Geomechanical optimization of hydraulic fracturing in unconventional reservoirs: a semi-analytical approach

Abstract

Unconventional drilling and completion architecture includes drilling multilateral horizontal wells in the direction of minimum horizontal stress and simultaneous multistage fracturing treatments perpendicular to the wellbore. This drilling and stimulation strategy is utilized in order to raise the connectivity of the reservoir to the wellbore, thereby remedying the low permeability problem, increasing reserve per well, enhancing well productivity, and improving project economics in this type of reservoir. However, in order to have the highest production with the least cost, an optimization technique should be used for the fracturing treatment. According to the fact that aperture, propagation direction, and propagation potential of hydraulic fractures are of paramount importance in optimization of the fracking treatment, in this research paper, these three major factors are studied in detail, the control variables on these three factors are examined, and the effect of each factor is quantified by proposing a complete set of equations. Using the proposed set of equations, one can make a good estimate about the fracture aperture (directly controlling the fracture conductivity), the stress shadow size (directly controlling the fracture path), and the change of stress intensity factor (directly controlling the fracture propagation potential). A geomechanical optimization procedure is then presented for toughness-dominated and viscosity-dominated regimes based on the proposed equations that can be used for estimation of different optimal fracturing patterns. The most efficient fracturing pattern can be determined afterward via considering the cumulative production using a reservoir simulator e.g. ECLIPSE, Schlumberger. This procedure is likely to offer an optimal simultaneous multistage hydraulic fracture treatment without deviation or collapse, with no fracture trapping, with the highest possible propagation potential in the hydrocarbon producing shale layer, and a predicted aperture for proppant type/size decision and conductivity of the fractures.

Appendices

Appendix A: Governing equations of a porothermoelastic problem

In order to solve a porothermoelastic problem, the governing equations as constitutive laws, balance laws, and the kinematic law that are presented as follows:

Constitutive equations are presented in Eqs. (A1–A4) which are respectively Hooke, pressure, Darcy, and Fourier:

$$\begin{aligned}&\sigma _{ij} =2G\varepsilon _{ij} +\left( {\frac{2G\nu }{1-2\nu } \varepsilon -\alpha p-K\beta _s T} \right) \delta _{ij} \end{aligned}$$

(A1)

$$\begin{aligned}&p= M(\zeta -\alpha \varepsilon \epsilon +\beta T) \end{aligned}$$

(A2)

$$\begin{aligned}&q_i^f =-\kappa _{ij}^f ({p_{,j} -f_j}) \end{aligned}$$

(A3)

$$\begin{aligned}&q_i^T =-\kappa _{ij}^T T_{,j} \end{aligned}$$

(A4)

The balance laws are shown in Eqs. (A5–A7) that are respectively momentum, mass, and energy:

$$\begin{aligned}&\sigma _{ij,j} +F_i =0 \end{aligned}$$

(A5)

$$\begin{aligned}&\frac{\partial \zeta }{\partial t}+\nabla \cdot q^{f}=0 \end{aligned}$$

(A6)

$$\begin{aligned}&\rho C_v \frac{\partial T}{\partial t}+\nabla \cdot q^{T}=0 \end{aligned}$$

(A7)

The kinematic relation is also written as:

$$\begin{aligned} \varepsilon _{ij} =\frac{1}{2}({u_{i,j} +u_{j,i}}) \end{aligned}$$

(A8)

in which $\sigma ,\varepsilon = \hbox {total stress and strains}, G,\nu = \hbox {rock moduli}, p= \hbox {pore pressure}, T= \hbox {temperature}, \delta _{ij} = \hbox {kronecker delta}, \alpha = \hbox {Biot pore pressure} \hbox {coefficient}, K= \hbox {dry bulk modulus}, \beta _s =$ expansion coefficient of the solid skeleton, $F_i = \hbox {solid body force}, u= \hbox {displacement}, \zeta = \hbox {variation of fluid content}, q_i^f = \hbox {fluid flux}, q_i^T = \hbox {heat flux}, M=2G({\nu _u -\nu })/[{\alpha ^{2}({1-2\nu _u})({1-2\nu })}]$ is Biot modulus, $f_i = \hbox {fluid} \hbox {body force}, \kappa =$ mobility of the fluid (rock permeability divided by fluid viscosity), and $\beta =$ undrained thermal expansion coefficient and is determined as $\beta =3[({\alpha -\varphi })\beta _s +\varphi \beta _f]$ where $\beta _f =$ fluid expansion coefficient. Tensile stresses are considered as positive under this convention.

