Multi-continuum Approach to Modelling Shale Gas Extraction

Abstract

Production rates in horizontal shale gas wells display declines that are controlled by the low permeability and the intrinsic heterogeneity of the shale matrix. We present an original multi-continuum approach that yields a physical model able to reproduce the complexity of the decreasing gas rates. The model describes the dynamics of gas rate as function of the physical reservoir parameters and geometry, while the shale matrix heterogeneity is accounted for by a stochastic description of transmissivity field. From the 3D (Dimensional) problem setting, including the heterogeneous shale matrix, the fractures generated by the hydrofracking operations, as well as the production well characteristics, we establish an effective upscaled 1D model for the gas pressures in fracture and matrix as well as the volumetric flux. We analyse the decline curves behaviour, and we identify the time scales that characterize the dynamics of the gas rate decline using explicit analytical Laplace space solutions of the upscaled process model. Asymptotically, the flux curves decrease exponentially, while in an intermediate regime we find a power-law behaviour, in which the flux scales with a power law in time as $t^{-\beta }$, where $\beta$ reflects the medium heterogeneity. We use this solution to fit a set of real data displaying distinctly different decline trends and study the sensitivity of the model to the reservoir parameters in order to identify their respective controls at the different stages of the decline curve dynamics. Results indicate that the initial value of the gas rate is determined by the transmissivity of the fractures and the initial pressure of the gas in the shale matrix. The latter causes mainly a shift in the entire decline curve. The early time of decline curve shape is primarily controlled by the fracture properties (compressibility and transmissivity). During the main part of the economically valuable production times, i.e. before the production rate drops exponentially, the decline curve is strongly controlled by the properties of the shale rocks including their heterogeneity, which is modelled by two parameters describing the non-Fickian pressure diffusion effects in a stochastic framework.

Appendices

Appendix 1: Derivation of the Memory Function

We assume that the lower cut-off scale $\tau _1$ in (19) is much smaller than the upper cut-off $\tau _2$ so that $\tau _1/\tau _2 \ll 1$ and we approximate

$$\begin{aligned} P_{im}(\tau ) = \frac{1 - \beta }{\tau _2} \left( \frac{\tau }{\tau _2}\right) ^{-\beta } \Theta (\tau _2 - \tau ). \end{aligned}$$

(41)

Inserting (41) in (18), the memory function for a multi-continuum model with a constant retardation factor reads:

$$\begin{aligned} \varphi (t) = S_{im} \frac{1 - \beta }{\tau _2} \left( \frac{t}{\tau _2}\right) ^{-\beta } \int \limits _{t/\tau _2}^\infty \mathrm{d}z \, z^{\beta - 1} g'(z), \end{aligned}$$

(42)

where $g'(z)$ is defined as: $g(t,\tau ) = \frac{1}{\tau } \, g' \left( \frac{t}{\tau } \right) .$ The local memory function $g(t,\tau )$ can be expressed as above from the definition of $\tilde{g}(\omega ,\tau )$ given in (11), and we note that $\tilde{g}(\omega ,\tau )$ is precisely a function of the product $\omega \tau$. The function $g'(z)$ behaves as $z^{-1/2}$ for $z < 1$ and decays exponentially fast for $z \gg 1$. Thus, we approximate it here by the truncated power-law distribution: $g'(z) =z^{-1/2} \exp (-z)/\Gamma (1/2).$ Inserting the previous approximation into (42) and executing the integral for the memory function, $\varphi _{\kappa }(t)$ reads

$$\begin{aligned} \varphi (t) = S_{im}\frac{1 - \beta }{\tau _2 \Gamma (1/2)} \left( \frac{t}{\tau _2}\right) ^{-\beta } \Gamma (\beta - 1/2,t/\tau _2). \end{aligned}$$

(43)

This function behaves as a power law according to $t^{-\beta }$ for $t \ll \tau _2$ and is cut-off exponentially fast for $t \gg \tau _2$. Thus, we approximate it by the truncated power law given in (20).

Appendix 2: General Solution of the Model

The general solution of Eq. (15) is given by a combination of modified Bessel functions of first kind $\mathcal {I}_0 (\cdot )$ and second kind $\mathcal {K}_0 (\cdot )$ (Carslaw and Jaeger 1947):

$$\begin{aligned} \langle \overline{p}_\mathrm{f}(r, \lambda )\rangle = N(\lambda ) \; \mathcal {K}_0[r \sqrt{\lambda \tau ^\prime (\lambda )} ] + M(\lambda ) \, \mathcal {I}_0 [r \sqrt{\lambda \tau ^\prime (\lambda )} ] + \frac{\overline{p}_\mathrm{f}(r,0)}{\lambda }, \end{aligned}$$

(44)

with $\tau ^\prime (\lambda )$ defined in Eq. (25). The functions $N(\lambda )$ and $M(\lambda )$ do not depend on the radial coordinate $r$ and are determined by imposing the given boundary condition in the general solution (44). Considering no radial flux at $r=R$, Eq. (27), we obtain: $N(\lambda )= M(\lambda ) \, \mathcal {I}_1 [r \sqrt{\lambda \tau ^\prime } ] / \mathcal {K}_1 [r \sqrt{\lambda \tau ^\prime } ]$, and by imposing Cauchy-type boundary condition at the well, Eq. (26), we obtain $M(\lambda )$. By inserting $N(\lambda )$ and $M(\lambda )$ in Eq. (44) and rearranging the terms, we obtain Eq. (23).

Appendix 3: Asymptotic Expansion

In order to evaluate the asymptotic behaviour of the gas flux $Q(t)$, we consider the expansion of the Bessel functions in (32) for small arguments $x$,

$$\begin{aligned}&\mathcal {K}_0\left( x\right) = - \ln (x/2) + \cdots \end{aligned}$$

(45)

$$\begin{aligned}&\mathcal {I}_0\left( x \right) = 1 + \cdots \end{aligned}$$

(46)

$$\begin{aligned}&\mathcal {K}_1\left( x \right) = \frac{1}{x} + \frac{1}{2} \ln x \ldots \end{aligned}$$

(47)

$$\begin{aligned}&\mathcal {I}_1\left( x \right) = \frac{x}{2} \ldots , \end{aligned}$$

(48)

where the dots denote sub-leading contributions. Substituting these expansion into (32), we obtain:

$$\begin{aligned} Q^*(\lambda ) \simeq \frac{ \pi T_\mathrm{f} \left( {R'}^2 - {r'}^2_\mathrm{w} \right) \left[ \frac{p_\mathrm{f}(0)}{\lambda } -p^*_\mathrm{w}(\lambda )\right] }{1 + \frac{{R'}^2}{2} \ln \frac{R'}{r_\mathrm{w}'} + \frac{A}{2} \left( {R'}^2 - {r'}^2_\mathrm{w} \right) }. \end{aligned}$$

(49)

Setting the pressure in the extraction well constant, i.e. $p_\mathrm{w}(t) = p_\mathrm{w} =$ constant, which implies $p^*_\mathrm{w}(\lambda ) = \lambda ^{-1} p_\mathrm{w}$, and substituting $r', R'$ and $r_\mathrm{w}'$ by (24) and (29), we obtain Eq. (33).