The relationships among the real velocity of gas/water phase ( g *v*

g g *v vv*

gas/water phase under the influence of reservoir skeleton deformation:

Gas phase: g g g g

g

Water phase: w w w w

w

equations, (28) and (29), have two extra parameters, *ml* and *S v*

of gas/water phase relative to the rock skeleton ( *r*g *v*

 *S*

Where, *g*

skeleton ( *s v*

(21), we get:

Because *Kij* is equal to *Kji*

has three independent coefficients.

) are as follows:

physical and chemical seepage process with phases changed; The latter one, *S v lls* , reflects the influence of reservoir skeleton deformation on the seepage field, meanwhile, that the porosity and permeability changed with reservoirs stress state reflects the coupling effect of seepage field and deformation field.

To simplify the questions, we assume that the deformation of reservoir skeleton is a steady deformation with small displacement and ignore the inertia force of rock skeleton. Then the individual derivative is similar to the partial derivative. So:

$$\frac{d(\\_\rangle}{dt \mathbf{x}\_{\text{i}}} = \frac{\partial(\\_\\_)}{\partial \mathbf{x}\_{\text{i}}}, \frac{d(\\_)}{dt} = \frac{\partial(\\_)}{\partial t}, \nabla = \stackrel{\circ}{\mathbf{v}}^{\prime}, \mathbf{v}\_{\text{j}} = \frac{\partial u\_{\text{i}}}{\partial t}, a\_{\text{j}} = \frac{\partial v\_{\text{i}}}{\partial t} \approx 0, \,\,\varepsilon\_{\stackrel{\circ}{\mathbf{i}}j} = \frac{1}{2} (u\_{\text{i}}{\mathbf{i}}{\mathbf{i}}{\mathbf{j}} + u\_{\text{j}}{\mathbf{j}}{\mathbf{j}}) \tag{30}$$

Using Eq. (30), the continuity equation of reservoir rock matrix, namely Eq. (10), can be simplified as follows:

$$\frac{\partial(1-\rho)}{\partial t} + (1-\rho)\nabla \cdot \overline{v}\_{\rm s} = 0 \tag{31}$$

According to the solid small deformation theory, we get:

$$\nabla \cdot \vec{v}\_S = \nabla \cdot \left(\frac{\partial \vec{u}}{\partial t}\right) = \frac{\partial}{\partial t} (\nabla \cdot \vec{u}) = \frac{\partial}{\partial t} \left(\frac{\partial u\_{\infty}}{\partial x} + \frac{\partial u\_{\text{y}}}{\partial y} + \frac{\partial u\_{\text{zz}}}{\partial z}\right) = \frac{\partial}{\partial t} \left(\varepsilon\_{\text{xxx}} + \varepsilon\_{\text{yy}} + \varepsilon\_{\text{zz}}\right) = \frac{\partial \varepsilon\_{\text{xy}}}{\partial t} \tag{32}$$

Where, *uu u u* , , *x y z* is the displacement of rock skeleton, and *v* is the volume strain of rock skeleton.

Substituting Eq. (32) into Eq. (31), the continuity equation of rock skeleton can be got as follows:

$$\frac{\partial \mathcal{Q}}{\partial t} + \mathcal{Q} \frac{\partial \mathcal{E}\_{\mathcal{U}}}{\partial t} = \frac{\partial \mathcal{E}\_{\mathcal{U}}}{\partial t} \tag{33}$$

The third term of the left of Eq. (28) and Eq. (29) can be transformed as follows:

$$\nabla \cdot \left( \rho \rho \Box\_{\mathbb{S}} S\_{\mathbb{S}} \bar{v}\_{\mathbb{S}} \right) = \nabla \cdot \left( \rho \rho \Box\_{\mathbb{S}} S\_{\mathbb{S}} \right) \bar{v}\_{\mathbb{s}} + \left( \rho \rho \Box\_{\mathbb{S}} S\_{\mathbb{S}} \right) \nabla \cdot \bar{v}\_{\mathbb{S}} \tag{34}$$

 *wws ww ww Sv S v S v s s* (35)

The first terms of the right side of Eqs. (34) and (35) are fluid convection flow along with matrix displacement. According to the small displacement theory, these can be ignored. The gas/water two-phase seepage equations (28) and (29) can be simplified further as follows:

Gas phase: g g g g g g gg g *S Kr K P g S vm q s g g t* (36)

Fluid-Solid Coupling Numerical Simulation

g g

g

And: *w wS Sw w S*

then use Eq.(46) to simplify them and get the following equation.

 *w S*

condition must be satisfied in continuous domain.

1. Boundary flow continuity

area on bounding surface.

2. Constant pressure boundary condition

variables.

Where, *n*

on Natural Gas Production from Hydrate Reservoirs by Depressurization 157

*v g gS S w w qq mm S S gw g w gg ww t t*

g

 *Krg g K K rw w rw w K K gg K Pc gw w*

*t tt w w*

*v w S K w rw w*

Eqs. (45) and (47) are the general form of fluid-solid coupling seepage equations of gas hydrate reservoir with gas phase pressure and water saturation as their basic solution

Initial and boundary conditions should be complemented to solve seepage field equations of fluid-solid coupling model. Also, for flow equations, the continuity condition and boundary

g g

The reservoir boundary pressure or bottom-hole pressure are known as following[6]:

g g *Kr n K P gq*

0 *w rw K n K P gq w w w*

is bounding surface unit vector; g*q* and *wq* are gas and water phase flow per unit

*w w*

*Krw w q m KP g w w w*

**2.2.4 Initial and boundary conditions of seepage field equations**

 *w*

Substitute Eq.(44) into Eq.(41) and change the water phase pressure into gas phase pressure,

*KP S c w tt t w*

g g

g g 0

(48)

(49)

