Numerical Simulation of Proppant Transport Coupled with Multi-Planar-3D Hydraulic Fracture Propagation for Multi-Cluster Fracturing

Abstract

A proppant transport simulator coupled with multi-planar 3D (multi-PL3D) fracture propagation has been developed to examine the proppant distribution among multiple hydraulic fractures during multi-cluster fracturing in a horizontal well. The multi-PL3D fracture model considers wellbore friction, multi-fracture stress interaction, fluid leak-off, and multi-scale propagation regimes. The proppant transport is described by the two-phase (slurry/proppant) flow equations that consider proppant settling, jamming and flow regime transition. A high-resolution weighted essentially non-oscillatory (WENO) finite difference (FD) scheme is adopted to solve the nonlinear proppant transport equations. An efficient time-stepping scheme is developed to solve the solid/fluid coupling equations and moving boundaries for the multi-PL3D model. The proppant transport model and multi-PL3D model are both validated against previously published results. Using the model, we examine the proppant distributions under different injection schedules, proppant sizes, proppant density, and fluid viscosity. Results show that proppant distribution among multiple fractures is different as the flow rate and fracture width distribution vary due to multi-fracture stress interaction. The proppant in the middle cluster settles remarkably as the flow rate is lowest among the multiple clusters. The proppant is usually jammed at the pinch point, where the fracture width reduces sharply. Proppant adding schedule has a significant effect on the proppant distribution. A constant-concentration results in a proppant stack at the fracture front. In contrast, an increasing concentration favors the prop of the near-wellbore fracture. The proppant distribution area ratio (defined as the proppant distribution area divided by the fracture area) is only 20% for 20/40 mesh proppant, while the ratio is 45% for 100 mesh proppant. Slick water can increase the fracture area but not favor promoting the proppant distribution area ratio. The results can be helpful for proppant design for multi-cluster fracturing in a horizontal well.

Appendices

Appendix A: The Critical Width of the Fracture Tip Using Universal Asymptotical Solutions

Define dimensionless variables (Dontsov and Peirce 2015b)

$$\tilde{K} = \frac{{K^{\prime}d^{0.5} }}{{wE^{^{\prime}} }},{ }\tilde{C} = \frac{{4C_{l} s^{0.5} }}{{v^{0.5} w}}, \tilde{d} = \frac{{\mu Vd^{2} }}{{12Ew^{3} }}, \tilde{x} = \frac{{\mu Vd^{2} }}{{12Ew^{3} }}, \chi = \frac{{4C_{l} E^{\prime}}}{{K^{\prime}V^{0.5} }} ,$$

(26)

where $V$ is fracture growth velocity.

The critical width satisfies

$$\tilde{w}^{3} - 1 - \frac{3}{2}b\left( {\tilde{w}^{2} - 1} \right) + 3b^{2} \left( {\tilde{w} - 1} \right) - 3b^{2} \ln \left( {\frac{{b + \tilde{w}}}{b + 1}} \right) = 3C_{1} \left( \delta \right)\tilde{x},$$

(27)

where $b=\frac{{C}_{2}\left(\delta \right)}{{C}_{1}\left(\delta \right)}\chi$, ${C}_{1}\left(\delta \right)=\frac{4\left(1-2\delta \right)}{\delta \left(1-\delta \right)}\mathrm{tan}\left(\pi \delta \right)$, and ${C}_{2}\left(\delta \right)=\frac{16\left(1-3\delta \right)}{3\delta \left(2-3\delta \right)}\mathrm{tan}\left(\frac{3\pi }{2}\delta \right).$

Equation (27) can be solved iteratively to find the critical width to activate a new element.

Appendix B: Influence Matrix of PL3D Displacement Discontinuity Method (DDM)

Displacement discontinuity method is an indirect boundary element method. For PL3D HFs, only normal displacement discontinuity is considered, and the normal stress induced by a normal displacement discontinuity is (Crouch and Starfield 1983)

$$\sigma_{zz} = \frac{G}{{4\pi \left( {1 - v} \right)}}\left( {I_{1} - zI_{2} } \right)w,$$

(28)

where

$$\left. {I_{1} = \frac{{\overline{x}\overline{y}\left( {z^{2} + r^{2} } \right)}}{{r\left( {z^{2} + \overline{y}^{2} } \right)\left( {z^{2} + \overline{x}^{2} } \right)}}} \right\|,$$

