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# Open Physics

### formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

Managing Editor: Lesna-Szreter, Paulina

IMPACT FACTOR 2018: 1.005

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ICV 2017: 162.45

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Volume 15, Issue 1

# Analytical modeling of coupled flow and geomechanics for vertical fractured well in tight gas reservoirs

Ruifei Wang
/ Xuhua Gao
• School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 10083, China
• Other articles by this author:
/ Hongqing Song
• Corresponding author
• College of Petroleum Engineering, Xi’an Shiyou University, Xi’an Shanxi 710065, China
• School of Civil and Environmental Engineering, University of Science and Technology Beijing, Beijing, 10083, China
• Email
• Other articles by this author:
/ Xinchun Shang
• School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 10083, China
• Other articles by this author:
Published Online: 2017-12-29 | DOI: https://doi.org/10.1515/phys-2017-0093

## Abstract

The mathematical model of coupled flow and geomechanics for a vertical fractured well in tight gas reservoirs was established. The analytical modeling of unidirectional flow and radial flow was achieved by Laplace transforms and integral transforms. The results show that uncoupled flow would lead to an overestimate in performance of a vertical fractured well, especially in the later stage. The production rate decreases with elastic modulus because porosity and permeability decrease accordingly. Drawdown pressure should be optimized to lower the impact of coupled flow and geomechanics as a result of permeability decreasing. Production rate increases with fracture half-length significantly in the initial stage and becomes stable gradually. This study could provide a theoretical basis for effective development of tight gas reservoirs.

PACS: 47.56.+r; 62.25.Mn; 89.30.an

## 1 Introduction

Nowadays, the demand for energy has greatly increased and conventional reservoirs cannot satisfy this situation [1, 2]. The tight gas reservoir, as one of important unconventional reservoirs, has a wide distribution worldwide and many major oil companies pay more and more attention to developing it [3, 4].

Geomechanics has a huge impact on tight gas exploitation due to ultra-low porosity and permeability [5]. In the period of tight gas extraction, the porosity and permeability decrease somewhat with pressure depletion in the tight gas reservoir. Coupled flow and geomechanics should be involved in the numerical simulation for tight gas reservoir development [6, 7].

There are two research aspects on coupled flow and geomechanics. On the one hand, many researchers and reservoir engineers utilize modified porosity and permeability to form analytical formula for simulation and evaluation in different conditions [8, 9]. For example, Wei proposed a triple-porosity/dual-permeability model, which considered the effective stress, gas slippage and micropore shrinkage/swelling [10]. Zheng proposed a two-part Hooke’s model to consider the geomechanical effects [11]. Wang incorporated the effects of geomechanics into permeability by considering its impact on pore throats [12]. This approach makes it easy to understand reservoir variations, and is useful for industry application. On the other hand, the researchers focus on conductivity of fractures with numerical simulations using the finite volume method (FVM), finite element method (FEM) and Newton method (NM), which uses the linear part of Taylor’s formula to solve nonlinear problems [13, 14]. Ren et al. proposed an extended finite element method-embedded discrete fracture model (XFEM-EDFM) and a proppant model to analyze the influence of stress-dependent fracture permeability on cumulative production [15]. Wei Yu studied and evaluated the effect of geomechanics on the stress-dependent fracture conductivity using FEM [16]. However, numerical simulation is more complicated and timeconsuming compared to the theoretical method, and the simulation results are unreliable because of the uncertainty of input parameters [17, 18].

Due to the ultra-low porosity and permeability, fracturing technology was widely utilized for tight gas reservoir development to enlarge controlled volume and increase porosity and permeability. At present, the commonly used extraction of natural gas from tight reservoirs adopts the technology of the fractured well, which includes the vertical fractured well and the horizontal fractured well. Compared with horizontal well fractured technology, the vertical fractured technology is more economical [19, 20, 21, 22].

In this paper, a new mathematical model of coupled flow and geomechanics for a vertical fractured well in tight gas reservoirs was initially established. In addition, the analytical modeling of unidirectional flow and radial flow was achieved by Laplace transforms and integral transforms. Moreover, the calculated results were compared with actual production data from the Sulige gas reservoir in China to verify the model. Finally, the influences of elastic modulus and half fracture length to well performances were analyzed. This study could provide a theoretical basis for effective development of tight gas reservoirs.

## 2.1 Physical description for vertical fractured well

The flow patterns of a controlled area in the tight gas reservoir can be categorized into unidirectional flow in the middle area and radial flow in the rest of the area [23]. The sketch of a vertical well for tight gas production is shown in Figure 1. For gas flow, the initial pressure and constant outer boundary are both pe. The pressure of the fracture connected with the well maintains pw.

Figure 1

Sketch of vertical well for tight gas production

## 2.2.1 Equation of stress

According to the principle of effective stress [24], the effective stress of the formation can be expressed as: $σij′=σij−αBpδij,$(1)

where σij is the solid stress, p is the fluid pressure, αB is Biot’s coefficient, and δij is defined as when i = j, δij = 1 and when ij, δij = 0.

