A global sensitivity analysis and reduced-order models for hydraulically fractured horizontal wells

Abstract

Several factors affect the performance and stimulation design of hydraulically fractured wells. Moreover, the dominant factors vary for different quantities of interest, and vary based on the spatial location and with the time of interest. Thus, it will be beneficial if there is a systematic procedure to identify the dominant factors affecting the quantities of interest. To this end, we present a systematic global sensitivity analysis using the Sobol method which can be utilized to rank the variables that affect two quantity of interests—pore pressure depletion and stress change—around a hydraulically fractured horizontal well based on their degree of importance. These variables include rock properties and stimulation design variables. A fully coupled poroelastic hydraulic fracture model is used to account for pore pressure and stress changes due to production. To ease the computational cost of a simulator, we also provide reduced-order models (ROMs), which can be used to replace the complex numerical model with a rather simple analytical model, for calculating the pore pressure and stresses at different locations around hydraulic fractures. The two main reasons for choosing the Sobol method are that it can capture the individual and interaction effects of input variables on the variance of outputs (which is not the case with local sensitivity analysis techniques). It also furnishes a systematic procedure with strong mathematical underpinning to generate ROMs for various quantities of interests for a given mathematical model and for a given set of input variables. The main findings of this research are as follows: (i) mobility, production pressure, and fracture half-length are the main contributors to the changes in the quantities of interest. The percentage of the contribution of each parameter depends on the location with respect to pre-existing hydraulic fractures and the quantity of interest. (ii) As time progresses, the effect of mobility decreases and the effect of production pressure increases. (iii) These two variables are also dominant for horizontal stresses at large distances from hydraulic fractures. (iv) At zones close to hydraulic fracture tips or inside the spacing area, other parameters such as fracture spacing and half-length are the dominant factors that affect the minimum horizontal stress. The results of this study will provide useful guidelines for the stimulation design of legacy wells and secondary operations such as refracturing and infill drilling.

Appendices

Appendix: A. Definition of the kernels

For the reader’s benefit, we present the equations related to the influence of a point displacement discontinuity and a fluid source distributed over a straight line of the source points. For a list of complete equations, one may refer to [12] and [69].

1.1 A.1 Influence of a point fluid source

The required integral for constant temporal distribution and constant spatial distribution of a point source displacement discontinuity is given by:

$$ \begin{array}{@{}rcl@{}} {{\int}_{0}^{t}}{\int}_{-a}^{a}\mathcal{F}\text{dx}^{\prime}d\tau \end{array} $$

where $\mathcal {F}\equiv \mathcal {F}(x-x^{\prime },y;t-\tau )$ denotes the singular solution (Green’s function). Solving the above integral for a piecewise constant displacement discontinuity element and a constant time discretization gives the pore pressure p_p, x component of stress σ_xx, y component of stress σ_yy, and shear stress σ_xy at the target point due to a fluid source at the source point as follows:

$$ \begin{array}{@{}rcl@{}} {p_{p}^{q}} & =& \frac{1}{4\pi\kappa}{\int}_{-a}^{a} \text{Ei}(\zeta^{2})\text{dx}^{\prime} \end{array} $$

(A.1a)

$$ \begin{array}{@{}rcl@{}} \sigma_{xx}^{q} &=& \frac{\alpha (1-2\nu)}{8\pi\kappa(1-\nu)}\left\{\left[-(x-x^{\prime})\frac{1}{\zeta^{2}}(1-e^{-\zeta^{2}})-(x-x^{\prime})\text{Ei}(\zeta^{2})\right]^{a}_{-a}-2{\int}_{-a}^{a}\text{Ei}(\zeta^{2})\text{dx}^{\prime}\right\} \end{array} $$

(A.1b)

$$ \begin{array}{@{}rcl@{}} \sigma_{yy}^{q} & =& \frac{\alpha (1-2\nu)}{8\pi\kappa(1-\nu)}\left[(x-x^{\prime})\frac{1}{\zeta^{2}}(1-e^{-\zeta^{2}})+(x-x^{\prime})\text{Ei}(\zeta^{2})\right]^{a}_{-a} \end{array} $$

(A.1c)

$$ \begin{array}{@{}rcl@{}} \sigma_{xy}^{q} & =& \frac{\alpha (1-2\nu)}{8\pi\kappa(1-\nu)} \left[-y\frac{1}{\zeta^{2}}(1-e^{-\zeta^{2}})-y\text{Ei}(\zeta^{2})\right]_{-a}^{a} \end{array} $$

(A.1d)

