗, Gk =K′`1/2∗E ′w∗.Locating the free boundary using the tipasymptoticsAdmitting GE = Gm = Gv = O(1), the above equa-tions correspond to the viscosity dominate propaga-tion regime (Peirce and Detourney, 2008), in whichthe viscous flow is the main dissipation mechanism,the tip fracture (within the spatial scale introducedby the numerical method) follows the asymptotictrendlimξ→0Ω∼ βm0V 1/3ξ 2/3, (5)where βm0 = 21/333/6(µ′E ′ ), and V the normal veloc-ity of the front, what is explored in the numericalscheme proposed in Peirce and Detourney (2008).Besides the space and time coordinates χ andτ , we introduce above ϖ , a third argument for thefields above, in order to represent the random di-mension needed for describing uncertainties in theinput parameters within a probabilistic perspective.Here, we follow the same approach presented in(Zio and Rochinha, 2012)3. NONSTATIONARY MULTI-OUTPUTGAUSSIAN PROCESSES (MGPS)In this section, we briefly reproduce main con-cepts and building blocks proposed in Bilionis andZabaras (2008) for developing local Gaussian sur-rogates considering multiple outputs and employ-ing a Bayesian approach. The statistics of a Gaus-sian Process f(.) is completely specified through itsmean and covariance functions. The first deals withthe general trend of the random field and the covari-ance expresses the smoothness of the dependenceon the arguments and the correlation between thedifferent components.In order to make the presentation simpler andmore compact, we adopt a simplified formal nota-tion, frequently found in the literature. The simu-lator (here corresponding to the numerical schemeused to solve the nonlinear partial differential equa-tions presented in the previous section), is to berepresented as a mapping connecting inputs x (ma-terial and control parameters, initial conditions,...)and outputs y (variables or quantities of interestlike, for instance, fracture length and aperture,velocity of propagation), is denoted as f(x) witha K-dimensional input x ∈ X ⊂ RK , i.e. X =×Kk=1[ak,bk], −∞ ≤ ak < bk ≤ ∞. To reflect un-certainties in the input parameters, x is considereda random vector with probability density p(x) de-fined as ∏Kk=1 pk(xk), assuming independence be-tween the individual inputs.The multiple outputs of the simulator are orga-nized in a vector form: y ∈ Y ⊂ RM, where M rep-resents the number of outputs. At this point, it isimportant to realize that the components of this vec-tor might have different physical meanings (pres-sure, velocity or length), which requires normaliza-tion before the construction of the simulator surro-gate. The simulator is ran at selected training in-puts X= (x1, ...,xN), leading to the training outputsY= (y1r = f(x1), ...,yNr = f(x)N) where r = 1, ..,M.We assume that we have observed a fixed numberN ≥ 1 and training set D= {(X,Y)}.From a Bayesian perspective, we regard f(.) asan unknown function to be inferred from the dataproduced at the training points. The motivation isto obtain a mapping f(.) which produces the simu-lator outputs at any inputs different from the train-ing points efficiently. That means that we seeka significant reduction in the computational costswhen compared with running the simulator itself.Therefore, we are seeking for replacing the orig-inal computational simulator that is referred to asa surrogate, many times also deemed as an emula-tor. As we are going to use limited data, the surro-gate is at most only an approximation of the sim-ulator, what, therefore, entails uncertainties in thedesired predictions. This uncertainty in the outputsis not to be confused with the one induced by theparameters (inputs) of the model. We cast the un-certainty in a probabilistic framework and build thesurrogate as statistical approximation of the deter-ministic simulator by combining a priori informa-tion with the data produced with the training. Weconsider as prior information that f(.) is a Gaus-sian Process (MGP) conditional on some hyper-312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015parameters. Baye’s theorem and a gaussian likeli-hood function guarantees that the posterior distribu-tion of f(.), regarded as a surrogate to the expensivecomputational simulator, is also Gaussian processconditional on the hyper-parameters, formally de-scribed by:f(.)