The strain-displacement relation is expressed as: $εij=12(ui,j+uj,i),$(2)

where ui is the component of the displacement in the i direction.

The constitutive relationship of stress and strain can be expressed as: $σij=Ev(1+v)(1−2v)εkkδij+E(1+v)εij,$(3)

where E is the elastic modulus, and v is Poisson’s ratio.

Without the body force, the momentum balance equation can be expressed as: $σji,j′=0.$(4)

Substituting (1)-(3) into (4), the equation of stress in terms of displacement can be derived as: $E2(1+v)(1−2v)∇(∇⋅u)+E2(1+v)∇2u−αB∇p=0$(5)

## 2.2.2 Equation of seepage

When gas transports in the porous media, gas and solid particles both have speed. The velocity of the gas can be given as: $vg=vr+vs,$(6)

where Vs is the absolute velocity of solid particles, vg is the absolute velocity of gas; vr is the relative velocity of them, namely, the Darcy velocity. According to definitions, the velocities mentioned can be given as: $vs=∂w∂t,vr=−kμ∇p.$(7)

The continuity equation of the solid can be given as: $∇(ρsvs)+∂(ρs(1−φ))∂t=0,$(8)

where ρs is the density of the formation and is considered as constant, so (8) can be simplified as: $∇vs−∂φ∂t=0.$(9)

The continuity equation of gas can be given as: $∇(ρgvg)+∂(ρgφ)∂t=0,$(10)

where ρgis the gas density. Substituting (6) and (9) into (10), the following formula can be derived: $∇(ρgvr)+2ρg∇vs+vs∇ρg+φ∂ρg∂t=0.$(11)

Since vs is too small relative to vr, and neglecting 2ρgvs+vsρg, then the equation of seepage field can be given as: $∇(ρgvr)+φ∂ρg∂t=0.$(12)

## 2.2.3 Equations of parameters

Vs, Vb, and Vp are used to represent the volume of solid skeleton, the whole volume of the formation, and the pore volume, respectively. And ΔVs, ΔVb, and ΔVp represent their increment, respectively. θ is the volumetric strain. According to the definition, the porosity of the formation can be expressed as: $φ=Vp+ΔVpVb+ΔVb=1−Vs+ΔVsVb+ΔVb=1−Vs(1+ΔVs/Vs)Vb(1+ΔVb/Vb)=1−1−φ01+θ(1+ΔVs/Vs)$(13)

It is assumed that the deformation of the formation is just caused by the recombination of the incompressible solid particles, thus ΔVs = 0, (13) can be simplified as: $φ=θ+φ01+θ.$(14)

The cubic relationship [24] between permeability and porosity can be expressed as: $k=k0⋅(φφ0)3.$(15)

Gas state equation can be given as: $ρg=TscZscρgscpsc⋅pTZ,$(16)

where φ0 is the initial porosity; ko is the initial permeability; T is the temperature, and Z is the compressibility factor; Tsc, Zsc, and psc represent the temperature, compressibility factor, and pressure at standard state, respectively.

## 3 Analytical solution

To derive an analytical solution of this mathematical model, we need to obtain the relationship between displacement and pressure first. For planar radial flow, (5) can be written as: $E(1−v)(1+v)(1−2v)(d2urdr2+1rdurdr−urr2)=αBdpdr,$(17)

where Ur is the axisymmetric radial displacement. For planar unidirectional flow, (5) can be written as: $E(1−v)(1+v)(1−2v)d2uxdx2=αBdpdx,$(18)

where Ux is the displacement in the x direction.

Integrating on both sides of (17) and (18), we then obtain:

$E(1−v)(1+v)(1−2v)(durdr+urr)=αB(p+c)$(19a)

$E(1−v)(1+v)(1−2v)duxdx=αB(p+c).$(19b)

Considering θ equals $\frac{du}{dr}+\frac{u}{r}$ in planar radial flow, and equals $\frac{d{u}_{x}}{dx}$ in planar unidirectional flow, (19) can be unified as: $E(1−v)(1+v)(1−2v)θ=αB(p+c).$(20)

Since the outer boundary is fixed, namely, θ equals 0 when r = re, thus c = −pe and the following formula can be derived: $θ=αB(1+v)(1−2v)E(1−v)(p−pe).$(21)

Define m = $\left(p+\frac{\mu D}{k}{\right)}^{2}$ and substitute (7) and (16) into (12), then define ${c}_{t}^{\ast }=\frac{\mu \phi }{kp}$ and pseudo time ${t}_{a}^{\ast }={c}_{ti}^{\ast }\underset{0}{\overset{t}{\int }}\frac{dt}{{c}_{t}^{\ast }},$ we can obtain $∇2m=cti∗∂m∂ta∗,$(22)

where ${c}_{ti}^{\ast }$ is the value in the initial condition.