(29)

$$\left. {I_{2} = - z\overline{x}\overline{y}\frac{{\left( {z^{2} + \overline{x}^{2} } \right)^{2} \left( {z^{2} + \overline{y}^{2} + 2r^{2} } \right) + \left( {z^{2} + \overline{y}^{2} } \right)^{2} \left( {z^{2} + \overline{x}^{2} + 2r^{2} } \right)}}{{r^{3} \left( {z^{2} + \overline{x}^{2} } \right)^{2} \left( {z^{2} + \overline{y}^{2} } \right)^{2} }}} \right\|,$$

(30)

$$r = \sqrt {\left( {x - \xi } \right)^{2} + \left( {y - \eta } \right)^{2} + z^{2} } ,$$

(31)

$$\overline{x} = x - \xi ,$$

(32)

$$\overline{y} = y - \eta .$$

(33)

The symbol || denotes Chinnery’s notation to represent the substitution

$$I\left( {\xi ,\eta } \right) = I\left( {a,b} \right) - I\left( {a, - b} \right) - I\left( { - a,b} \right) + I\left( { - a, - b} \right),$$

(34)

where $a=0.5\Delta x$, $b=0.5\Delta y$.

Appendix C: Second-Order RKL Numerical Scheme

The numerical scheme of a second-order RKL is (Meyer et al. 2014; Chen et al. 2020a)

$$\begin{gathered} {\varvec{W}}_{0} = {\varvec{w}}_{m} \hfill \\ {\varvec{W}}_{1} = {\varvec{W}}_{0} + \tilde{\mu }_{1} \Delta t{\varvec{M}}\left( {{\varvec{W}}_{0} } \right){\varvec{W}}_{0} \hfill \\ \begin{array}{*{20}c} {{\varvec{W}}_{k} = \mu_{k} {\varvec{M}}\left( {{\varvec{W}}_{k - 1} } \right){\varvec{W}}_{k - 1} { } + \nu_{k} {\varvec{M}}\left( {{\varvec{W}}_{k - 2} } \right){\varvec{W}}_{k - 2} + \left( {1 - \mu_{k} - \nu_{k} } \right){\varvec{M}}\left( {{\varvec{W}}_{0} } \right){\varvec{W}}_{0} { }} \\ { + \tilde{\mu }_{k} \Delta t{\varvec{M}}\left( {{\varvec{W}}_{k - 1} } \right){\varvec{W}}_{k - 1} + \tilde{\gamma }_{k} \Delta t{\varvec{M}}\left( {{\varvec{W}}_{0} } \right){\varvec{W}}_{0} { },{ }2 \le k \le s } \\ \end{array} \hfill \\ {\varvec{w}}_{m + 1} = {\varvec{W}}_{s} , \hfill \\ \end{gathered}$$

(35s)

where ${{\varvec{w}}}_{m}$ is the fracture width at the previous time step, ${{\varvec{w}}}_{m+1}$ is the fracture width for the current time step, $\Delta t$ is the time step of the RKL scheme, and $s$ is the internal stage number. The coefficients in Eq. (34) are provided

$$\begin{gathered} b_{0} = b_{1} = b_{2} = \frac{1}{3} \hfill \\ b_{j} = \frac{{j^{2} + j - 2}}{{2j\left( {j + 1} \right)}} \hfill \\ \tilde{\mu }_{1} = \frac{4}{{3\left( {s^{2} + s - 2} \right)}} \hfill \\ \tilde{\mu }_{j} = \frac{{4\left( {2j - 1} \right)b_{j} }}{{j\left( {s^{2} + s - 2} \right)b_{j - 1} }} \hfill \\ \mu_{j} = \frac{{\left( {2j - 1} \right)b_{j} }}{{jb_{j - 1} }} \hfill \\ \tilde{\gamma }_{j} = - a_{j - 1} \tilde{\mu }_{j} , 2 \le j \le s. \hfill \\ \end{gathered}$$

(35)

Appendix D: WENO Reconstruction Scheme

For the convenience of writing, the subscript j in the y-direction and k in the z-direction are omitted in this subsection.

(1) Reconstruction ${u}_{i+1/2}^{-}$ and ${u}_{i-1/2}^{-}$

The approximation ${u}_{i+1/2}^{-}$ is WENO reconstruction from the total stencil one point biased to the left, that is ${S}^{T}=\left\{{u}_{i-2},{u}_{i-1},{u}_{i},{u}_{i+1},{u}_{i+2}\right\}$, as shown in Fig. 28.