∼GP(m(x),C(x,x’;Θ)), (6)where m(x) and C(x,x’;Θ) are the mean and thecovariance function of the GP and Θ the unknownhyper-parameters. The choice of covariance mod-els for multiple outputs has been discussed in theliterature, where one can find many different op-tions. Here, we adopt the non isotropic Square Ex-ponential function:cSE(x,x’) = s2f exp(−12K∑k=1(xk− x′k)2`2k), (7)In order to keep a compromise between accuracyand easiness on the use of a surrogate, we do notadopt a fully Bayesian approach. In that same line,all the outputs share the same correlation function,which does not allow an explicit modeling of thecorrelation between outputs. The hyper-parametersΘ= (s f , `1, ..., `k), having as components the signalstrength and the correlation lengths in each input di-rection, are obtained by maximizing the logarithmof the marginal likelihood. Here, we use the Con-jugate Gradient method(CG) to maximize the log-likelihood, obtaining the optimal value Θ∗. Usingthe correlation function defined with Θ∗ and the loworder statistics (mean and variance) of the trainingdata, we compute a GP surrogate and, therefore, weare able to make predictions at any input points.Having this surrogate, we can address an Uncer-tainty Quantification study. We can compute thestatistics of outputs such as mean, variance, and theprobability distribution(PDF) due to uncertaintiesin the input parameters. That can be done either bysampling or, if possible, by computing the corre-sponding integrals analytically. Moreover, the un-certainty on those predictions due to the limited in-formation used in the construction of the surrogatecan be estimated by the final variance of the MGP.Here, we employ the adaptive strategy designed inBilionis and Zabaras (2008) that explores this vari-ance to guide the decomposition of the input spaceinto disjoint subsets. For each subset, a local MGPis built to capture the evolving dynamics of the frac-ture as we use time dependent outputs. In the nextsection, along the numerical examples, the natureand choice of outuputs will be presented in furtherdetail in the next section. A broader discussion ofemulators to be employed as surrogates for simu-lators of dynamical systems lies outside the scopeof this work and can be found in Stefano Conti(2010). Moreover, the adaptivity process also re-sorts, if needed, to the inclusion of more trainingpoints within a subdomain (until a prescribed num-ber) such that an a priori accuracy threshold δ forthe variance of the MGP is achieved. Technical de-tails about the algorithm in its implementation canbe found in Bilionis and Zabaras (2008) At the endof this adaptive process, the whole input space iscovered with local approximations, which allows tocapture nonstationarity of the emulator, which is ofcrucial importance in our application, as it will bedemonstrated in the next section.4. NUMERICAL RESULTSWe study here a challenging problem in which thefracture propagates vertically within a three-layeredmedium, as described schematically in (Fig.1). Inthis complex geological formation, the fracturegrowth tends to experiment a non-symmetric evolu-tion due to abrupt changes in the confining stresseswhich assumes different values at each layer ( Σ10for the layer 1, Σ20 layer 2 and Σ30 for layer 3). Thelayers are separated by interfaces χ1 and χ2, thatthe fracture will pass through at unknown injectiontimes. In such a scenario, numerical simulationsof the fracture propagation might help the opera-tion in the field, trying to reduce the risks or to im-prove the stimulation. However, non accurate de-scriptions of the input parameters obtained by in-direct measurements might hamper the ability ofthe numerical model to deliver trustworthy predic-tions. Here, we present a numerical modelling ofHF, which also takes into account uncertainties in-herited from imprecise measurement of rock prop-erties. We combine the robust numerical methoddeveloped in (Peirce and Detourney, 2008); that412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015takes care of the deterministic aspects of the prob-lem, with the MGPs described before. Those un-certainties are represented in the model by consid-ering some of the input parameters as random vari-ables. We elected the Elasticity Modulus and theconfining stresses on each layer as those uncertainparameters. Indeed, those choices entails a reason-able first scenario to be analyzed, as those parame-ters have an important impact in the fracturing, andtheir values in the technical literature show a sub-stantial dispersion that could be represented withthe help of a probabilistic model.The fracture takes place in a rocky medium con-sisting of three layers nominated hereafter as layer1, layer2 and layer3. The fracture is initiatedintermediary layer and evolves towards the othertwo. Taking in consideration typical values ob-served in this type of rock formation and with avolumetric injection rate Q0 = 0.11m3/s. We ob-tain the dimensionless values of the confining stressΣ10 = 1.25±0.02, Σ20 = 1.41±0.041 , Σ30 = 1.45±0.031. In the probabilistic model we consider thatΣ10, Σ20, Σ30 have a uniform distribution with meanΣ10=1.25, Σ20=1.41, Σ30=1.45 and the standard devi-ation σΣ10=0.02, σΣ20=0.041, σΣ30=0.031. The Elas-ticity Modulus, also modeled as an uniform ran-dom variable, takes the following values: GE = 1and σGE =0.15. The quantities of interest directlycomputed from the simulation are: fracture lengthin upper and lower parts of the fracture `up(τ,ϖ),`low(τ,ϖ), the fracture aperture Ω(χ,τ,ϖ) and thefluid pressure Π(χ,τ,ϖ) along the fracture.The HF process begins at the initial time τ0 =11.72 and stops at time τ f = 240. The interfacesare located at the position χ1 = 4.2 and χ2 = −7.5with respect to the well position χ = 0 (Fig.1). Us-ing the expected value as nominal parameters, wesimulate the fracture evolution and observe that thefracture reaches the interfaces 1 and 2 at the approx-imate time τp1 = 100 and τp2 = 200,respectively, asshown in (Fig.2). That figure also presents the frac-ture aperture evolution at the interfaces. In (Fig.2),we present the fracture length on both sides. Weobserve that the evolution of the length in the upperand lower is the symmetric until the fracture propa-gation reaches the first interface. When the fracturereaches the interface 2, the length of the fracture inthe lower part feels this effect and changes its trendof growing abruptly. To have a better understand-ing of the nonlinearity and the non smoothness ofHF response, we perform a sensitivity analysis byplotting outputs with respect to certain inputs. Inthis analysis, we consider outputs such as the upperlengths of the fracture in time τ `up(τ = 47) and`up(τ = 240), the fracture aperture at the interfaceχ2, Ω(χ2,τ = 240). These outputs are plotted withrespect to the pair of inputs (Σ30,GE), (Σ30,Σ20). In(Figs. 3 - 4). Below, we will construct a surrogateto approximate efficiently the HF outputs having asinputs the four uncertain parameters, and we inves-tigate its ability to handle the non smooth responsereported in the previous figures.Figure 1: Schematic view of the layered mediumFigure 2: Fracture aperture at the interfaces evolution512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Figure 3: Fracture length `up(τ = 47)Figure 4: Fracture aperture Ω(χ2,τ = 240)4.1. Surrogate computational performanceIn order to assess the accuracy of the MGPs,we employ a reference solution obtained by MonteCarlo (MC) simulations. This reference solution isobtained after checking the convergence of somerelevant outputs such as the final length of the frac-ture and the aperture at the interfaces positions de-picted at (Figs. 5- 6). In these figures, we observedthat 30,000 samples were enough for achievingan accurate approximation with the Monte Carlomethod for the mean and variance of the outputs.Then, we verify the efficiency of the MGP surro-gates by comparing its predictions with the onesprovided by the MC simulations.The surrogate depends on the definition of whatinputs and outputs are relevant for a specific anal-ysis. In a hydraulic fracture treatment, the finalextension of the fracture and volume occupied byit make important quantities for the analysts. Al-though we are dealing with a dynamic scenariothat develops in a spatial scenario, neither time orspatial positions were considered as inputs. Wedecided to have them only as indexes of the out-puts in order to make the computational surrogateconstruction a tractable problem. We built theMGP electing as outputs the lower length of thefracture along the whole timeframe `low(τ0 : τ f ),the fracture aperture after the fracture reaches thefirst layer. Therefore, in this particular applica-tion we have 4 inputs and a total of 108 outputs:{`low(τ0 : τ f ),Ω(χ1,τp1 : τ f ),Ω(χ2,τp2 : τ f )}. Wefixed the maximum number of training points ineach stochastic element Nmax = 20 and used withthree different values of the accuracy threshold:δ = 10−3,10−4,10−5. In (Figs.7-8) we comparethe results obtained with the surrogate with thosecomputed with MC. The HF code was called 22;34 and 99 times to construct the surrogate usingthe different levels of accuracy. We observe thatthe surrogate constructed with different values of δhave a mean value close to the reference solution(MC 30,000). The mean and standard deviationof outputs are close to the reference solution whenδ = 10−5 correspond to 99 called of HF numeri-cal code. These results show that, the efficiency ofthe MGP as surrogate in this example by achiev-ing accuracy with a very low runs of the expensiveoriginal simulator.Figure 5: Convergence in mean and standard deviationof the the lower length of the fracture `low at τ = 240using MC4.2. MGPs capturing local features of HydraulicFracture evolutionThe objective of this section is to see how MGPscan cope with abrupt changes in the simulator re-sponse, like the ones featured in our previous sen-sitivity analysis that reveals different output behav-iors for small changes in the input parameters. That612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Figure 6: Convergence in mean and standard deviationof fracture aperture in the interface 2 Ω(χ2,τ = 240)using MCFigure 7: Fracture length mean and standard deviationMGP vs MCFigure 8: Fracture aperture Ω(χ2,τ) mean and stan-dard deviation MGP vs MCcan accommodate by dividing the input domain intosubregions, such that inside each one the responseis smooth. That is by far a non trivial task in the sit-uation we are now analysing, as the abrupt changesin the response, are time dependent. To emphasizethis, we construct two different