Coupling of Eqs. (A1–A8) creates Navier Type and diffusion equations which can be shown as follows:

$$\begin{aligned} Gu_{i,jj} +\frac{G}{1-2\nu }\varepsilon _{kk,i} -\alpha p_{,i} -K\beta _s T_{,i} +F_i= & {} 0 \nonumber \\ \end{aligned}$$

(A9)

$$\begin{aligned} \frac{1}{M}\frac{\partial p}{\partial t}-\frac{k_f }{\mu }p_{,ii} +\alpha \frac{\partial \varepsilon _{kk} }{\partial t}-\beta \frac{\partial T}{\partial t}+\kappa f_{i,i}= & {} 0\nonumber \\ \end{aligned}$$

(A10)

These two field equations are governing equations for a porothermoelastic medium by using some appropriate boundary and initial conditions, each field as stress, pore pressure, temperature, strain, and variation of fluid content can exactly be determined. In order to solve a porothermoelastic problem, an analytical approach is utilized in case of simple geometry, having constant properties, and simple boundary and initial conditions. For complex geometries, having variant properties, and challenging boundary conditions, however, numerical schemes are preferred to solve the problem.

Appendix B: Numerical determination of stress shadow size

In order to calculate the deviation of principal stresses around a pressurized hydraulic fracture, stress field on plane Q (see Fig. 2b) is studied. We have only three stress components on this plane as $\sigma _{xx} ,\sigma _{yy} ,\tau _{xy}$ (see Fig. 1). In this method, having the total stress field from numerical simulations (obtained from ABAQUS), the change in magnitude and direction of the principal stresses can be obtained from Eqs. (B1) and (B2), respectively.

$$\begin{aligned} S_{22,33}= & {} \frac{\sigma _{xx} +\sigma _{yy} }{2}\pm \sqrt{\left( {\frac{\sigma _{xx} -\sigma _{yy}}{2}}\right) ^{2}+\tau _{xy}^2} \end{aligned}$$

(B1)

$$\begin{aligned} \tan 2\theta _p= & {} \frac{2\tau _{xy} }{| {\sigma _{xx} -\sigma _{yy}}|} \end{aligned}$$

(B2)

in which $\sigma _{xx} ,\sigma _{yy} ,\tau _{xy}$ are induced stresses in the shadow region, $S_{22}$ and $S_{33}$ are changed principal stresses in this region, and $\theta _p$ is the direction of principal stresses. Deviation angle of principal stresses ($\theta _D$) can also be obtained based on the following criteria:

$$\begin{aligned} \theta _D =\left\{ {{\begin{array}{ll} {\theta _p }&{} {\sigma _{xx} >\sigma _{yy} } \\ {90-\theta _p }&{} {\sigma _{xx} <\sigma _{yy} } \\ {45}&{} {\sigma _{xx} =\sigma _{yy} } \\ \end{array} }} \right. \end{aligned}$$

(B3)

It is worth mentioning that maximum principal stress deviation, depending on the magnitude of internal pressure, can be near the crack tip or crack center and it vanishes away from the crack. Therefore, by defining a threshold angle ($\theta _T$) the area around the crack with stress deviations more than this threshold angle can be determined. Considering Eq. (B2), it is also evident that maximizing the stress contrast means lowering the deviation angle. Using this numerical method, there is no limitation on the fracture geometry and boundary condition. Thus, a wide range of problems can be solved satisfactorily using this approach.

Appendix C: Methodology of determining the SIF change in different fracturing patterns

The methodology of estimating the SIF ratio of any hydraulic fracturing pattern over that of a single hydraulic fracture in an infinite domain is explained herein. It is known that stress at a point in the vicinity of the crack tip in the plane where the fracture is located inside ($r=R, \theta =0$), is related to the SIF using the following equation:

$$\begin{aligned} \sigma _y \sqrt{2\pi R}=K_I \end{aligned}$$

(C1)

in which R is the distance from the crack tip, $K_I$ is linearly related to the internal pressure ($P_N$), directly related to a function of the geometry of the fracture, and a function incorporating the effect of boundaries on the fracture. In this way, having the ratio of stress for a fracturing scenario under study over that of the basic scenario 1, one can have the ratio of the SIF for the fracturing technique over that of the basic scenario 1 (see Fig. 3a). Therefore, we can write:

$$\begin{aligned} \frac{({K_I})_{{ scenario}\ n}}{( {K_I})_{{ scenario}\ 1}} =\frac{({\sigma _y })_{{ scenario}\ n}}{({\sigma _y})_{{ scenario}\ 1}};\quad n=2-4 \end{aligned}$$

(C2)

Equation (C2) shows the direct relationship between the ratio of stresses along fracture edge and the ratio of SIF for the scenario under study over that of the basic scenario 1. Therefore, stresses along the length and height of the fractures can be indicative of SIF of the fracture edge. The important point is that SIF changes along the length and height of the fracture. This means that SIF of a rectangular crack along every point on the half-length and half-height of the fractures is different. Hence, in order to have a representing value of the SIF for length or height of a fracture, a fitting function is used to be fitted on the stresses along the length of the fracture ($\sigma _{zz} -x$) and the other on stresses along the height of the fracture ($\sigma _{zz} -y$) and the ratio of fitting function coefficients for the designed scenario over a single fracture is used as a representative of SIF change. The proposed functions, which give satisfactory fit on the numerical stress values, are given in Eq. (C3).

$$\begin{aligned} \frac{({\sigma _{zz}})_L -\sigma _{zz0} }{P_N }= & {} a_{L1} \left( {1-\exp \left( {-a_{L2} \frac{x}{c}} \right) } \right) \end{aligned}$$