*P f xyzt* ,,, *G P* (50)

(46)

'

(45)

(47)

  *Kr Krw w K KP w*

Water phase: *w wS Krw w t KP g S vm q w w ww s ww w* (37)

From Eqs. (32) and (33), the follow relationship can be deduced:

$$\frac{\partial \left(\rho \rho\_{\mathcal{G}} S\_{\mathcal{G}}\right)}{\partial t} = \rho\_{\mathcal{G}} S\_{\mathcal{G}} \frac{\partial \rho}{\partial t} + \rho \frac{\left(\rho\_{\mathcal{G}} S\_{\mathcal{G}}\right)}{\partial t} = \rho\_{\mathcal{G}} S\_{\mathcal{G}} \left(\frac{\partial \varepsilon\_{\mathcal{U}}}{\partial t} - \rho \frac{\partial \varepsilon\_{\mathcal{U}}}{\partial t}\right) + \rho \frac{\left(\rho\_{\mathcal{G}} S\_{\mathcal{G}}\right)}{\partial t} \tag{38}$$

$$(\rho \rho\_{\mathbf{g}} S\_{\mathbf{g}}) \nabla \cdot \bar{v}\_{\mathbf{s}} = \rho \rho\_{\mathbf{g}} S\_{\mathbf{g}} \frac{\partial \varepsilon\_{\mathbf{U}}}{\partial t} \tag{39}$$

Using Eqs. (33), (38) and (39) to simplify Eq. (36), we can get:

$$\text{Gas phase:} \qquad \rho\_{\mathcal{S}} S\_{\mathcal{S}} \frac{\partial \varepsilon\_{\mathcal{U}}}{\partial t} + \rho \frac{\partial \left(\rho\_{\mathcal{S}} S\_{\mathcal{S}}\right)}{\partial t} - \nabla \cdot \left(\frac{K\_{\text{rg}} \rho\_{\mathcal{S}}}{\mu\_{\mathcal{S}}} [K] \left(\nabla P\_{\mathcal{S}} + \rho\_{\mathcal{S}} \bar{g}\right)\right) = m\_{\mathcal{S}} + q\_{\mathcal{S}} \tag{40}$$

Water phase: *v w wS Krw w S K Pw wg m q w w t t w w w* (41)

Plus Eq. (40) and Eq.(41), then it can be got:

$$\begin{split} & -\nabla \cdot \left( \frac{K\_{\text{reg}} \rho\_{\text{g}}}{\mu\_{\text{g}}} [K] \left( \nabla P\_{\text{g}} + \rho\_{\text{g}} \bar{\varrho} \right) \right) - \nabla \cdot \left( \frac{K\_{\text{Tw}} \rho\_{\text{w}}}{\mu\_{\text{w}}} [K] \left( \nabla P\_{\text{w}} + \rho\_{\text{w}} \bar{\varrho} \right) \right) = \\ & = q\_{\text{g}} + q\_{\text{w}} + m\_{\text{g}} + m\_{\text{w}} - \left( \rho\_{\text{g}} S\_{\text{g}} + \rho\_{\text{w}} S\_{\text{w}} \right) \frac{\partial \varepsilon\_{\text{U}}}{\partial t} - \rho \frac{\partial \left( \rho\_{\text{g}} S\_{\text{g}} + \rho\_{\text{w}} S\_{\text{w}} \right)}{\partial t} \end{split} \tag{42}$$

The relationship between gas phase pressure *P*g and water phase pressure *P*w is as follows:

$$P\_w = P\_g - P\_c + \left(\rho\_g - \rho\_w\right) g\_i h\_i \tag{43}$$

Where, *P*c is capillary pressure, which is the function of water saturation.

The derivation of Eq.(43) is:

$$
\nabla P\_{\rm w} = \nabla P\_{\rm g} - \nabla P\_{\rm c} + \left(\rho\_{\rm g} - \rho\_{\rm w}\right)\bar{\mathbf{g}} = \nabla P\_{\rm g} - P\_{\rm c}^{\ \ \text{'}} \nabla S\_{\rm w} + \left(\rho\_{\rm g} - \rho\_{\rm w}\right)\bar{\mathbf{g}}\tag{44}
$$

Where, ' *P PS c cw* , and ' 0 *Pc P c Sw* , which can be determined by capillary pressure curve.

Substitute Eq.(44) into Eq.(42) and simplify it, we get a pressure equation with gas phase pressure as its solution variables.

$$-\nabla \cdot \left( \left( \frac{K\_{\rm r\overline{g}} \rho\_{\overline{g}}}{\mu\_{\overline{g}}} [K] + \frac{K\_{\rm rw} \rho\_{\overline{w}}}{\mu\_{\overline{w}}} [K] \right) \nabla P\_{\overline{g}} \right) = $$

$$= q\_{\mathcal{S}} + q\_{\mathcal{U}} + m\_{\mathcal{S}} + m\_{\mathcal{w}} - \left( \rho\_{\mathcal{S}} S\_{\mathcal{S}} + \rho\_{\mathcal{W}} S\_{\mathcal{w}} \right) \frac{\partial \varepsilon\_{\mathcal{V}}}{\partial t} - \rho \frac{\partial \left( \rho\_{\mathcal{S}} S\_{\mathcal{G}} + \rho\_{\mathcal{W}} S\_{\mathcal{W}} \right)}{\partial t}$$

$$+ \nabla \cdot \left( \left( \frac{K\_{\rm r\underline{\mathcal{S}}} \rho\_{\mathcal{S}}}{\mu\_{\mathcal{S}}} [K] + \frac{K\_{\rm rw} \rho\_{\mathcal{W}}}{\mu\_{\mathcal{W}}} [K] \right) \rho\_{\mathcal{S}} \bar{g} \right) - \nabla \cdot \left( \frac{K\_{\rm rw} \rho\_{\mathcal{W}}}{\mu\_{\mathcal{W}}} [K] \nabla P\_{\mathcal{C}} \right) \tag{45}$$

$$\mathcal{H} = -\mathcal{S}\_{\rm r} \tag{46}$$

g g

*SS S g g g g v v S S t t t tt t*

*w*

*w*

*Kr Krw w KP g KP g w w*

*v g gS S w w qq mm S S gw g w gg ww t t*

The relationship between gas phase pressure *P*g and water phase pressure *P*w is as

