Hydraulic Fracture Initiation and Propagation from Wellbore with Oriented Perforation

Abstract

Considering the influence of casing, analytical solutions for stress distribution around a cased wellbore are derived, based on which a prediction model for hydraulic fracture initiation with the oriented perforation technique (OPT) is established. Taking well J2 of Z5 oilfield for an example, the predicted initiation pressure with the OPT of our model is about 4.2 MPa higher than the existing model, which neglects the influence of casing. In comparison with the results of laboratory fracturing experiments with OPT on a 400 × 400 × 400 mm³ rock sample for a cased well with the deviation of 45°, the fracture initiation pressure of our model has an error of 3.2 %, while the error of the existing model is 6.6 %; when the well azimuth angle is 0° and the perforation angle is 45°, the prediction error of the fracture initiation pressure of the existing model and our model are 3.4 and 7.7 %, respectively. The study verifies that our model is more applicable for hydraulic fracturing prediction of wells with OPT completion; while the existing model is more suitable for hydraulic fracturing with conventional perforation completion.

Appendix

The governing equations are cited from the 3D model proposed by Clifton and Abou-Sayed (1979), whose assumptions include: (1) the rock formation acts as isotropic and linear elastic solid; (2) the porosity and permeability of the reservoir are so low that the poroelastic effect is negligible; (3) the fluid flow between the porous parallel plates is non-compressible, non-Newtonian, and laminar; (4) the fluid velocity gradient in the direction of fracture width is neglected in comparison with those along the fracture height and length, due to the smallness of the width; and (5) the fracture propagation is governed by the cracking opening criteria from LEFM. To facilitate the application of similarity principle, the fracturing fluid here is considered to be a Newtonian fluid.

Clifton and Abou-Sayed’s governing equations for hydraulic fracture propagation consist of an equilibrium equation, a fluid continuity equation and a pressure gradient equation (1979).

1. The elastic equilibrium equation.

$$p(x,y) - \sigma_{zz}^{0} (x,y,0) = {\text{Ee}}\iint\limits_{A} {\left\{ {\frac{\partial }{\partial x}\left[ {\frac{1}{R}\frac{\partial w(x,y)}{\partial x}} \right] + \frac{\partial }{\partial y}\left[ {\frac{1}{R}\frac{\partial w(x,y)}{\partial y}} \right]} \right\}{\text{d}}x{\text{d}}y}$$

(63)

where ${\text{Ee}} = {G \mathord{\left/ {\vphantom {G {\left( {4\pi \left( {1 - v} \right)} \right)}}} \right. \kern-0pt} {\left( {4\pi \left( {1 - v} \right)} \right)}}$ is the equivalent elastic modulus; $R = [(x - x_{0} )^{2} + (y - y_{0} )^{2} ]^{1/2}$ is the distance between integral point $(x,y)$ of the integrand and the pressure point $(x_{0} ,y_{0} )$.

2. The fluid continuity equation.

$$\frac{{\partial q_{x} }}{\partial x} + \frac{{\partial q_{y} }}{\partial y} + \frac{\partial w}{\partial t} + \frac{{2K_{\text{L}} \frac{{p_{i} - p_{0} }}{{\sigma_{zz}^{0} (0,0) - p_{0} }}}}{{\sqrt {t - \tau (x,y} )}} - q_{1} = 0$$

(64)

where $p_{i}$ is the fluid pressure inside the fracture; $\sigma_{zz}^{0}$ is the normal pressure on fracture surface before hydraulic fracturing; $K_{\text{L}}$ is the total leak-off coefficient; $w$ is the fracture width; $q_{x}$ is the volume flow rate per unit length along x direction; $q_{y}$ is the volume flow rate per unit length along y direction; $\tau$ is the contact time between the fracture and fracturing fluid; $t$ is the fracture propagation time; $q_{1}$ is the injection rate per unit area of the fracture.

3. The pressure gradient equation.

$$\left\{ \begin{gathered} \frac{\partial p}{\partial x} + \eta \frac{{q_{x} }}{{w^{3} }} = 0 \hfill \\ \frac{\partial p}{\partial y} + \eta \frac{{q_{y} }}{{w^{3} }} = \rho F_{y} \hfill \\ \end{gathered} \right.$$

(65)

where $\eta$ is the viscosity coefficient of hydraulic fracturing fluid; $\rho$ is the density of fracturing fluid, the $\rho F_{y}$ is the force per unit volume.

4. Fracture propagation condition.

$$w_{a} (s) > w_{c} = K_{Ic} \left( {2a(s)/\pi } \right)^{1/2} /2\pi Ee$$

(66)

where $a$ is the width of vicinity area at the fracture tip; $K_{Ic}$ is the critical stress intensity factor for fracture propagation; and $w_{c}$ is the critical fracture width for fracture propagation. When $w_{a} (s) < w_{c}$, fracture propagation terminates.

5. Single-value conditions include the geometric condition, medium conditions, boundary conditions, and the initial conditions.

The geometric condition: the dimension of the hydraulic fracture is restricted by the rock sample geometry.

The medium conditions: parameters $\rho F_{y}$, $\eta$, $Ee$, $K_{Ic}$ and $K_{L}$ are classified as the medium conditions.

The boundary conditions: parameters $\sigma_{zz}^{0}$, $p_{i}$, $q_{1}$ and the flow rate $Q = 2\int_{ - h/2}^{h/2} {q_{x} } (0,y,t){\text{d}}y$ are classified as the boundary conditions.

The initial conditions: the hydraulic fracturing time is $t = T$, the effective fracture length achieved is $x_{\hbox{max} } = L$.

Then, to obtain the experimental parameters, the similarity principle is employed to derive the governing equations. First, some units of measurement are used as temporary variables, which are

$$l_{0} ,p_{0} ,\sigma_{0} ,{\text{Ee}}_{ 0} ,q_{0} ,q_{I} ,t_{0} ,Q_{0} ,K_{L0} ,\eta_{0} ,\rho F_{y0} ,K_{{{\text{IC}}0}}$$

(67)

Then, the identical equations can be obtained as follows:

$$\begin{gathered} p = \frac{p}{{p_{0} }}p_{0} = \overline{p} p_{0} ,\quad p_{f} = \frac{{p_{f} }}{{p_{0} }}p_{0} = \overline{p}_{f} p_{0} ,\quad \sigma_{zz}^{0} = \frac{{\sigma_{zz}^{0} }}{{\sigma_{0} }}\sigma_{0} = \overline{\sigma }_{zz}^{0} \sigma_{0} , \hfill \\ {\text{Ee}} = \frac{\text{Ee}}{{{\text{Ee}}_{0} }}{\text{Ee}}_{0} = \overline{\text{Ee}} {\text{Ee}}_{0} ,\quad x = \frac{x}{{l_{0} }}l_{0} = \overline{x} l_{0} ,\quad y = \frac{y}{{l_{0} }}l_{0} = \overline{y} l_{0} , \hfill \\ w = \frac{w}{{l_{0} }}l_{0} = \overline{w} l_{0} ,\quad w_{c} = \frac{{w_{c} }}{{l_{0} }}l_{0} = \overline{w}_{c} l_{0} ,\quad \rho F_{y} = \frac{{\rho F_{y} }}{{\rho F_{y0} }}\rho F_{y0} = \overline{{\rho F_{y} }} \rho F_{y0} , \hfill \\ R = \frac{R}{{l_{0} }}l_{0} = \overline{R} l_{0} ,\quad h = \frac{h}{{l_{0} }}l_{0} = \overline{h} l_{0} ,\quad a = \frac{a}{{l_{0} }}l_{0} = \overline{a} l_{0} ,\quad q_{\text{I}} = \frac{{q_{\text{I}} }}{{q_{0} }}q_{0} = \overline{{q_{I} }} q_{0} , \hfill \\ q_{x} = \frac{{q_{x} }}{{q_{0} }}q_{0} = \overline{q}_{x} q_{0} ,q_{y} = \frac{{q_{y} }}{{q_{0} }}q_{0} = \overline{q}_{y} q_{0} ,Q = \frac{Q}{{Q_{0} }}Q_{0} = \overline{Q} Q_{0} , \hfill \\ t = \frac{t}{{t_{0} }}t_{0} = \overline{t} t_{0} ,\quad \tau = \frac{\tau }{{\tau_{0} }}\tau_{0} = \overline{\tau } \tau_{0} ,\quad K_{\text{L}} = \frac{{K_{\text{L}} }}{{K_{{{\text{L}}0}} }}K_{{{\text{L}}0}} = \overline{K}_{\text{L}} K_{{{\text{L}}0}} , \hfill \\ K_{IC} = \frac{{K_{\text{IC}} }}{{K_{{{\text{IC}}0}} }}K_{{{\text{IC}}0}} = \overline{K}_{\text{IC}} K_{{{\text{IC}}0}} ,\quad \eta = \frac{\eta }{{\eta_{0} }}\eta_{0} = \overline{\eta } \eta_{0} , \hfill \\ \end{gathered}$$