In the above equations, the superscript q is referring to the changes due to fluid source. κ is the mobility and is defined as the ratio of the absolute permeability of the rock to the fluid viscosity (i.e., $\frac {k}{\mu }$). α is the Biot’s poroelastic coefficient, ν is the drained Poisson ratio, and ζ is a dimensionless parameter defined as follows:

$$ \begin{array}{@{}rcl@{}} \zeta = \frac{r}{2\sqrt{\text{ct}}} \end{array} $$

(A.2)

where c is the diffusivity coefficient, r is the direct distance between source and target displacement discontinuity elements, x and y are the relative coordinates of the source point with respect to the local coordinate system of the target element, $x^{\prime }$ takes the values of − a, and a that is fracture half-length. Also, Ei(x) is the exponential integral and is defined as follows:

$$ \begin{array}{@{}rcl@{}} \text{Ei}(x) = {\int}_{x}^{\infty} \frac{e^{-u}}{u}\text{du} \end{array} $$

(A.3)

where erf(x) is defined as the error function of x and is defined as follows:

$$ \begin{array}{@{}rcl@{}} \text{erf}(x) = \frac{2}{\sqrt{\pi}}{{\int}_{0}^{x}}e^{-u^{2}}\text{du} \end{array} $$

(A.4)

1.2 A.2 Influence of a point displacement discontinuity

The required integral for constant temporal distribution and constant spatial distribution is given by:

$$ \begin{array}{@{}rcl@{}} {{\int}_{0}^{t}}{\int}_{-a}^{a}\mathcal{F}dx^{\prime}d\tau \end{array} $$

where $\mathcal {F} \equiv \mathcal {F}(x-x^{\prime },y;t-\tau )$ denotes the singular solution (Green’s function). Solving the above integral for a piecewise constant displacement discontinuity element and a constant time discretization gives the pore pressure p_p, x component of stress σ_xx, y component of stress σ_yy, and shear stress σ_xy at the target point due to a normal displacement discontinuity at the source point as [12].

1.2.1 A.2.1 Normal displacement discontinuity at the source point

$$ \begin{array}{@{}rcl@{}} p_{p}^{\text{dn}} &=& \frac{\mu({\nu}_{u}-\nu)}{2\pi\alpha(1-2\nu)(1-{\nu}_{u})}\left[-\frac{2(x-x^{\prime})}{r^{2}}\left( 1-e^{-{\zeta}^{2}}\right)\right]_{-a}^{a} \end{array} $$

(A.5a)

$$ \begin{array}{@{}rcl@{}} \sigma_{\text{xx}} & =& \frac{\mu}{2\pi(1-\nu)}\left\{\frac{(x-x^{\prime})^{3}-(x-x^{\prime})y^{2}}{r^{4}}+\left( \frac{\nu_{u}-\nu}{1-\nu_{u}}\right) \left[\frac{(x-x^{\prime})^{3}-(x-x^{\prime})y^{2}}{r^{4}} \right. \right. \\ && -\left. \left. \frac{(x-x^{\prime})^{3}-3(x-x^{\prime})y^{2}}{r^{4}}\frac{1}{\zeta^{2}}(1-e^{-\zeta^{2}}) - \frac{2(x-x^{\prime})y^{2}}{r^{4}}e^{-\zeta^{2}} \right]\right\}_{-a}^{a} \end{array} $$

(A.5b)

$$ \begin{array}{@{}rcl@{}} \sigma_{\text{yy}}^{\text{dn}} &=& \frac{\mu}{2\pi(1-\nu)}\left\{\frac{(x-x^{\prime})^{3}+3(x-x^{\prime})y^{2}}{r^{4}}+\left( \frac{\nu_{u}-\nu}{1-\nu_{u}}\right) \left[\frac{(x-x^{\prime})^{3}-3(x-x^{\prime})y^{2}}{r^{4}} \right. \right. \\& &+ \left. \left. \frac{(x-x^{\prime})^{3}-3(x-x^{\prime})y^{2}}{r^{4}}\frac{1}{\zeta^{2}}(1-e^{-\zeta^{2}}) - \frac{2(x-x^{\prime})^{3}}{r^{4}}e^{-\zeta^{2}} \right]\right\}_{-a}^{a} \end{array} $$

(A.5c)

$$ \begin{array}{@{}rcl@{}} \sigma_{\text{xy}}^{\text{dn}} &=& \frac{\mu}{2\pi(1-\nu)}\left\{\frac{(x-x^{\prime})^{2}y-y^{3}}{r^{4}}+\left( \frac{\nu_{u}-\nu}{1-\nu_{u}}\right) \left[\frac{(x-x^{\prime})^{2}y-y^{3}}{r^{4}} \right. \right. \\ & &- \left. \left. \frac{3(x-x^{\prime})^{2}y-y^{3}}{r^{4}}\frac{1}{\zeta^{2}}(1-e^{-\zeta^{2}}) + \frac{2(x-x^{\prime})^{2}y}{r^{4}}e^{-\zeta^{2}} \right]\right\}_{-a}^{a} \end{array} $$

(A.5d)

Table 4 Coefficients of the reduced-order model for pore pressure at point 1 (5.2a)–(5.2d)

In the above equations,the superscript d_n is referring to the changes due to a normal displacement discontinuity.