*P PP w g c g w ii*

 ' *P PP* g g g g *w w g P PS g w c c w*

Substitute Eq.(44) into Eq.(42) and simplify it, we get a pressure equation with gas phase

*t KP g S vm q w w ww s ww*

>

, which can be determined by capillary pressure

*g h* (43)

(39)

(38)

(37)

(40)

(41)

(42)

(44)

Water phase: *w wS Krw w*

Using Eqs. (33), (38) and (39) to simplify Eq. (36), we can get:

From Eqs. (32) and (33), the follow relationship can be deduced:

>

g g

g

Plus Eq. (40) and Eq.(41), then it can be got:

The derivation of Eq.(43) is:

*c cw*

pressure as its solution variables.

, and ' 0 *Pc P*

Where, ' *P PS*

curve.

follows:

*w*

g g g g

Gas phase: *g gS Krg g v S K P g m q g g g g g g t t g*

Water phase: *v w wS Krw w S K Pw wg m q w w t t w w*

g g

Where, *P*c is capillary pressure, which is the function of water saturation.

*c Sw*

 g g g g *v Sv S s t*

$$\text{And:}\tag{4}\tag{4}\tag{5}\tag{6}\tag{6}\tag{6}\tag{6}\tag{6}\tag{6}\tag{7}\tag{7}\tag{6}\tag{7}\tag{10}\tag{10}\tag{10}\tag{11}\tag{11}\tag{11}\tag{11}\tag{11}\tag{12}\tag{12}\tag{12}\tag{12}\tag{12}\tag{13}\tag{13}\tag{13}\tag{13}\tag{13}\tag{13}\tag{13}\tag{14}\tag{14}\tag{14}\tag{14}\tag{15}\tag{15}\tag{15}\tag{15}\tag{15}\tag{16}\tag{16}\tag{16}\tag{16}\tag{16}\tag{17}\tag{17}\tag{17}\tag{17}\tag{17}\tag{18}\tag{18}\tag{18}\tag{18}\tag{19}\tag{19}\tag{19}\tag{11}\tag{11}\tag{11}\tag{11}\tag{11}\tag{11}\tag{11}\tag{11}\tag{11}\tag{12}\tag{12}\tag{12}\tag{12}\tag{13}\tag{13}\tag{13}\tag{13}\tag{13}\tag{14}\tag{14}\tag{14}\tag{14}\tag{15}\tag{15}\tag{15}\tag{15}\tag{15}\tag{15}\tag{16}\tag{16}\tag{16}\tag{16}\tag{17}\tag{17}\tag{17}\tag{17}\tag{17}\tag{17}\tag{18}\tag{18}\tag{18}\tag{18}\tag{19}\tag{19}\tag{19}\tag{19}\tag{19}\tag{11}\tag{11}\tag{11}\tag{11}\tag{11}\tag{11}\tag{11}\tag{11}\tag{11}\tag{12}\tag{12}\tag{12}\tag{12}\tag{12}\tag{13}\tag{13}\tag{13}\tag{13}\tag{13}\tag{13}\tag$$

Substitute Eq.(44) into Eq.(41) and change the water phase pressure into gas phase pressure, then use Eq.(46) to simplify them and get the following equation.

 ' g g *v w S K w rw w w S w w KP S c w tt t w Krw w q m KP g w w w* (47)

Eqs. (45) and (47) are the general form of fluid-solid coupling seepage equations of gas hydrate reservoir with gas phase pressure and water saturation as their basic solution variables.

#### **2.2.4 Initial and boundary conditions of seepage field equations**

Initial and boundary conditions should be complemented to solve seepage field equations of fluid-solid coupling model. Also, for flow equations, the continuity condition and boundary condition must be satisfied in continuous domain.

1. Boundary flow continuity

$$-\bar{n}\frac{\rho\_{\text{g}}K\_{r\text{g}}}{\mu\_{\text{g}}}[K]\Big(\nabla P\_{\text{g}}+\rho\_{\text{g}}\bar{g}\Big)-\overline{\eta}\_{\text{g}}=0\tag{48}$$

$$-i\hbar \frac{\rho\_W K\_{rw}}{\mu\_W} [K] (\nabla P\_{\bar{w}} + \rho\_{\bar{w}} \bar{\varrho}) - \overline{q}\_{\bar{w}} = 0 \tag{49}$$

Where, *n* is bounding surface unit vector; g*q* and *wq* are gas and water phase flow per unit area on bounding surface.

2. Constant pressure boundary condition

The reservoir boundary pressure or bottom-hole pressure are known as following[6]:

$$P\_{\mathbf{G}} = f\_{\mathbf{P}}(\mathbf{x}, y, z, t) \tag{50}$$

Fluid-Solid Coupling Numerical Simulation

Where, 111000 *T ij*

.