surrogates. The sur-rogate 1 is built for describing the fracture evolutionbefore it reaches the interfaces χ1 or χ2. The secondone takes into consideration the whole timeframeof the evolution. The surrogate 1 is built usingthe outputs obtained at the injection time between[τ0,τp], such as the fracture length at lower and up-per, the fracture aperture at the well and the aper-ture at the tips, {`up(τ0 : τp1), `low(τ0 : τp1),Ω(χ =0,τ p1),Ω(χtipr ,τp1),Ω(χtipl ,τp1)} with the totalof M= 41 outputs, where τp < τp1 < τp2. Thesurrogate 2 is built using the outputs obtained atthe injection interval [τp1,τ f ]. At these time. Thesurrogate 2 has as outputs {`r(τp1 : τ f ), `l(τp2 :τ f ),Ω(χ = 0,τ f ),Ω(χtipr ,τ f ),Ω(χtipl ,τ f )} with atotal of M= 46 outputs. The maximum number oftraining points was established as Nmax = 20. Thesurrogates were constructed with the degree of pre-diction δ = 10−5. We observe that, to construct thesurrogate 1 only two stochastic divisions (two localsurrogates) were enough to get δ = 10−6 with thetotal of Nt=22 calls of the HF code . The surrogate2 was constructed after 25 divisions of input spaceto get a δ = 10−5 with the total of Nt=296 calls ofthe HF code. The input domain decomposition isillustrated in in Fig.9. These results show the ef-fects of the interfaces on the prediction of HF frac-ture statistics and the ability of MGPs to deal withthis kind of challenge. When the fracture passesthrough the interfaces, the uncertainty in statisticprediction (error bars) becomes larger and does notsatisfy the criterion of statistic prediction δ thatwas imposed. In this case, the MGPs refines thestochastic space adaptively to achieve the criterionthat we impose, but still keeping the computationaleffort afordable.Figure 9: Final pairwise input space decompositioncorresponding to the surrogate 2712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20154.3. Application in Uncertainty QuantificationA surrogate can be used for many different ap-plications, such as optimization, uncertainty quan-tification, risk analysis, and global sensibility anal-ysis. In the next subsection, we use surrogates con-structed using MGPs to uncertainty quantificationand risk analysis (George Shu Heng Pau, 2013).In this section, we present the impact of uncer-tainties in HF evolution. The surrogates 2 obtainedwith δ = 10−4 is interrogated with 4000 samples tocalculate the output statistics. In (Figs.10 - 11), wepresent the mean + uncertainty and the PDF of frac-ture aperture at the well and at the tips of fracture.Figure 10: Mean + Uncertainty of fracture aperture Ωat the well in tips right and leftFigure 11: probability distribution function (PDF) ofthe aperture Ω at the tips of the fracture at differenttimes5. CONCLUSIONSIn this work, we show the efficiency of MGPs to ap-proximate(surrogate) the nonlinear and nonsmoothbehaviors of hydraulic fracture dynamics. The sur-rogate is used to compute the statistics of the re-sponse in the context of an Uncertainty Quantifi-cation analysis at a low computational cost whencompared with the standard technique the MonteCarlo method. In the next future, we intend to em-ploy MGPs as surrogates in the context of optimiza-tion of an HF process.6. REFERENCESAdachi, J. I. and Detournay, E. (2008). “Plane strainpropagation of a hydraulic fracture in a permeablerock.” Engineering Fracture Mechanics, 4666–4694.Bilionis, I. and Zabaras, N. (2008). “Multi-output lo-cal gaussian process regression: Applications to un-certainty quantificaction.” Journal of CompuationalPhysics.George Shu Heng Pau, Y. Z. (2013). “Reduced ordermodels for many-query subsurface flow applications.”Computational Geosciences, DOI 10.1007/s10596-013-9349-z.Ghanem, R. and Spanos, P. (1991). “Stochastic fi-nite element method: A spectral approach.” Springer-Verlag.Mingjie Chen, Y. S. e. a. “Surrogate-based optimiza-tion of hydraulic fracturing in pre-existing fracturenetworks.” Computer and Geoscience, 58, 69–79.Peirce, A. (2014). “Modeling multi-scale processesin hydraulic fracture propagation using the implicitlevel set algorithm.” Department of Mathematics,University of British Columbia, Vancouver, BritishColumbia, V6T 1Z2, Canada.Peirce, A. and Detourney, E. (2008). “An implicit levelset method for modeling hydraulically driven frac-tures.” Computer Methods in Applied Mechanics andEngineering 197, 2858-2885.Stefano Conti, A. O. (19 March 2010). “Bayesian em-ulation of complex multi-output and dynamic com-puter models.” Journal of Statistical Planning and In-ference, 140(3), 640–651.Xiaoliang Wan, G. E. K. (2005). “An adaptive multi-element generalized polynomial chaos method forstochastic differential equations.” Journal of Compu-ational Physics.Zio, S. and Rochinha, F. A. (2012). “A stochastic collo-cation approach for uncertainty quantification in hy-draulic fracture numerical simulation.” InternationalJournal for Uncertainty Quantification, 145–160.8


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