For planar radial flow, (22) can be written as: $∂2m∂r2+1r∂m∂r=cti∗∂m∂ta∗.$(23)

Now equation (23) has a certain solution: $m(r,ta)=−c1Ei−cti∗r24ta+c2,$(24)

where Ei is the exponential integral function and Ei(−x) = $-{\int }_{x}^{\mathrm{\infty }}\frac{{e}^{-t}}{t}dt.$

For planar unidirectional flow, (22) can be written as: $∂2m∂x2=cti∗∂m∂ta∗.$(25)

With Laplace transforms, defineL[m(x, ta)] = M(x, s), and (25) can be transformed into: $∂2M(x,s)∂x2=cti∗(sM(x,s)−pe2).$(26)

The solution of (26) can be easily derived: $M(x,s)=c3ecti∗s⋅x+c4e−cti∗s⋅x+pe2s,$(27)

where c3 and c4 are constant, which satisfy the following equations: ${c}_{3}+{c}_{4}+\frac{{p}_{e}^{2}}{s}=0{c}_{4}=\frac{\frac{{p}_{e}^{2}}{s}\left(1-{e}^{\sqrt{{c}_{ti}^{\ast }s}\cdot {r}_{e}}\right)-M\left({r}_{e},s\right)}{{e}^{\sqrt{{c}_{ti}^{\ast }s}\cdot {r}_{e}}-{e}^{-\sqrt{{c}_{ti}^{\ast }s}\cdot {r}_{e}}}$

Then m(r, ta) can be derived with Laplace inverse transforms: $m(r,ta)=L−1[M(x,s)].$(28)

## 4.1 Validation

According to the established mathematical model of coupled flow and geomechanics for vertical fractured well, the simulation was carried out to describe coupled phenomenon.

Basic parameters from one of the actual wells in Sulige tight gas reservoir in China were picked up for simulation as follows: reservoir thickness 20m; fracture halflength 50m; wellbore radius 0.1m; drainage radius 1000m; permeability 0.2mD; porosity 6.5%; pressure at boundary 20MPa; bottom hole and artificial fracture pressure 6Mpa; reservoir temperature 383K; temperature at standard state 293K; gas viscosity 0.01mPa·s; elastic modulus 3GPa; Poisson’s ratio 0.2; and Biot’s coefficient 0.66.

Figure 2 is the comparison between simulation results and actual production data. The results show that the calculation results in terms of coupled flow and geomechanics have a good agreement with actual data from the Sulige gas reservoir in China and uncoupled flow would lead to overestimate the performance of the vertical fractured well especially in the later stage. This means that the present model and approximate analytical solution are more accurate than traditional approaches.

Figure 2

The comparison between simulation results and actual production data

## 4.2 Elastic modulus

Figure 3 shows variations of porosity (a) and permeability (b) with different elastic modulus. Figure 4 shows variations in production rate and cumulative production with different elastic modulus. The results show that the production rate and cumulative production decrease with the decrease of elastic modulus. The elastic modulus has little effect on production rate after 3Gpa. When elastic modulus decreases, porosity and permeability decrease accordingly. So, the geomechanical effect should be taken into consideration for calculation and evaluation in tight gas reservoirs.

Figure 3

Variations of porosity (a) and permeability (b) with different elastic modulus

Figure 4

Variations of production rate and cumulative production with different elastic modulus

Figure 5 shows variations of permeability with different drawdown pressures. As shown, permeability decreases with drawdown pressure and the gap between the current permeability and the initial permeability becomes greater with the time. The drawdown pressure should be optimized to lower the impact of coupled flow and geomechanics.

Figure 5

Variations of permeability with different drawdown pressures

## 4.3 Fracture half-length

The effect of the fracture half-length on production rate and cumulative production is studied considering the geomechanical effect (E = 3Gpa). Figure 6 shows production rate and cumulative production under different fracture half lengths. The results show that production rate increases with fracture half length. And the decline of production rate is fast in the initial stage, then it becomes stable gradually with the time.

Figure 6

Production rate and cumulative production under different fracture half lengths

## 5 Conclusion

We proposed the mathematical model of coupled flow and geomechanics for vertical fractured well in tight gas reservoirs. The analytical modeling of unidirectional flow and radial flow was achieved by Laplace transforms and integral transforms. The influences of elastic modulus and half fracture length to well performances were figured out.

The calculated results have a good agreement with actual production data from the Sulige gas reservoir in China, verifying the models. An uncoupled flow would lead to an overestimate in the performance of a vertical fractured well especially in the later stage. The production rate decreases with elastic modulus because porosity and permeability decreases accordingly. The geomechanical effect should be taken into consideration for calculation and evaluation in tight gas reservoirs. Drawdown pressure should be optimized to lower the impact of coupled flow and geomechanics as a result of permeability decrease. Production rate increases with fracture half-length significantly in the initial stage and becomes stable gradually. This study could a provide theoretical basis for effective development of tight gas reservoirs.

## Acknowledgement

We gratefully acknowledge the PetroChina Innovation Foundation under Grant No. 2015D-5006-0106, the National Nature Science Foundations of China under Grant 51404024 and 51104119 for financial support. We also express gratitude to researchers of the Sulige Gas field for their cooperation.

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Accepted: 2017-09-20

Published Online: 2017-12-29

Citation Information: Open Physics, Volume 15, Issue 1, Pages 797–802, ISSN (Online) 2391-5471,

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