# Stimulated Reservoir Volume Estimation and Analysis of Hydraulic Fracturing in Shale Gas Reservoir

• Lan Ren
• Ran Lin
• Jinzhou Zhao
Research Article - Petroleum Engineering

## Abstract

Hydraulic fracturing in horizontal well is the key technology for the commercial exploitation of shale gas reservoir. Stimulated reservoir volume (SRV) is an important indicator to evaluate the fracturing performance. However, estimating the SRV has been a long-standing challenge due to its complex forming mechanism. Most current SRV estimation methods are either expensive or time-consuming. This paper developed a 3D mathematical model to estimate the SRV by simulating the four main processes during shale fracturing—multiple hydraulic fractures propagation, formation stress variation, reservoir pressure lifting and natural fractures failure. In this model, hydraulic fractures propagation is calculated by pseudo-three-dimensional model, coupling with formation stress model; formation stress and reservoir pressure are obtained by displacement discontinuity method and Green’s function approach, respectively; natural fracture failure criterion is derived from Warpinski’s theory. This model not only considers the stress interference effect of multiple fractures, but also subdivides the SRV into shear-SRV and tensile-SRV according to the failure type of natural fractures network. This model was first implemented to a pilot well in the FL gas field in southwest China to estimate a SRV that matches well with the on-site monitoring microseismic signals. Then, this model was applied to FL gas field on a large scale to evaluate the overall fracturing effects. Finally, a sensitivity study was conducted to analyze the impact of engineering parameters on the SRV. This research explores an efficient method to estimate the SRV without high cost or complicated process and provides the theoretical basis and guidelines for pre-fracturing design and post-fracturing evaluation in shale gas reservoir.

## Keywords

Shale gas Hydraulic fracturing Stimulated reservoir volume Mathematical modeling Field application

## List of symbols

$$(A_\mathrm{nn})_{ij}$$

Plane-strain, elastic-influence coefficient matrix representing the normal stress at element i induced by normal-displacement discontinuity at element $$j; i, j \in \{1, 2, {\ldots }, N\}$$

$$(A_\mathrm{nt})_{ij}$$

Plane-strain, elastic-influence coefficient matrix representing the normal stress at element i induced by shear-displacement discontinuity at element $$j; i, j \in \{1, 2, {\ldots }, N\}$$

$$(A_\mathrm{tn})_{ij}$$

Plane-strain, elastic-influence coefficient matrix representing the shear stress at element i induced by normal-displacement discontinuity at element $$j; i, j \in \{1, 2, {\ldots }, N\}$$

$$(A_\mathrm{tt})_{ij}$$

Plane-strain, elastic-influence coefficient matrix representing the shear stress at element i induced by shear-displacement discontinuity at element $$j; i, j \in \{1, 2, {\ldots }, N\}$$

$$c_\mathrm{L}$$

Filtration coefficient ($$\hbox {m/s}^{0.5}$$)

E

Young’s modulus of formation rock ($$\hbox {Pa}^{-1}$$)

Open image in new window

Force vector imposed on the unit area of fracture surface (Pa)

$$F_{i}$$

Derivatives functions, $$i \in \{3, 4, 5, 6\}$$,

G

Formation shear modulus ($$\hbox {Pa}^{-1}$$)

$$h_\mathrm{f}$$

Fracture height (m)

$$h_\mathrm{r}$$

Thickness of reservoir (m);

$$h_\mathrm{rD}$$

Dimensionless thickness of reservoir

$$K_{0}$$

Zeroth-order Bessel function

$$K_\mathrm{f}$$

Friction coefficient of natural fracture

$$K_\mathrm{IC}$$

Fracture toughness ($$\hbox {Pa}\,\hbox {m}^{0.5}$$)

$$k_\mathrm{m}$$

Average permeability of matrix system (D)

$$k_\mathrm{mx}$$

x-Directional permeability of matrix system (D)

L

Arbitrary reference length (m)

$$L_\mathrm{f}$$

Fracture half-length (m)