(C3a)

$$\begin{aligned} \frac{({\sigma _{zz}})_H -\sigma _{zz0} }{P_N }= & {} a_{H1} \left( {1-\exp \left( {-a_{H2} \frac{y}{b}} \right) } \right) \end{aligned}$$

(C3b)

in which $P_N$ is the applied net pressure, $\sigma _{zz0}$ is the normal stress at the corners, $({\sigma _{zz}})_L$ and $({\sigma _{zz}})_H$ are respectively normal stresses along the length and height of the fracture, x and y axes originate from the corner toward fracture length and height. c, b are respectively the fracture half-length and half-height, $a_{Hi}$ and $a_{Li}$ are coefficients of the function ($i=1,2$). Observing the behavior of the proposed function, it was revealed that stress change is highly controlled by $a_1$ rather than $a_2$. This is because the exponential part is directly multiplied by $a_{H1}$ and $a_{L1}$, while $a_{H2}$ and $a_{L2}$ are only components of the exponential part with negative sign. Therefore, $a_{H1}$ and $a_{L1}$ coefficients were considered as sufficient variables for showing the SIF change along the edge of a rectangular hydraulic fracture (see Taghichian et al. 2018).

Appendix D: Verifications of the developed equations for the fracture aperture (FA), stress shadow effect (SSE), and stress intensity factor (SIF)

See Figs. 11, 12, 13, 14 and 15.

Fig. 11

Prediction of aperture of a contained hydraulic fracture with different aspect ratios (see Eq. 2)

Fig. 12

Prediction of aperture change of a contained hydraulic fracture as a result of multistage fracturing

Fig. 13

Prediction of stress shadow size of a contained hydraulic fracture with different aspect ratios

Fig. 14

Prediction of stress shadow change of a contained hydraulic fracture as a result of simultaneous fracturing

Fig. 15

Prediction of SIF change of a contained hydraulic fracture in simultaneous multistage fracturing scheme

Fig. 16

Elliptical fracture geometry and conventions for a and b values, and angle for SIF determination

Fig. 17

Geometrical view for l value used in the formulations based on a and b values

Fig. 18

Numerical integration of E(k) depicted with respect to AR of the fractures

Appendix E: Approximating fracture net pressure of a single fracture ($P_N$)

In order to have an estimation on the fracturing pressure, the pressure which makes the fracture prone to propagate, all the fractures are assumed to be elliptical and their stress intensity factor is determined according to the available analytical solutions from Sadowsky and Sternberg (1949), Green and Sneddon (1950) and Irwin (1962) (Figs. 16, 17).

The stress intensity factor for an elliptical fracture with a as half-length and b as half-height of the fracture can be determined as:

$$\begin{aligned} P_N =\frac{K_{IC} E(k)}{\sqrt{\pi \ell }} \end{aligned}$$

(E1)

in which $P_N$ is the net pressure applied in the fracture, and $\ell$ and E(k) are determined in the following Eqs. (E2, E3) as:

$$\begin{aligned} \ell= & {} b\sqrt{\left( {\sin ^{2}\theta +\frac{b^{2}}{a^{2}}\cos ^{2}\theta } \right) } \end{aligned}$$

(E2)

$$\begin{aligned} E(k)= & {} \int _0^{\pi /2} \sqrt{1-k^{2}\sin ^{2}\phi }d\phi ; \quad k^{2}=1-\frac{b^{2}}{a^{2}}\nonumber \\ \end{aligned}$$

(E3)

Since Eq. (E3) is only dependent on the aspect ratio of the fracture, it can be simply determined for different aspect ratios (Fig. 18).

Table 7 Reservoir and fracture properties for optimization

Table 8 Properties of some optimized fracturing patterns

Appendix F: Practical examples for fracturing optimization in Barnett shale

Based on the developed formulations for optimization of hydraulic fracturing in unconventional shales, some examples of optimization procedure for Barnett shale is presented herein. In order to do so, the following shale characteristics are assumed. Only a portion of the reservoir is considered for stimulation, the areal extent of which is $533 \times 740\ \hbox {m}^{2}$ (Table 7).

Using the set of formulations for FA, SSE, and SIF change, one can obtain the following optimized patterns for toughness-dominated and viscosity-dominated regimes which are reported in Table 8.

The optimization of toughness-dominated and viscosity-dominated fractures are written in an *.m file herein. The optimized fracturing patterns obtained in Table 8, are also depicted in Figs. 19 and 20. It is worth indicating that the average fracturing pressure for viscosity-dominated regime has been assumed to be 1.2 MPa. The cyan lines represent horizontal lateral wells, the blue and red surfaces exhibit fracture face for toughness-dominated and viscosity-dominated regimes (Figs. 19, 20).

Fig. 19

Toughness-dominated fracturing patterns for a shale reservoir with properties from Table 7

Fig. 20

Viscosity-dominated fracturing patterns for a shale reservoir with properties from Table 7