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Fracture Height Growth Prediction Using Fluid Velocity Based Apparent Fracture Toughness Model

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Abstract

This paper outlines the development of a fracture height growth prediction model that uses the modified rock mechanical properties obtained from a combination of fluid flow and rock material property behavior and applies them during the calibration process to match actual treatment data resulting fracture height growth evolution and profile. The traditional fracture growth models are closely linked with the fracture toughness parameter which is not only difficult to estimate but is also considered dynamic in nature as it may assume various geometry-dependent values rather than a fixed input value. To account for this limitation, this study uses the fluid tip velocity to estimate apparent fracture toughness during the initial model setup process and continuously scales the model during the fracture growth prediction phase. The influence of controllable parameters such as injection rate and fluid viscosity on vertical growth of fractures was studied and applied to field examples. Analysis from the modeling work on two of the case histories presented, indicated a rapid fracture growth in the first few minutes of the treatment that was proportional to the rate of net pressure gain. The influence of the fracture tip on the fracture pressures in the tip region is visible only in the early part of the treatment and are relatively short-lived especially once the contribution from viscous flow pressures dominates the pressure behavior. The findings are presented in detail and are helpful from a treatment planning and execution viewpoint as they provide a detailed insight into fracture propagation. Relevant theory, equations and workflow adopted during the study are presented for ready adaptation in the field.

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Fig. 1

(adapted from Anderson 2005)

Fig. 2

(adapted from Gulliot and Dunand 1985)

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Abbreviations

a :

Radius of crack, crack half-length, fracture half-height, M0L1t0, ft (m)

a v :

Viscosity calibrating parameter, dimensionless

A :

Amplitude in Eq. (18), strength of singularity, M1L−0.5t−2, psi-in.0.5 (Pa.m0.5)

b :

Exponent for the proportionality of width to distance from the fracture tip, unitless

c F :

Fracture compliance, M−1L2t2, ft/psi (m/Pa)

C L :

Leakoff coefficient, M0L1t−0.5, ft/min0.5 [m/s0.5]

c 2(n):

Constant, function of power law index, Eq. (26), dimensionless

d :

Characteristic length, M0L1t0, ft (m)

dΔp/dx:

Flowing pressure gradient along length of the fracture. Equation (19), M1L−2t−2, psi/ft (Pa/m)

D h :

Hydraulic diameter, M0L1t0, ft (m)

e :

Function of power law index, e = n + 2, dimensionless

E :

Young’s modulus, M1L−1t−2, psi (Pa)

E’ :

Plane strain modulus, M1L−1t−2, psi (Pa)

F b :

Ratio of maximum to average width of the fracture, dimensionless

g :

Acceleration due to gravity, M0L1t−2, ft/s2 (m/s2)

g(t):

Normal stresses at t along crack surface in Eq. (10), M1L−1t−2, psi (Pa)

G :

Shear modulus, M1L−1t−2, psi (Pa)

h :

Fracture height, M1L1t0, ft (m)

h cp :

Elevation from the bottom tip of the fracture to center of perforations, M1L1t0, ft (m)

h f :

Fracture height, M1L1t0, ft (m)

h i :

Elevation of ith layer top from bottom tip of the fracture, M1L1t0, ft (m)

h L :

Fracturing fluid leakoff, height, M1L1t0, ft (m)

h s :

Fracture penetration in surrounding layers, M1L1t0, ft (m)

i :

Imaginary portion of equation, unitless

j :

Layer count, number

k :

Layer count, number

K I :

Stress intensity factor, M1L−0.5t−2, psi-in.0.5 (Pa.m0.5)

K I top,bot :

Stress intensity factor for top and bottom layers, M1L−0.5t−2, psi-in.0.5 (Pa.m0.5)

K Ic :

Critical stress intensity factor or fracture toughness, M1L−0.5t−2, psi-in.0.5 (Pa.m0.5)

K :

Power law fluid consistency index, M1L−1tn−2, lbf-secn/ft2 (Pa.sn)

l :

Fracture half-height, M0L1t0, ft (m)

L :

Half of payzone thickness, crack half-length, M0L1t0, ft (m)

L h :

Characteristic length, M0L1t0, ft (m)

m :

Variable – function of power law index shown in Eq. (20), unitless

n :

Power law fluid flow behavior index, dimensionless

p :

Pressure in the fracture, M1L−1t−2, psi (Pa)

p cp :

Pressure at the center of the perforations, M1L−1t−2, psi (Pa)

p F :

Viscous flow pressure, M1L−1t−2, psi (Pa)

p o :

Fracture tip pressure, M1L−1t−2, psi (Pa)

p T :

Influence of facture tip on fracture pressure, dimensionless

p(x):

Pressure changes along the length of the crack, M1L−1t−2, psi (Pa)

p(t):

Crack face traction (Fig. 1) and Eq. (12), M1L−1t−2, psi (Pa)

p(y):

Pressure in the crack in vertical direction, M1L−1t−2, psi (Pa)

P f :

Fracture pressure, M1L−1t−2, psi (Pa)

q :

Injection rate, M0L3t−1, bbl/min (m3/s)

q w :

Injection rate, M0L3t−1, bbl/min (m3/s)