M

Number of hydraulic fractures

N

Number of discontinuous fracture elements

$$n_{i}$$

Component of

$$N_\mathrm{IL}$$

An even number between 6 and 18

p

Current reservoir pressure (Pa)

$$p_{0}$$

Initial reservoir pressure (Pa)

$$p_\mathrm{f}$$

Fluid pressure in fracture (Pa)

$$p_{{\mathrm{f}, i}}$$

Fluid pressure in fracture, $$i \in \{1, 2,\ldots {M}\}\,(\hbox {m}^{3}/\hbox {s}$$)

$$p_\mathrm{fnet}$$

Net pressure in fracture (Pa)

$$p_\mathrm{nf}$$

Fluid pressure in natural fracture (reservoir pressure) (Pa)

$${\tilde{q}}$$

Filtration rate per unit area (m/s)

q

Flow rate in fracture ($$\hbox {m}^{3}/\hbox {s}$$)

$$Q_{i}$$

Inlet flow rate of each fracture $$i \in \{1, 2,\ldots {M}\}(\hbox {m}^{3}/\hbox {s}$$)

$$Q_\mathrm{pump}$$

Pump rate ($$\hbox {m}^{3}/\hbox {s}$$)

s

Laplace transform variable

$$S_\mathrm{t}$$

Tensile strength of natural fracture (Pa)

t

Time (s)

u

Defined function

$$\left( {\hat{{u}}_{\mathrm{n}} }\right) _i$$

Normal strain of element i in local coordinate, $$i \in \{1, 2,\ldots {N}\}\,(\hbox {m})$$

$$\left( {\hat{{u}}_{\mathrm{t}} }\right) _i$$

Shear strain of element i in local coordinate, $$i \in \{1, 2,\ldots {N}\}\,(\hbox {m})$$

$$W_\mathrm{f}$$

Fracture width (m)

$$x_\mathrm{D}$$

Dimensionless x value

$$x_\mathrm{wD}$$

Dimensionless x value at perforation point

$$y_\mathrm{D}$$

Dimensionless y value

$$y_\mathrm{wD}$$

Dimensionless y value at perforation point

$$z_\mathrm{w}$$

z value at cluster point (m)

$$\alpha$$

Integration variable

$$\delta$$

Kronecker delta

$$\Delta p$$

Pressure increment field in the real domain (Pa)

$${\Delta }\bar{{p}}$$

Reservoir pressure increment in Laplace domain (Pa)

$$\Delta p_{i}$$

Pressure increment, induced by fluid leak-off from fracture $$i, i \in \{1, 2,\ldots {M}\}\,(\hbox {Pa})$$

$$\Delta \sigma _{ij}$$

Components of induced stress tensor, $$i, j \in \{x, y, z\}\,(\hbox {Pa})$$;

$$\theta$$

Approaching angle of natural fracture ($${}^{\circ }$$)

$$\varphi$$

Dip angle of natural fracture ($${}^{\circ }$$)

$$\mu$$

Viscosity of fracturing fluid ($$\hbox {Pa}\,\hbox {s}$$)

$$\nu$$

Poisson ratio

$$\sigma _{ij}$$

Components of current formation stress tensor, $$i, j \in \{x, y, z\}$$ (Pa)

$$\sigma _{ij}^{\left( 0\right) }$$

Components of original formation stress tensor, $$i, j \in \{x, y, z\}$$ (Pa)

Open image in new window

Formation stress tensor (Pa)

$$({\sigma }_\mathrm{n})_{i}$$

Normal stress on element i in local coordinate, $$i \in \{1, 2,\ldots N\}$$ (Pa)

$$({\sigma }_\mathrm{t})_{i}$$

Shear stress on element i in local coordinate, $$i \in \{1, 2,\ldots N\}$$ (Pa)

$${\sigma }_\mathrm{n}$$

Normal stress imposed on fracture surface (Pa)

$${\sigma }_{\tau }$$

Shear stress imposed on natural fracture (Pa)

$${\tau }$$

Start time of filtration in fracture (s)

$${\tau }_\mathrm{o}$$

Cohesive strength of natural fracture (Pa)

## Notes

### Acknowledgements

The authors would like to acknowledge the financial support of the Major Program of the National Natural Science Foundation of China (51490653), National Natural Science Foundation of China (51404204), and the National Science and Technology Major Project of the Ministry of Science and Technology of China (2016ZX05023005-001-002).

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© King Fahd University of Petroleum & Minerals 2018

## Authors and Affiliations

• Lan Ren
• 1
• Ran Lin
• 1
• Jinzhou Zhao
• 1
1. 1.State Key Laboratory of Oil and Gas Reservoir Geology and ExploitationSouthwest Petroleum UniversityChengduChina