Terzaghi effective stress formula is combined with:

1/ *K K*

calculated by the formula 1 *SSg*

**2.3.2 Geometric equations of reservoir skeleton**

**2.3.3 Constitutive equations of reservoir skeleton**

**<IP_ADDRESS> Linear elastic constitutive equations**

follows:

on Natural Gas Production from Hydrate Reservoirs by Depressurization 159

As the total stress supported by formation are composed of pore pressure and skeleton stress which is known as skeleton effective stress. The deformation and strength properties

We appoint that the tensile stress is positive and the compressive stress is negative. The

*T P ij ij ij*

(55)

(56)

(57)

(58)

(59)

(60)

As to water-gas two-phase flow, the calculation of equivalent pore pressure is as follows:

g g *P PS P S*

Substituting the modified effective stress equation (55) into Eq. (54), we can obtain the fluidsolid coupling deformation equilibrium equations based on the effective stress, which are

0 , , *f Pij ij j i j*

Where, *f*i is the gravity term, in which *f*x is equal to *f*y, and the value of them is zero, *f*z is

Based on the solid small deformation theory, the relationship between strain component and displacement component can be described by following geometric equations with tensor form:

For linear elastic media, stress and strain have a linear corresponding relation. The linear elastic constitutive relation can be expressed by generalized Hook's law with matrix form as

*D ij e ij*

 1 , , 2 *u u i j j i ij*

Zinkiewicz pointed that, for saturated fluid, the Biot coefficient can be calculated by:

Where, *K* is rock volume modulus, and *Ks* is rock skeleton volume modulus.

the basic equations to solve the porous media deformation field.

 *s oo ww* .

*w w*

*s*

of reservoirs are controlled by the skeleton effective stress instead of the total stress.

Where, *f xyzt* ,,, *P* is the given pressure function on boundary G at moment t.

3. Constant flow boundary condition

This means that the pressure derivative of the reservoir external boundary or the production in internal boundary is known, as following:

$$\frac{\partial P}{\partial \mathbf{n}}\Big|\_{\mathbf{G}} = f\_q(\mathbf{x}, y, z, t) \tag{51}$$

Where, *n* is the direction of normal line; *f x*,,, *y z t q* is the known function on specify boundary.

#### 4. Initial condition

The initial conditions of seepage field are mainly the initial pressure and saturation of gas hydrate reservoirs, as following[6]:

$$P(\mathbf{x}, y, z)|\_{\mathbf{f} = \mathbf{0}} = P\_I(\mathbf{x}, y, z) \tag{52}$$

$$S|\_{\mathfrak{f}=\mathbf{0}} = S\_{\mathbf{f}}(\mathbf{x}, \mathbf{y}, \mathbf{z}) \tag{53}$$

Where, *Pl*(*x*, *y*, *z*) is the known pressure function, and *Sl*(*x*, *y*, *z*) is the known saturation function.

The water-gas two-phase fluid-solid coupling seepage equations of gas hydrate reservoirs, namely Eqs. (47), can't be solved separately. Several simultaneous equations should be cited, including the kinetic equation of gas hydrates decomposition (1), the continuity equation solid phase hydrate (22), temperature field equations (17) and fluid-solid coupling deformation field equation in the next section. What's more, relevant state equation, assistant equations and the initial and boundary conditions, etc still need to be supplemented.

#### **2.3 Deformation field equations of fluid-solid coupling model**

The unknown parameters or unknown item in the seepage equations are affected by rock skeleton deformation, so the fluid-solid coupling seepage equations have highly nonlinear characteristics, and need to simultaneity the deformation field equations to solve the question. For porous media deformation field, the basic component equations mainly include equilibrium equation, geometric equation and constitutive equation, and the corresponding definite conditions. In this portion, the porous media deformation field mathematical model is constructed.

#### **2.3.1 Equilibrium equations of reservoir skeleton**

On the basis of the conventional force equilibrium principle of cell cube, we can get the deformation field balance equation with tensor form as follows:

$$
\sigma\_{\stackrel{\frown}{\dot{\text{ij}}},\stackrel{\frown}{\dot{\text{j}}}}^T + f\_{\stackrel{\frown}{\dot{\text{i}}}} = 0 \tag{54}
$$

As the total stress supported by formation are composed of pore pressure and skeleton stress which is known as skeleton effective stress. The deformation and strength properties of reservoirs are controlled by the skeleton effective stress instead of the total stress.

We appoint that the tensile stress is positive and the compressive stress is negative. The Terzaghi effective stress formula is combined with:

$$
\sigma\_{\stackrel{\circ}{ij}} T = \sigma\_{\stackrel{\circ}{ij}} - \alpha \overline{P} \delta\_{\stackrel{\circ}{ij}} \tag{55}
$$

Where, 111000 *T ij* .

158 Advances in Natural Gas Technology

This means that the pressure derivative of the reservoir external boundary or the production

 ,,, *P G f xyzt q n*

Where, *n* is the direction of normal line; *f x*,,, *y z t q* is the known function on specify

The initial conditions of seepage field are mainly the initial pressure and saturation of gas

Where, *Pl*(*x*, *y*, *z*) is the known pressure function, and *Sl*(*x*, *y*, *z*) is the known saturation

The water-gas two-phase fluid-solid coupling seepage equations of gas hydrate reservoirs, namely Eqs. (47), can't be solved separately. Several simultaneous equations should be cited, including the kinetic equation of gas hydrates decomposition (1), the continuity equation solid phase hydrate (22), temperature field equations (17) and fluid-solid coupling deformation field equation in the next section. What's more, relevant state equation, assistant equations and the initial and boundary conditions, etc still need to be

The unknown parameters or unknown item in the seepage equations are affected by rock skeleton deformation, so the fluid-solid coupling seepage equations have highly nonlinear characteristics, and need to simultaneity the deformation field equations to solve the question. For porous media deformation field, the basic component equations mainly include equilibrium equation, geometric equation and constitutive equation, and the corresponding definite conditions. In this portion, the porous media deformation field

On the basis of the conventional force equilibrium principle of cell cube, we can get the

*T f ij j i*

**2.3 Deformation field equations of fluid-solid coupling model**

(51)

*P xyz P xyz* , , *t* 0 , , *l* (52)

*S S xyz t* 0 , , *l* (53)

(54)

Where, *f xyzt* ,,, *P* is the given pressure function on boundary G at moment t.