1.2.2 A.2.2 Shear displacement discontinuity at the source point

$$ \begin{array}{@{}rcl@{}} p_{p}^{\text{ds}} &=& \frac{\mu(\nu_{u}-\nu)}{2\pi\alpha(1-2\nu)(1-\nu_{u})}\left[\frac{2y}{r^{2}}\left( 1-e^{-\zeta^{2}}\right)\right]_{-a}^{a} \end{array} $$

(A.6a)

$$ \begin{array}{@{}rcl@{}} \sigma_{\text{xx}}^{\text{ds}} &=& \frac{\mu}{2\pi(1-\nu)}\left\{\frac{-3(x-x^{\prime})^{2}y-y^{3}}{r^{4}}+\left( \frac{\nu_{u}-\nu}{1-\nu_{u}}\right) \left[\frac{-3(x-x^{\prime})^{2}y-y^{3}}{r^{4}} \right. \right. \\&&+ \left. \left. \frac{3(x-x^{\prime})^{2}y-y^{3}}{r^{4}}\frac{1}{\zeta^{2}}(1-e^{-\zeta^{2}}) + \frac{2y^{3}}{r^{4}}e^{-\zeta^{2}} \right]\right\}_{-a}^{a} \end{array} $$

(A.6b)

$$ \begin{array}{@{}rcl@{}} \sigma_{\text{yy}}^{\text{ds}} &=& \frac{\mu}{2\pi(1-\nu)}\left\{\frac{(x-x^{\prime})^{2}y-y^{3}}{r^{4}}+\left( \frac{\nu_{u}-\nu}{1-\nu_{u}}\right) \left[\frac{(x-x^{\prime})^{2}y-y^{3}}{r^{4}} \right. \right. \\&&- \left. \left. \frac{3(x-x^{\prime})^{2}y-y^{3}}{r^{4}}\frac{1}{\zeta^{2}}(1-e^{-\zeta^{2}}) + \frac{2(x-x^{\prime})^{2}y}{r^{4}}e^{-\zeta^{2}} \right]\right\}_{-a}^{a} \end{array} $$

(A.6c)

$$ \begin{array}{@{}rcl@{}} \sigma_{\text{xy}}^{\text{ds}} &=& \frac{\mu}{2\pi(1-\nu)}\left\{\frac{(x-x^{\prime})^{3}-(x-x^{\prime})y^{2}}{r^{4}}+\left( \frac{\nu_{u}-\nu}{1-\nu_{u}}\right) \left[\frac{(x-x^{\prime})^{3}-(x-x^{\prime})y^{2}}{r^{4}} \right. \right. \\&&+ \left. \left. \frac{3(x-x^{\prime})y^{2}-(x-x^{\prime})^{3}}{r^{4}}\frac{1}{\zeta^{2}}(1-e^{-\zeta^{2}}) - \frac{2(x-x^{\prime})y^{2}}{r^{4}}e^{-\zeta^{2}} \right]\right\}_{-a}^{a} \end{array} $$

(A.6d)

In the above equations,the superscript d_s is referring to the changes due to a shear displacement discontinuity.

B. Coefficients of the reduced-order models

Table 5 Coefficients of the reduced-order model for the minimum horizontal stress at point 1 (5.3a)–(5.3g)

Table 6 Coefficients of the reduced-order model for the maximum horizontal stress at point 1 (5.5a)–(5.5g)

Table 7 Coefficients of the reduced-order model for pore pressure at point 5 (5.6a)–(5.6e)

Table 8 Coefficients of the first-order Sobol functions for the minimum horizontal stress at point 5 (5.7a)–(5.7g)

Table 9 Coefficients of the second-order Sobol functions for the minimum horizontal stress at point 5 (5.7h)–(5.7l)

Table 10 Coefficients of the reduced-order model for the maximum horizontal stress at point 5 (5.8a)–(5.8g)

Table 11 Coefficients of the reduced-order model for the pore pressure at point 6 (5.9a)–(5.9g)

Table 12 Coefficients of the reduced-order model for the minimum horizontal stress at point 6 (5.10a)–(5.10c)

Table 13 Coefficients of the reduced-order model for the maximum horizontal stress at point 6 (5.11a)–(5.11h)