R :

Constant described in Eq. (20), unitless

S:

Stiffness, M1L−2t−2, psi/ft (Pa/m)

t :

Distance along the crack, M0L1t0, in. (m)

t e :

Distance along the crack, M0L1t0, in. (m)

t p :

Injection time or pump time, M0L0t1, min (s)

t* :

Reduced time function, M0L0t1, min (s)

u 1 , u 2 :

Displacements in respective directions, M0L1t0, in. (m)

v, v w :

Velocity in y direction, M0L1t−1, ft/sec (m/s)

V s :

Volumetric spurt loss, M0L1t0, gal/ft2 [m3/m2]

w, w w :

Width of fracture, M0L1t0, in. (mm)

\(\overline{w }\) :

Average width of the fracture, M0L1t0, in. (mm)

w(x):

Width along the crack in x direction, M0L1t0, in. (mm)

w(y):

Width in vertical direction y, M0L1t0, in. (mm)

x :

Any value in displacement along horizontal x-axis, M0L1t0, ft (m)

x :

Distance from the tip at the axis of the crack Eqs. (16) to (18), and (23), M0L1t0, ft (m)

x f :

Fracture half-length, M0L1t0, ft (m)

x* :

Difference between arbitrary distance along x-axis and crack half-length, M0L1t0, ft (m)

y :

Vertical displacement, M0L1t0, in. (m)

z :

Any value in displacement along the axis of the crack, M0L1t0, ft (m)

\(\overline{z }\) :

Conjugate of complex variable z presented in Eq. (8).

Z :

Complex stress function Eq. (2), M1L−1t−2, psi (Pa)

\(\overline{Z },\overline{\overline{Z }}\) :

Integrals with respect to arbitrary variable z.

α, α(n):

Exponent, function of n, dimensionless

β :

Calibration constant/coefficient in Eq. (13), unitless

β p :

Net pressure ratio, dimensionless

Δp(x):

Pressure at any point x in the fracture measured from the tip, M1L−1t−2, psi (Pa)

Δp f :

Fracture net pressure, M1L−1t−2, psi (Pa)

ΔP :

Fracture net pressure, M1L−1t−2, psi (Pa)

Φ(n):

Dimensionless width over fracture cross-section, dimensionless

Φ(z):

Potential function resulting in stress, M1L−1t−2, psi (Pa)

Φ,ω :

Analytical functions, various

Φ(x,y):

Airy stress function in x and y plane, M1L−1t−2, psi (Pa)

\(\dot{\gamma }\) :

Shear rate, M0L0t−1 (1/s)

\(\left(z\right)\) :

Analytical function described in Eq. (8), M1L−1t−2, psi (Pa)

η :

Fluid efficiency, fraction or percent

μ :

Fluid viscosity, M1L−1t−1, cP (Pa.s)

ν :

Poisson’s ratio, unitless

θ :

Angle, degrees

ρ,ρ f :

Fluid density, M1L−3t0, lbm/gal (kg/m3)

σ :

Stress, M1L−1t−2, psi (Pa)

σ min :

Minimum in-situ stress in horizontal direction, M1L−1t−2, psi (Pa)

σ xx , σ yy :

Normal stresses in x and y directions respectively, M1L−1t−2, psi (Pa)

iσ xy :

Imaginary stress component Eq. (8), M1L−1t−2, psi (Pa)

σ zz :

Normal stresses in z direction, M1L−1t−2, psi (Pa)

σ θθ :

Stress in radial coordinates at angle θ, M1L−1t−2, psi (Pa)

σ j , σ k :

Stress in jth and kth layers, M1L−1t−2, psi (Pa)

σ o :

Far-field compressive stress acting on the fracture face, M1L−1t−2, psi (Pa)

σ a :

Stress in payzone, M1L−1t−2, psi (Pa)

σ b :

Stress in barriers or stress in upper barrier alone, M1L−1t−2, psi (Pa)

σ c :

Stress in lower barrier, M1L−1t−2, psi (Pa)

σ(Z):

Stresses distribution in vertical y-direction, M1L−1t−2, psi (Pa)

σ 3 :

Minimum in-situ stress in horizontal direction, M1L−1t−2, psi (Pa)

τ xx ,τ yy :

Shear stresses in x–y plane, M1L−1t−2, psi (Pa)

ψ(z):

Potential function resulting in stress, M1L−1t−2, psi (Pa)

\(\xi \) :

Distance from the fracture tip to the wellbore in near vicinity of tip, M0L1t0, in. (m)

ξ’ :

Dimensionless ratio of length, dimensionless

∇:

Divergence, mathematical operator

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Acknowledgements

The authors of the paper would like to thank the management of the University of North Dakota and ConocoPhillips Company for granting permission to publish this work.

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  1. University of North Dakota and ConocoPhillips Company, Houston, USA

    Vibhas J. Pandey

  2. University of North Dakota, Grand Forks, USA

    Vamegh Rasouli

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Pandey, V.J., Rasouli, V. Fracture Height Growth Prediction Using Fluid Velocity Based Apparent Fracture Toughness Model. Rock Mech Rock Eng 54, 4059–4078 (2021). https://doi.org/10.1007/s00603-021-02489-w

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