3. Constant flow boundary condition

boundary.

function.

supplemented.

4. Initial condition

hydrate reservoirs, as following[6]:

mathematical model is constructed.

**2.3.1 Equilibrium equations of reservoir skeleton**

0 ,

deformation field balance equation with tensor form as follows:

in internal boundary is known, as following:

As to water-gas two-phase flow, the calculation of equivalent pore pressure is as follows:

$$
\overline{P} = P\_{\mathbf{g}} S\_{\mathbf{g}} + P\_{w} S\_{w} \tag{56}
$$

Zinkiewicz pointed that, for saturated fluid, the Biot coefficient can be calculated by:

$$a = 1 - K / K\_{\rm s} \tag{57}$$

Where, *K* is rock volume modulus, and *Ks* is rock skeleton volume modulus.

Substituting the modified effective stress equation (55) into Eq. (54), we can obtain the fluidsolid coupling deformation equilibrium equations based on the effective stress, which are the basic equations to solve the porous media deformation field.

$$\left(\sigma\_{\vec{i}\vec{j},\vec{j}} + f\_{\vec{i}} - \left(a\delta\delta\_{\vec{l}\vec{j}}\overline{P}\right)\_{\prime\dot{\vec{j}}}\right)\_{\prime\dot{\vec{j}}} = 0 \tag{58}$$

Where, *f*i is the gravity term, in which *f*x is equal to *f*y, and the value of them is zero, *f*z is calculated by the formula 1 *SSg s oo ww* .

#### **2.3.2 Geometric equations of reservoir skeleton**

Based on the solid small deformation theory, the relationship between strain component and displacement component can be described by following geometric equations with tensor form:

$$
\varepsilon\_{\rm ij} = \frac{1}{2} (u\_{\hat{\mathbf{i}}, \hat{f}} + u\_{\hat{\mathbf{j}}, \hat{\mathbf{i}}}) \tag{59}
$$

#### **2.3.3 Constitutive equations of reservoir skeleton**

#### **<IP_ADDRESS> Linear elastic constitutive equations**

For linear elastic media, stress and strain have a linear corresponding relation. The linear elastic constitutive relation can be expressed by generalized Hook's law with matrix form as follows:

$$
\sigma\_{\text{ij}} = D\_{\text{e}} \varepsilon\_{\text{ij}} \tag{60}
$$

Fluid-Solid Coupling Numerical Simulation

material in the engineering territory. 1. Modified Mohr-Coulomb criterion

*F I*

2. Drucker-Prager yield criterion

criterion in this research.

second stress deviator *J*2 and Lode stress angle *θ*σ, as follows:

1 *I*

*arctg*

The expression of the Drucker-Prager yield criterion is as follows:

invariant as follows:

Where:

stresses.

on Natural Gas Production from Hydrate Reservoirs by Depressurization 161

Rock is friction type material, the plastic yield of which need to consider both the shear yield and the volume strain yield. Therefore, the yield function can be expressed using the stress

Currently the Mohr-Coulomb criterion and the Drucker-Prager criterion are used for rock

The Mohr-Coulomb criterion can be deduced using the first stress tensor invariant *I*1, the

1 1 sin cos sin sin cos 0 1 2 3 3

*xyz*

1 2 2 2 222 6 2 6 *J*

 *x y y z z x xy yz zx*

>

Where, *C* is the cohesion of rock; *φ* is internal friction angle; σ1,σ2,σ3 are three principle

0 *FI Jk*

Where, *α* and *K* are the yield function parameters, the expressions of which are as follows:

Comparing the two yield criterions above, the Mohr-Coulomb criterion neglects the effects of the middle principle stress *σ*2, and needs to determine the magnitude of each principle stress when it is used, which is not convenient. So we adopt the Drucker-Prager yield

Assume that the space region occupied by rock skeleton is *d* (the value of *d* is two or three), and the boundary of the region is , in which the displacement boundary is *u*

sin 2 9 3sin

**2.3.4 Initial and boundary conditions of deformation field equations**

1 2 3 2 1 3 1 3

; 3 cos

*c k*

2 9 3sin

*J C* (67)

1 2 (71)

*FI J J* 123 ,, 0 (66)

(68)

(70)

(72)

(69)

In the above equation, *De* is the elastic constitutive matrix, which has 36 elements in 3 D coordinate. Elastic mechanics has proved it is symmetric matrices with only 21 independent constant. For the isotropic orthogonal media, there are only two independent elastic material constants, with elastic modulus *E* and Poisson's ratio *ν* signified.

Then the elastic constitutive matrix of rock skeleton can be expressed as:

$$[D\_{\mathcal{C}}] = \frac{E(1-\nu)}{(1+\nu)(1-2\nu)} \begin{vmatrix} 1 \\ \frac{\nu}{1-\nu} & 1 \\ \frac{\nu}{1-\nu} & \frac{\nu}{1-\nu} & 1 \\ 0 & 0 & 0 & \frac{1-2\nu}{2(1-\nu)} \end{vmatrix} \tag{61}$$

$$\begin{vmatrix} 0 & 0 & 0 & 0 & \frac{1-2\nu}{2(1-\nu)} \\ 0 & 0 & 0 & 0 & 0 & \frac{1-2\nu}{2(1-\nu)} \end{vmatrix} \tag{62}$$

#### **<IP_ADDRESS> Elastoplastic constructive equations**

For elastic-plastic deformation, the strain is not only related to the current stress, but also to the loading history, loaded/unloaded condition, loading paths and microscopic structure of rock deformation, and so on.

