volume_integral module

This file is part of PyFrac.

Created by Haseeb Zia on Fri Oct 14 18:27:39 2016. Copyright (c) “ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE, Switzerland, Geo-Energy Laboratory”, 2016-2020. All rights reserved. See the LICENSE.TXT file for more details.

volume_integral.Area(dist, *param)

Gives Area under the tip depending on the regime identifier ; used in case of 0 or 90 degree angle; can be used for 1d case

volume_integral.FindBracket_w(dist, Kprime, Eprime, muPrime, Cprime, Vel, regime)

This function finds the bracket to be used by the Universal tip asymptote root finder.

volume_integral.FindBracket_w_HB(a, b, *args)

This function finds the bracket to be used by the Universal tip asymptote root finder.

volume_integral.Integral_over_cell(EltTip, alpha, l, mesh, function, frac=None, mat_prop=None, fluid_prop=None, Vel=None, Kprime=None, Eprime=None, Cprime=None, stagnant=None, KIPrime=None, dt=None, arrival_t=None, projMethod=None)

Calculate integral of the function specified by the argument function over the cell.

Parameters

• EltTip (ndarray) – – the tip cells over which the integral is to be evaluated

• alpha (ndarray) – – the angle alpha of the perpendicular drawn on the front from the zero vertex.

• l (ndarray) – – the length of the perpendicular drawn on the front from the zero vertex.

• mesh (CartesianMesh) – – the mesh object.

• function (string) –

  – the string specifying the type of function that is to be integreated. Possible options are:

  • ’A’ gives the area (fill fraction)

  • ’K’ gives tip volume according to the square root asymptote

  • ’M’ gives tip volume according to the viscocity dominated asymptote

  • ’Lk’ is used to calculate the leak off given the distance of the front l (note, its not tip volume)

  • ’Mt’ gives tip volume according to the viscocity, Leak-off asymptote

  • ’U’ gives tip volume according to the Universal asymptote (Donstov and Pierce, 2017)

  • ’MK’ gives tip volume according to the M-K transition asymptote

  • MDR (Maximum drag reduction asymptote, see Lecampion & Zia 2019)

  • M_MDR (Maximum drag reduction asymptote in viscosity sotrage

    regime, see Lecampion & Zia 2019)

  • HBF or HBF_aprox (Herschel-Bulkley fluid, see Bessmertnykh and Dontsov 2019; the tip volume is evaluated with a fast aproximation)

  • HBF_num_quad (Herschel-Bulkley fluid, see Bessmertnykh and Dontsov 2019; the tip volume is evaluated with numerical quadrature of the

    approximate function, which makes it very slow)

  • PLF or PLF_aprox (power law fluid, see Dontsov and Kresse 2017; the tip volume is evaluated with a fast aproximation)

  • PLF_num_quad (power law fluid, see Dontsov and Kresse 2017; the tip volume is evaluated with numerical quadrature of the

    approximate function, which makes it very slow)

  • PLF_M (power law fluid in viscosity storage regime; see Desroche et al.)

• frac (Fracture) – – the fracture object.

• mat_prop (MaterialProperties) – – the material properties object.

• fluid_prop (FluidProperties) – – the fluid properties object

• Vel (ndarray) – – the velocity of the front in the given tip cells.

• Kprime (ndarray) – – if provided, the toughness will be taken from the given array instead of taking it from the mat_prop object

• Eprime(ndarray – – plain strain TI modulus for current iteration. if not given, the Eprime from the given material properties object will be used.

• Cprime (ndarray) – – the Carter’s leak off coefficient multiplied by 2.

• stagnant (ndarray) – – list of tip cells where the front is not moving.

• KIPrime (ndarray) – – the stress intensity factor of the cells where the fracture front is not moving.

• dt (float) – – the time step, only used to calculate leak off.

• arrival_t (ndarray) – – the time at which the front passes the given point.

Returns

integral (ndarray) – the integral of the specified function over the given tip cells.

volume_integral.MomentsTipAssympGeneral(dist, Kprime, Eprime, muPrime, Cbar, Vel, stagnant, KIPrime, regime)

Moments of the General tip asymptote to calculate the volume integral (see Donstov and Pierce, 2017)

volume_integral.MomentsTipAssymp_HBF_approx(s, *HB_args)

Approximate moments of the Herschel-Bulkley fluid. Delta is taken to be 1/6.

volume_integral.Pdistance(x, y, slope, intercpt)

distance of a point from a line

volume_integral.TipAsym_MK_W_deltaC_Res(w, *args)

Residual function for viscosity to toughness regime with transition, without leak off

volume_integral.TipAsym_MK_W_zrthOrder_Res(w, *args)

Residual function for viscosity to toughness regime with transition, without leak off

volume_integral.TipAsym_UniversalW_delt_Res(w, *args)

The residual function zero of which will give the General asymptote

volume_integral.TipAsym_UniversalW_zero_Res(w, *args)

Function to be minimized to find root for universal Tip assymptote (see Donstov and Pierce 2017)

volume_integral.TipAsym_res_Herschel_Bulkley_d_given(w, *args)

Residual function for Herschel-Bulkley fluid model (see Besmertnykh and Dontsov, JAM 2019)

volume_integral.TipAsym_viscStor_Res(w, *args)

Residual function for viscosity dominate regime, without leak off

volume_integral.VolumeTriangle(dist, *param)

Volume of the triangle defined by perpendicular distance (dist) and em (em=1/sin(alpha)cos(alpha), where alpha is the angle of the perpendicular). The regime variable identifies the propagation regime.

volume_integral.find_corresponding_ribbon_cell(tip_cells, alpha, zero_vertex, mesh)

zero_vertex is the node index in the mesh.Connectivity The four vertices of an element have the following order ______ ______ ______

| | |

C | D | E |

|______3______2______| | | | | | B | i | F | |______0______1______| | | | | | A | H | G | |______|______|______|

case alpha = 0 -> B F F B

alpha = pi/2 -> H D D H alpha = any other -> A G E C

volume_integral.leak_off_stagnant_tip(Elts, l, alpha, vrtx_arr_time, current_time, Cprime, time_step, mesh)

This function evaluates leak-off in the tip cells with stagnant front. Its samples the leak-off midway from the zero vertex of the cell to the front and multiply it with the area of the fracture in the cell (filling fraction times the area of the cell). todo: can be more precise

volume_integral.width_HBF(s, *HB_args)

This function is used to evaluate the zeroth moment of HBF tip solution with numerical quadrature.

volume_integral.width_dist_product_HBF(s, *HB_args)

This function is used to evaluate the first moment of HBF tip solution with numerical quadrature.