Fig. 28

Illustration of 1D WENO reconstruction

The total stencil consists of three sub-stencil,${S}_{1}=\left\{{u}_{i-2},{u}_{i-1},{u}_{i}\right\},{S}_{2}=\left\{{u}_{i-1},{u}_{i},{u}_{i+1}\right\},{S}_{3}=\left\{{u}_{i},{u}_{i+1},{u}_{i+2}\right\}.$ Moreover, the reconstruction is a nonlinear combination of interpolations in each three sub-stencil. For the sub-stencil ${S}_{1}$, a unique polynomial ${P}_{1}\left(x\right)={a}_{1}{x}^{2}+{b}_{1}x+{c}_{1}$ of degree at most two can be obtained from the following equations

$$\left\{ {\begin{array}{*{20}c} {u_{i - 2} = \frac{1}{{{\Delta }x }}\mathop \int \limits_{{x_{i - 2.5} }}^{{x_{i - 1.5} }} \left( {a_{1} x^{2} + b_{1} x + c_{1} } \right)dx} \\ {u_{i - 1} = \frac{1}{{{\Delta }x }}\mathop \int \limits_{{x_{i - 1.5} }}^{{x_{i - 0.5} }} \left( {a_{1} x^{2} + b_{1} x + c_{1} } \right)dx} \\ {u_{i} = \frac{1}{{{\Delta }x }}\mathop \int \limits_{{x_{i - 0.5} }}^{{x_{i + 0.5} }} \left( {a_{1} x^{2} + b_{1} x + c_{1} } \right)dx} \\ \end{array} ,} \right.$$

(36)

where ${a}_{1}$, ${b}_{1}$, and ${c}_{1}$ are coefficients determined by solving Eq. (36). After simple algebra, the explicit formula for the approximation is

$$u_{i + 1/2}^{\left( 1 \right)} = \frac{1}{3}u_{i - 2} - \frac{7}{6}u_{i - 1} + \frac{11}{6}u_{i} .$$

(37)

Similarly, the interpolation at ${x}_{i+\frac{1}{2}}$ can be obtained from the S₂ and S_3, respectively

$$u_{i + 1/2}^{\left( 2 \right)} = - \frac{1}{6}u_{i - 2} + \frac{5}{6}u_{i - 1} + \frac{1}{3}u_{i} .$$

(38)

$$u_{i + 1/2}^{\left( 3 \right)} = \frac{1}{3}u_{i - 2} + \frac{5}{6}u_{i - 1} - \frac{1}{6}u_{i} .$$

(39)

When the solution is smooth, a 5th order interpolation can be calculated by a linear combination of each sub-stencil interpolation as

$$u_{i + 1/2} = \gamma_{1} u_{{i + \frac{1}{2}}}^{\left( 1 \right)} + \gamma_{2} u_{{i + \frac{1}{2}}}^{\left( 2 \right)} + \gamma_{3} u_{{i + \frac{1}{2}}}^{\left( 2 \right)} ,$$

(40)

where ${\gamma }_{1}=1/10$, ${\gamma }_{2}=3/5,\mathrm{ and }{\gamma }_{3}=3/10.$

When the solution is discontinuous around ${x}_{i+\frac{1}{2}}$, the WENO scheme is to reconstruct the approximation as a convex combination of each sub-stencil approximations ${u}_{i+\frac{1}{2}}^{\left(1\right)}$, ${u}_{i+\frac{1}{2}}^{\left(2\right)}$, and ${u}_{i+\frac{1}{2}}^{\left(3\right)}$

$$u_{i + 1/2} = \omega_{1} u_{i + 1/2}^{\left( 1 \right)} + \omega_{2} u_{i + 1/2}^{\left( 2 \right)} + \omega_{3} u_{i + 1/2}^{\left( 2 \right)} ,$$

(41)

where ${\omega }_{j}\ge 0$, ${\omega }_{1}+{\omega }_{2}+{w}_{3}=1$.

The nonlinear weights ${\omega }_{j}$ are determined based on the smoothness of $u(x)$, which is measured by the smoothness indicator

$$\left\{ {\begin{array}{*{20}l} {\beta_{1} = \frac{13}{{12}}\left( {u_{i - 2} - 2u_{i - 1} + u_{i} } \right)^{2} + \frac{1}{4}\left( {u_{i - 2} - 4u_{i - 1} + 3u_{i} } \right)^{2} } \\ {\beta_{2} = \frac{13}{{12}}\left( {u_{i - 1} - 2u_{i} + u_{i + 1} } \right)^{2} + \frac{1}{4}\left( {u_{i - 1} - u_{i + 1} } \right)^{2} ,} \\ {\beta_{3} = \frac{13}{{12}}\left( {u_{i} - 2u_{i + 1} + u_{i + 2} } \right)^{2} + \frac{1}{4}\left( {u_{i} - 4u_{i + 1} + 3u_{i + 2} } \right)^{2} } \\ \end{array} } \right.$$