In this study, the elastoplastic constructive relation is adopted to describe the relationship between strain and stress, which is expressed by the incremental form as follows:

$$d\sigma\_{\rm ij} = D\_{\rm ijkl} d\varepsilon\_{\rm kl} \tag{62}$$

Eq. (62) can be represented by the matrix form as follows:

$$\{d\sigma\} = \left[D\_{\mathcal{C}\mathcal{P}}\right] \{d\varepsilon\} \tag{63}$$

The elastoplastic matrix can be calculated by the following equation.

$$
\begin{bmatrix} D\_{\ell p} \end{bmatrix} = \begin{bmatrix} D\_{e} \end{bmatrix} - \begin{bmatrix} D\_{p} \end{bmatrix} \tag{64}
$$

The elastic matrix can be got by Eq. (61), and the plastic matrix can be deduced from the elastic-plastic associative flow rule, as follows:

$$
\begin{bmatrix} D\_{\mathcal{P}} \end{bmatrix} = \frac{\begin{bmatrix} D\_{\mathcal{E}} \end{bmatrix} \begin{Bmatrix} \frac{\partial F}{\partial \sigma} \end{Bmatrix} \begin{Bmatrix} \frac{\partial F}{\partial \sigma} \end{Bmatrix}^{T} \begin{bmatrix} D\_{\mathcal{E}} \end{bmatrix}}{A + \begin{Bmatrix} \frac{\partial F}{\partial \sigma} \end{Bmatrix}^{T} \begin{bmatrix} D\_{\mathcal{E}} \end{bmatrix} \begin{Bmatrix} \frac{\partial F}{\partial \sigma} \end{Bmatrix}} \tag{65}
$$

Where, *A* is the hardening index, and *F* is the yield function.

Rock is friction type material, the plastic yield of which need to consider both the shear yield and the volume strain yield. Therefore, the yield function can be expressed using the stress invariant as follows:

$$F(I\_1\mathbf{1}\_2\mathbf{1}\_2\mathbf{1}\_3) = 0\tag{66}$$

Currently the Mohr-Coulomb criterion and the Drucker-Prager criterion are used for rock material in the engineering territory.

#### 1. Modified Mohr-Coulomb criterion

The Mohr-Coulomb criterion can be deduced using the first stress tensor invariant *I*1, the second stress deviator *J*2 and Lode stress angle *θ*σ, as follows:

$$F = \frac{1}{3} I\_1 \sin \phi + \left( \cos \theta\_\sigma - \frac{1}{\sqrt{3}} \sin \theta\_\sigma \sin \phi \right) \sqrt{f\_2} - \mathcal{C} \cos \phi = 0 \tag{67}$$

Where:

160 Advances in Natural Gas Technology

In the above equation, *De* is the elastic constitutive matrix, which has 36 elements in 3 D coordinate. Elastic mechanics has proved it is symmetric matrices with only 21 independent constant. For the isotropic orthogonal media, there are only two independent elastic

material constants, with elastic modulus *E* and Poisson's ratio *ν* signified. Then the elastic constitutive matrix of rock skeleton can be expressed as:

1

1

1 1 1 1 2 0 00 1 12 2 1

1

>

1

For elastic-plastic deformation, the strain is not only related to the current stress, but also to the loading history, loaded/unloaded condition, loading paths and microscopic structure of

In this study, the elastoplastic constructive relation is adopted to describe the relationship

*d Dd ij ijkl kl*

*d Dd*

The elastic matrix can be got by Eq. (61), and the plastic matrix can be deduced from the

*Dp T F F A De*

*T F F D D e e*

between strain and stress, which is expressed by the incremental form as follows:

*ep*

The elastoplastic matrix can be calculated by the following equation.

1 2 0 00 0

1 2 0 00 0 0

2 1

(62)

(65)

(63)

*D DD ep e p* (64)

(61)

2 1

Eq. (62) can be represented by the matrix form as follows:

Where, *A* is the hardening index, and *F* is the yield function.

elastic-plastic associative flow rule, as follows:

*E De*

**<IP_ADDRESS> Elastoplastic constructive equations**

rock deformation, and so on.

$$I\_1 = \sigma\_{\chi} + \sigma\_{y} + \sigma\_{z} \tag{68}$$

$$J\_{2} = \frac{1}{6} \left[ \left( \sigma\_{\text{X}} - \sigma\_{\text{Y}} \right)^{2} + \left( \sigma\_{\text{Y}} - \sigma\_{\text{Z}} \right)^{2} + \left( \sigma\_{\text{Z}} - \sigma\_{\text{X}} \right)^{2} + 6 \left( \tau\_{\text{XY}} \, ^{2} + \tau\_{\text{Y}\mathcal{Z}} \, ^{2} + \tau\_{\text{Z}\mathcal{X}} \, ^{2} \right) \right] \tag{69}$$

$$\theta\_{\sigma} = \operatorname{arctg}\left[\frac{1}{\sqrt{3}} \left(2\frac{\sigma\_{2} - \sigma\_{3}}{\sigma\_{1} - \sigma\_{3}} - 1\right)\right] \tag{70}$$

Where, *C* is the cohesion of rock; *φ* is internal friction angle; σ1,σ2,σ3 are three principle stresses.

2. Drucker-Prager yield criterion

The expression of the Drucker-Prager yield criterion is as follows:

$$F = \alpha I\_1 - \sqrt{I\_2} + k = 0\tag{71}$$

Where, *α* and *K* are the yield function parameters, the expressions of which are as follows:

$$\alpha = \frac{\sin \phi}{\sqrt{9 + 3 \sin^2 \phi}} \; \because \; k = \frac{3c \cos \phi}{\sqrt{9 + 3 \sin^2 \phi}} \tag{72}$$

Comparing the two yield criterions above, the Mohr-Coulomb criterion neglects the effects of the middle principle stress *σ*2, and needs to determine the magnitude of each principle stress when it is used, which is not convenient. So we adopt the Drucker-Prager yield criterion in this research.

