Fluid leakoff determines hydraulic fracture dimensions: Approximate solution for non-Newtonian fracturing fluid

Mikhailov, D. N.; Economides, Michael J.; Nikolaevskiy, Victor N.

doi:10.1016/j.ijengsci.2011.03.021

Cited by 26 publications

(11 citation statements)

References 6 publications

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“…Function q l ¼ q l ðt; xÞ, in the right-hand side of the continuity equation (1), is the volumetric rate of fluid loss to the rock formation in the direction perpendicular to the crack surfaces per unit length of the fracture. Usually it is assumed to be given (local formulation) (Mikhailov, Economides, & Nikolaevskiy, 2011;Nordgren, 1972). More accurate analysis involves a nonlocal formulation where the mass transfer in the entire external domain should be taken into account (Kovalyshen, 2010).…”

Section: Governing Equations For 1-d Model Of Hydraulic Fracturementioning

confidence: 99%

Hydraulic fracture revisited: Particle velocity based simulation

Wróbel

Mishuris

2015

International Journal of Engineering Science

157

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Wrobel, M., Mishuris, G. (2015). Hydraulic fracture revisited: Particle velocity based simulation. International Journal of Engineering Science, 94, 23-58.We develop a new effective mathematical formulation and resulting universal computational algorithm capable of tackling various HF models in the framework of a unified approach. The scheme is not limited to any particular elasticity operator or crack propagation regime. Its basic assumptions are: (i) proper choice of independent and dependent variables (with the direct utilization of a new one ? the reduced particle velocity), (ii) tracing the fracture front by use of the Stefan condition (speed equation), which can be integrated in closed form and provides an explicit relation between the crack propagation speed and the coefficients in the asymptotic expansion of the crack opening, (iii) proper regularization techniques, (iv) improved temporal approximation, (v) modular algorithm architecture. The application of the new dependent variable, the reduced particle velocity, instead of the usual fluid flow rate, facilitates the computation of the crack propagation speed from the local relation based on the speed equation. This way, we avoid numerical evaluation of the undetermined limit of the product of fracture aperture and pressure gradient at the crack tip (or alternatively the limit resulting from ratio of the fluid flow rate and the crack opening), which always poses a considerable computational challenge. As a result, the position of the crack front is accurately determined from an explicit formula derived from the speed equation. This approach leads to a robust numerical scheme. Its performance is demonstrated using classical examples of 1D models for hydraulic fracturing: PKN and KGD models under various fracture propagation regimes. Solution accuracy is verified against dedicated analytical benchmarks and other solutions available in the literature. The scheme can be directly extended to more general 2D and 3D cases.publishersversionPeer reviewe

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Section: Governing Equations For 1-d Model Of Hydraulic Fracturementioning

confidence: 99%

Hydraulic fracture revisited: Particle velocity based simulation

Wróbel

Mishuris

2015

International Journal of Engineering Science

157

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“…Yet fluids used in hydraulic fracturing have a typically complex rheology, as this allows achieving objectives which are contradictory for Newtonian fluids, that is, (Barbati et al, 2016): (i) low-friction pressure-drop along the wellbore; (ii) suspend proppant in both dynamic and static conditions; (iii) exhibit low leak-off into the formation; (iv) flow back easily to the surface without interfering with gas or oil flow; and (v) adapt to variable temperatures and chemical environments in subsurface domains. The extension of existing conceptualizations and models of the different phases of fracking technology to non-Newtonian rheology is ongoing in the literature: Garagash (2006) and Mikhailov et al (2011) derived solutions for fracture growth driven by a power-law fluid; Lakhtychkin et al (2012) modeled transport of two proppant-laden immiscible non-Newtonian fluids through an expanding fracture; scaling laws for hydraulic fractures driven by a power-law fluid in homogeneous anisotropic rocks were derived by Dontsov (2019); further relevant references are cited in Section 5.2 of Osiptov (2017). In particular, as the rheology of fracturing fluids has been approximately described as power-law in many applications (Detournay, 2016;Montgomery & Smith, 2019), it seems timely to investigate the influence of such a constitutive relationship on the backflow phenomenon.…”

Section: Introductionmentioning

confidence: 99%

Non‐Newtonian Backflow in an Elastic Fracture

Chiapponi

Ciriello

Longo

et al. 2019

Water Resources Research

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Backflow phenomenon, as a consequence of hydraulic fracturing, is of considerable technical and environmental interest. Here, backflow of a non-Newtonian fluid from a disk-shaped elastic fracture is studied theoretically and experimentally. The fracture is of constant aperture h and the outlet section at constant pressure p e . We consider a shear-thinning power-law fluid with flow behavior index n. Fracture walls are taken to react with a force proportional to h , with a positive elasticity exponent; for = 1 linear elasticity holds. Constant overload f 0 , acting on the fracture, is also embedded in the model. A transient closed-form solution is derived for the (i) fracture aperture, (ii) pressure field, and (iii) outflow rate. The particular case of a Newtonian fluid (n = 1) is explicitly provided. For p e = 0 and f 0 = 0, the residual aperture and outflow rate scale asymptotically with time t as t −n/(n+ +1) and t −(2n+ +1)/(n+ +1) , respectively, thus generalizing literature results for n = 1 and/or = 1. For nonzero exit pressure and/or overload, the fracture aperture tends asymptotically to a constant value depending on , n, p e , f 0 , and other geometrical and physical parameters. The results are provided in dimensionless and dimensional form, including the time to achieve a given percentage of fluid recovery. In addition, an example application (with values of parameters derived from field scale applications) is included to further characterize the influence of fluid rheology. Experimental tests are conducted with Newtonian and shear-thinning fluids and different combinations of parameters to validate the model. Experimental results match well the theoretical predictions, mostly with a slight overestimation.

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“…For any fixed * ξ , the analytical solution of the initial value (Cauchy) problem (16), (17), (19), (20) is found in rapidly converging series (see Appendix). When having Y and V, the BC (18) defines the self-similar influx A, corresponding to the accepted * ξ .…”

Section: Formulation In Self-similar Variables Analytical Solution Fmentioning

confidence: 99%

On comparison of thinning fluids used for hydraulic fracturing

Linkov

2014

International Journal of Engineering Science

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Fluid leakoff determines hydraulic fracture dimensions: Approximate solution for non-Newtonian fracturing fluid

Cited by 26 publications

References 6 publications

Hydraulic fracture revisited: Particle velocity based simulation

Hydraulic fracture revisited: Particle velocity based simulation

Non‐Newtonian Backflow in an Elastic Fracture

On comparison of thinning fluids used for hydraulic fracturing

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