(42)

With the smoothness indicator, the nonlinear weights are defined as

$$\omega_{j} = \frac{{\tilde{\omega }_{j} }}{{\tilde{\omega }_{1} + \tilde{\omega }_{2} + \tilde{\omega }_{3} }},{ }\tilde{\omega }_{j} = \frac{{\gamma_{j} }}{{\left( {\varepsilon + \beta_{j} } \right)^{2} }}.$$

(43)

where $\varepsilon$ is a small positive number to avoid the denominator to be zero, chosen as $\varepsilon ={10}^{-6}$ in the study; ${\gamma }_{1}$=1/10, ${\gamma }_{2}$=3/5, ${\gamma }_{3}$=3/10.

In the above WENO reconstruction, one point is biased to the left for the ${u}_{i+1/2}$ approximation, thus the result of reconstruction is ${u}_{i+1/2}^{-}$ using ${S}^{T}=\left\{{u}_{i-2},{u}_{i-1},{u}_{i},{u}_{i+1},{u}_{i+2}\right\}$. Similarly, we can reconstruct ${u}_{i-1/2}^{-}$ using $\left\{{u}_{i-3},{u}_{i-2},{u}_{i-1},{u}_{i},{u}_{i+1}\right\}$ by translating one element to left of ${S}^{T}$.

(2) Reconstruction ${u}_{i-1/2}^{+}$ and ${u}_{i+1/2}^{+}$

Using the total stencil ${S}^{T}$, we can reconstruct ${u}_{i-1/2}$ from each sub-stencil as

$$\left\{ {\begin{array}{*{20}c} {u_{i - 1/2}^{\left( 1 \right)} = - \frac{1}{6}u_{i - 2} + \frac{5}{6}u_{i - 1} + \frac{1}{3}u_{i} } \\ {u_{i - 1/2}^{\left( 2 \right)} = - \frac{1}{6}u_{i - 2} + \frac{5}{6}u_{i - 1} + \frac{1}{3}u_{i} } \\ {u_{i - 1/2}^{\left( 3 \right)} = - \frac{1}{6}u_{i - 2} + \frac{5}{6}u_{i - 1} + \frac{1}{3}u_{i} } \\ \end{array} ,} \right.$$

(44)

WENO reconstruction approximates the u by a nonlinear combination of each approximation in sub-stencil

$$u_{i - 1/2} = \omega_{1}^{^{\prime}} u_{i - 1/2}^{\left( 1 \right)} + \omega_{2}^{^{\prime}} u_{i - 1/2}^{\left( 2 \right)} + \omega_{3}^{^{\prime}} u_{i - 1/2}^{\left( 2 \right)} ,$$

(45)

where ${\omega }_{j}^{^{\prime}}\ge 0$, ${\omega }_{1}^{^{\prime}}+{\omega }_{2}^{^{\prime}}+{\omega }_{3}^{^{\prime}}=1$.

The nonlinear weights are defined as

$$\omega_{j}^{^{\prime}} = \frac{{\tilde{\omega }_{j}^{^{\prime}} }}{{\tilde{\omega }_{1}^{^{\prime}} + \tilde{\omega }_{2}^{^{\prime}} + \tilde{\omega }_{3}^{^{\prime}} }},{ }\tilde{\omega }_{j}^{^{\prime}} = \frac{{\gamma_{j}^{^{\prime}} }}{{\left( {\varepsilon + \beta_{j} } \right)^{2} }},$$

(46)

where $\varepsilon$ is a small positive number $\varepsilon ={10}^{-6}$ in the study; ${\gamma }_{1}^{^{\prime}}$=3/10, ${\gamma }_{2}^{^{\prime}}$=3/5, ${\gamma }_{3}^{^{\prime}}$=1/10.

In the above WENO reconstruction, one point is biased to the right for the ${u}_{i-1/2}$ approximation, thus the result of reconstruction is ${u}_{i-1/2}^{+}$ using ${S}^{T}=\left\{{u}_{i-2},{u}_{i-1},{u}_{i},{u}_{i+1},{u}_{i+2}\right\}$. Similarly, we can reconstruct ${u}_{i+1/2}^{+}$ using $\left\{{u}_{i-1},{u}_{i},{u}_{i+1},{u}_{i+2},{u}_{i+3}\right\}$ by translating one element to right of ${S}^{T}$.