#### **2.3.4 Initial and boundary conditions of deformation field equations**

Assume that the space region occupied by rock skeleton is *d* (the value of *d* is two or three), and the boundary of the region is , in which the displacement boundary is *u*

Fluid-Solid Coupling Numerical Simulation

*co* , 0.45 *m* , 0.3 *S*

Fig. 2-1. Curve of gas/water relative permeability

Fig. 2-2. Capillary pressure curve

equation[11-13].

**2.4.2 Thermal physical parameters**

**<IP_ADDRESS> Effective coefficient of heat conduction**

 (1 ) ( *K K S K SK S K* ) *c r HH g g w w*

Where, 1 *P KPa*

on Natural Gas Production from Hydrate Reservoirs by Depressurization 163

When the saturation of hydrates is 0.4, the relative permeability curve of gas and water

The coefficient of heat conduction for gas hydrate reservoirs is related to the component. Assume that the heat conductions coefficient of rock skeleton, hydrates, water and gas are constant, so the effective coefficient of heat conduction *K*c can be calculated by the following

*gr* , 0.5 *K*

(82)

(83)

*rgo* .

(84)

*rwo* , 1.0 *K*

1 *SSS w H wr SwH S S wr gr*

1 1 *SS S w H gr S g S S wr gr*

*wr* , 0.05 *S*

phase and the capillary pressure curve are shown as Figs. 2-1 and 2-2.

and the stress boundary is , what's more, *u* and 0 *u* . So we get the following boundary conditions:

The known stress boundary: *T n i ij j X* (73)

The known displacement boundary: *u ui i u X* (74)

If the effective stress is adopted, so the expressions are:

$$\text{Stress boundary:} \qquad \qquad T\_{\stackrel{\circ}{\cdot}} = \left( \sigma\_{\stackrel{\circ}{\cdot\cdot}} \! -\alpha \overline{P} \delta\_{\stackrel{\circ}{\cdot\cdot}} \right) n\_{\stackrel{\circ}{\cdot}} \quad \forall \overline{X} \in \Gamma^{\underline{\sigma}} \tag{75}$$

Displacement boundary: *u ui i u X* (76)

#### **2.4 Basic property parameters calculation of gas hydrate reservoirs**

#### **2.4.1 Permeability and capillary pressure**

The effective permeability of gas hydrate reservoir is the function of hydrate saturation, and it meets the model proposed by Masuda (1997)[7,8], as follows:

$$K = K\_0 \left(1 - S\_H\right)^N \tag{77}$$

Where, *K*0 is the absolute permeability of the formation, of which the hydrate saturation is zero; *SH* is the hydrate saturation; N is the decline exponent of permeability, which is related to the types of hydrates combined in porous space, and the value is 2~15.

The relative permeability and capillary pressure curve can be obtained from the modified models proposed by Van Genuchten (1980)[9] and Parker (1987)[10].

The relative permeability of the water phase:

$$\mathbf{K}\_{rw} = \mathbf{K}\_{rwo} \overline{\mathbf{S}}\_w^{1/2} \left[ \mathbf{1} \text{--} (\mathbf{1} - \overline{\mathbf{S}}\_w^{1/m})^m \right]^2 \tag{78}$$

The relative permeability of the gas phase:

$$\mathbf{K}\_{r\mathbf{g}} = \mathbf{K}\_{r\mathbf{g}\sigma} \overline{\mathbf{S}}\_{\mathbf{g}}^{1/2} \left[ 1 - (1 - \overline{\mathbf{S}}\_{wH}^{1/m})^m \right]^{2m} \tag{79}$$

The capillary pressure between liquid phase and gas phase:

$$P\_e = P\_{co} \left[ (\overline{S}\_w)^{-1/m} - 1 \right]^{1-m} \tag{80}$$

Where:

$$\overline{S}\_{w} = \frac{S\_{w} - S\_{wr}}{1 - S\_{wr} - S\_{gr}} \tag{81}$$

*X*

*X*

The known displacement boundary: *u ui i u X* (74)

Displacement boundary: *u ui i u X* (76)

The effective permeability of gas hydrate reservoir is the function of hydrate saturation, and

(1 ) 0

Where, *K*0 is the absolute permeability of the formation, of which the hydrate saturation is zero; *SH* is the hydrate saturation; N is the decline exponent of permeability, which is related

The relative permeability and capillary pressure curve can be obtained from the modified

2 1/2 1/ 1 (1 ) *m m K KS S rw rwo w w*

*m m K KS S rg rgo g wH*

> *m PP S e co w*

> > 1 *S S w wr S w S S wr gr*

2 1/2 1/ 1 (1 )

1 1/ () 1

and 0 *u*

(73)

(75)

*N KK SH* (77)

(78)

*m*

(79)

(80)

(81)

*m*

. So we get the

, what's more, *u*

and the stress boundary is

following boundary conditions:

The known stress boundary: *T n i ij j*

Stress boundary: ' *T Pn ij ij i j*

**2.4.1 Permeability and capillary pressure**

The relative permeability of the water phase:

The relative permeability of the gas phase:

Where:

If the effective stress is adopted, so the expressions are:

**2.4 Basic property parameters calculation of gas hydrate reservoirs**

to the types of hydrates combined in porous space, and the value is 2~15.

models proposed by Van Genuchten (1980)[9] and Parker (1987)[10].