(68)

According to the Gaussian rule of absolute measurement units, the expressions of the governing equations should not be affected by the selection of different measurement units. Substituting Eq. (68) into the governing equations, nine constraints are obtained:

$$\left\{ \begin{gathered} \frac{{p_{0} }}{{{\text{Ee}}_{0} }} = 1;\frac{{\sigma_{0} }}{{{\text{Ee}}_{0} }} = 1;\frac{{l_{0}^{2} }}{{t_{0} q_{0} }} = 1;\frac{{t_{0} K_{L0} }}{{q_{0} \sqrt {t_{0} } }} = 1;\frac{{q_{10} l_{0} }}{{q_{0} }} = 1 \hfill \\ \frac{{q_{0} \eta_{0} }}{{p_{0} l_{0}^{2} }} = 1;\frac{{l_{0} \rho F_{y0} }}{{p_{0} }} = 1;\frac{{K_{{{\text{IC}}0}} }}{{{\text{Ee}}_{0} \sqrt {L_{0} } }} = 1;\frac{{Q_{0} }}{{q_{0} l_{0} }} = 1 \hfill \\ \end{gathered} \right.$$

(69)

Selecting $l_{0} ,{\text{Ee}}_{0} ,Q_{0}$as the free measurement units, $l_{0} = L,{\text{Ee}}_{0} = {\text{Ee}},Q_{0} = Q$, and substituting these expressions into Eq. (69), nine measurement units are achieved:

$$\left\{ \begin{gathered} p_{0}^{(k)} = {\text{Ee}},\sigma_{0}^{(k)} = {\text{Ee}},q_{0}^{(k)} = \frac{Q}{L},t_{0}^{(k)} = \frac{{L^{3} }}{Q},K_{L0}^{(k)} = \sqrt {\frac{Q}{L}} \hfill \\ q_{I0}^{(k)} = \frac{Q}{{L^{2} }},\eta_{0}^{(k)} = \frac{{{\text{Ee}}L^{3} }}{Q},\rho F_{y0}^{(k)} = \frac{\text{Ee}}{{L^{{}} }},K_{{{\text{IC}}0}}^{(k)} = {\text{Ee}}\sqrt L \hfill \\ \end{gathered} \right.$$

(70)

Substituting Eq. (70) into the governing equations, and non-dimensionalizing the governing equations:

$$\left\{ \begin{gathered} \bar{p}^{(k)} - \bar{\sigma }_{zz}^{0(k)} = \iint\limits_{{\bar{A}}} {\left\{ {\frac{\partial }{{\partial \bar{x}}}\left[ {\frac{1}{{\bar{R}}}\frac{{\partial \bar{w}}}{{\partial \bar{x}}}} \right] + \frac{\partial }{{\partial \bar{y}}}\left[ {\frac{1}{{\bar{R}}}\frac{{\partial \bar{w}}}{{\partial \bar{y}}}} \right]} \right\}{\text{d}}\bar{x}{\text{d}}\bar{y}} \hfill \\ \frac{{\partial \bar{q}_{x}^{(k)} }}{{\partial \bar{x}}} + \frac{{\partial \bar{q}_{y}^{(k)} }}{{\partial \bar{y}}} + \frac{{\partial \bar{w}}}{{\bar{\partial }t^{( - k)} }} + \frac{{2\bar{K}_{L}^{(k)} \frac{{\bar{p}_{i}^{(k)} - \bar{p}_{0}^{(k)} }}{{\bar{\sigma }_{zz}^{0(k)} - \bar{p}_{f}^{(k)} }}}}{{\sqrt {\bar{t}^{(k)} - \bar{\tau }^{(k)} } }} - \bar{q}_{1}^{(k)} = 0 \hfill \\ \frac{{\partial \bar{p}^{(k)} }}{{\partial \bar{x}}} + \bar{\eta }^{(k)} \frac{{\bar{q}_{x}^{(k)} }}{{\bar{w}^{3} }} = 0 \hfill \\ \frac{{\partial \bar{p}^{(k)} }}{{\partial \bar{y}}} + \bar{\eta }^{(k)} \frac{{\bar{q}_{y}^{(k)} }}{{\bar{w}^{3} }} = \overline{{\rho F_{y} }}^{(k)} \hfill \\ \bar{w}_{c} = \frac{{\bar{K}_{1C}^{(k)} }}{2\pi }\left[ {\frac{{2\bar{a}}}{\pi }} \right]^{1/2} \hfill \\ 1 = 2\int_{{ - \bar{h}/2}}^{{\bar{h}/2}} {\bar{q}_{x}^{(k)} (0,\bar{y},\bar{t}^{(k)} )} {\text{d}}\bar{y} \hfill \\ \bar{t}^{(k)} = \bar{T}^{(k)} \hfill \\ \bar{x}_{\hbox{max} } = 1 \hfill \\ \end{gathered} \right.$$

(71)

where the dimensionless variables are

$$\left\{ \begin{gathered} \bar{L} = 1,\bar{E}_{e} = 1,\bar{Q} = 1,\bar{x} = \frac{x}{L},\bar{y} = \frac{y}{L} \hfill \\ \bar{w} = \frac{w}{L},\bar{w}_{c} = \frac{{w_{c} }}{L},\bar{R} = \frac{R}{L},\bar{h} = \frac{h}{L},\bar{a} = \frac{a}{L},\bar{p}^{(k)} = \frac{p}{\text{Ee}} \hfill \\ \bar{p}_{f}^{(k)} = \frac{{p_{f} }}{\text{Ee}},\bar{\sigma }_{zz}^{0(k)} = \frac{{\sigma_{zz}^{0} }}{\text{Ee}},\overline{\rho Fy}^{(k)} = \frac{\rho FyL}{\text{Ee}},\overline{q}_{1}^{(k)} = \frac{{q_{1} L^{2} }}{Q} \hfill \\ \overline{q}_{x}^{(k)} = \frac{{q_{x} L^{{}} }}{Q},\overline{q}_{y}^{(k)} = \frac{{q_{y} L^{{}} }}{Q},\overline{\eta }_{{}}^{(k)} = \frac{\eta Q}{{{\text{Ee}}L^{3} }},\overline{t}_{{}}^{(k)} = \overline{\tau }_{{}}^{(k)} = \frac{\tau Q}{{L^{3} }} \hfill \\ \overline{K}_{L}^{(k)} = \frac{{K_{L} }}{{K_{L0}^{(k)} }} = K_{L} \sqrt {\frac{{L^{{}} }}{Q}} ,\overline{K}_{\text{IC}}^{(k)} = \frac{{K_{IC} }}{{K_{{{\text{IC}}0}}^{(k)} }} = \frac{{K_{IC} }}{{{\text{Ee}}\sqrt L }} \hfill \\ \end{gathered} \right.$$

(72)

Substituting the single-value condition (67) into dimensionless Eq. (72), we get the dimensionless similarity principle:

$$\left\{ \begin{gathered} \frac{p}{\text{Ee}} = {\text{idem}},\frac{{p_{f} }}{\text{Ee}} = {\text{idem}},\frac{{\sigma_{zz}^{0} }}{\text{Ee}} = {\text{idem}},\frac{{L\rho F_{y} }}{\text{Ee}} = {\text{idem}}, \hfill \\ \frac{TQ}{{L^{3} }} = {\text{idem}},K_{L} \sqrt{\frac{L}{Q}} = {\text{idem}},\frac{{K_{IC} }}{{{\text{Ee}}\sqrt L }} = {\text{idem}},\frac{\eta Q}{{{\text{Ee}}L^{3} }} = {\text{idem}} \hfill \\ \end{gathered} \right.$$

(73)

where ${\text{idem}}$ is the criterion numeral. After simplifying (73), we obtain the dimensionless similarity index:

$$\frac{{C_{p} }}{{C_{\text{Ee}} }} = \frac{{C_{{\sigma_{zz}^{0} }} }}{{C_{\text{Ee}} }} = \frac{{C_{T} C_{Q} }}{{C_{{^{{_{L} }} }}^{3} }} = C_{{K_{L} }} \sqrt {\frac{{C_{L} }}{{C_{Q} }}} = \frac{{C_{{K_{\text{IC}} }}^{2} }}{{C_{\text{Ee}}^{2} C_{L} }} = \frac{{C_{\eta } C_{Q} }}{{C_{p} C_{L}^{3} }} = 1$$

(74)

where the similarity coefficient is

$$C_{V} = V_{\text{model}} /V_{\text{field}}$$

(75)

where subscript $V = L,{\text{Ee}},Q,T,K_{L} ,\eta ,p,\sigma_{ZZ}^{0} ,K_{\text{IC}}$, and $C_{V}$ is the ratio of the model quantities to the field quantities.