The capillary pressure between liquid phase and gas phase:

it meets the model proposed by Masuda (1997)[7,8], as follows:

$$\overline{S}\_{wH} = \frac{S\_{w} + S\_{H} - S\_{wr}}{1 - S\_{wr} - S\_{gr}} \tag{82}$$

$$\overline{S}\_{\mathcal{g}} = \frac{1 - S\_{\mathcal{w}} - S\_H - S\_{\mathcal{gr}}}{1 - S\_{\mathcal{w}\mathcal{r}} - S\_{\mathcal{gr}}} \tag{83}$$

Where, 1 *P KPa co* , 0.45 *m* , 0.3 *S wr* , 0.05 *S gr* , 0.5 *K rwo* , 1.0 *K rgo* .

When the saturation of hydrates is 0.4, the relative permeability curve of gas and water phase and the capillary pressure curve are shown as Figs. 2-1 and 2-2.

Fig. 2-1. Curve of gas/water relative permeability

Fig. 2-2. Capillary pressure curve

#### **2.4.2 Thermal physical parameters**

#### **<IP_ADDRESS> Effective coefficient of heat conduction**

The coefficient of heat conduction for gas hydrate reservoirs is related to the component. Assume that the heat conductions coefficient of rock skeleton, hydrates, water and gas are constant, so the effective coefficient of heat conduction *K*c can be calculated by the following equation[11-13].

$$\boldsymbol{K}\_c = (\mathbf{1} - \boldsymbol{\rho})\mathbf{K}\_r + \boldsymbol{\rho}(\mathbf{S}\_H \mathbf{K}\_H + \mathbf{S}\_g \mathbf{K}\_g + \mathbf{S}\_w \mathbf{K}\_w) \tag{84}$$

Fluid-Solid Coupling Numerical Simulation

2 2 0.457235 ( , ) (T) *RT T c r Pc*

2 0.5 ( , ) 1 (1 ) *T kTr r*

2 *k* 0.37464 1.54226 0.26992

critical pressure, and *ω* is the acentric factor.

**<IP_ADDRESS> Fugacity coefficient of methane**

**2.4.4 Phase equilibrium equation**

with temperature and pressure.

(1997) model[17-19], and the expression is:

minimum one is the liquid compressibility factor.

Where:

*bP B RT*

( ) 2 2 *T P A R T*

0.07779607*RTc b Pc*

on Natural Gas Production from Hydrate Reservoirs by Depressurization 165

Where, *α*(T) and *b* are the parameters of PR equation; *T*c is the critical temperature; *P*c is the

Eq. (91) has three real roots, in which the maximum one is the gas compressibility factor, the

After obtaining the methane compressibility factor under the conditions of temperature *T*i and pressure *P*g, the density of methane *ρ*g can be calculated by the following equation:

> *P Mg g ZRTi*

(1 2) exp ( 1) ln( ) ln 2 2 (1 2) *f A Z B Z ZB P BZ B*

The phase equilibrium of methane hydrates, water and methane is calculated by Makogon

Where, *P*e is the equilibrium pressure, Pa; the value of *T*0 is 27.15K; *A*, *B*, *C* are experiment

What's more, the viscosities of water and gas are assumed as constant, which don't change

coefficients, and the value is 0.0342K-1, 0.0005K-2, 6.4804, respectively.

(92)

2 log10 *P AT T BT T C e oo* (94)

(93)

g

The fugacity coefficient of methane is calculated by PR state equation, as follows:

#### **<IP_ADDRESS> Specific heat capacity**

The specific heat capacities of rock skeleton and hydrates are seemed as constants, while the specific heat capacities of water and gas change with temperature. When the pressure is 15MPa, and the temperature range from 273.15K~373.15K, we can get the following relationship (American National Standard)[14]:

$$C\_{pg} = 90907.40529 - 912.13019T + 3.59612T^2 - 0.00638.T^3 + 4.30251 \times 10^{-6}T^4 \tag{85}$$

$$C\_{pwv} = 4733.58633 - 3.99122T - 0.00677T^2 \tag{86}$$

$$\mathbf{C}\_{\text{ph}} = \mathbf{1600} \tag{87}$$

$$\mathcal{C}\_{\rm ps} = 835 \,\text{V} \tag{88}$$

Where, *C*pg and *C*pw are the specific heat capacities of methane and water, J/Kg/K, which are related to temperature; *C*ph and *C*ps are the specific heat capacities of hydrates and rock skeleton, J/Kg/K.

#### **<IP_ADDRESS> Decomposition heat of methane hydrate**

The decomposition heat of methane hydrate can be calculated by Masuda (1999)[15] model as follows:

$$Q\_H = -\frac{m\_H \left(c + d \cdot T\right)}{M\_H} \tag{89}$$

Where, *Q*H is the decomposition of hydrates; *M*H is the molar mass of hydrates; *c* and *d* is the experiment coefficients, the values are proposed 56599J/mol and -16.744J/mol/K for methane hydrates[16].

#### **2.4.3 Fugacity coefficient and density of methane**

#### **<IP_ADDRESS> Density of methane**

The density of methane is calculated by Peng-Robinson state equation as follows:

$$P = \frac{RT}{V - b} - \frac{a(T)}{V(V + b) + b(V - b)}\tag{90}$$

PR state equation is a mostly used cubic equation of state, which can calculate the gas volume and gas compression factor. PR equation can be expressed with compressibility factor *Z*, as follows:

$$(Z^{\mathfrak{Z}} + (\mathbb{B} - 1)Z^{\mathfrak{Z}} + (A - 2B - 3B^{\mathfrak{Z}})Z + (B^{\mathfrak{Z}} + B^{\mathfrak{Z}} - AB) = 0 \tag{91}$$

Where:

164 Advances in Natural Gas Technology

The specific heat capacities of rock skeleton and hydrates are seemed as constants, while the specific heat capacities of water and gas change with temperature. When the pressure is 15MPa, and the temperature range from 273.15K~373.15K, we can get the following

2 3 6 4 *C* 90907.40529 912.13019 3.59612 0.00638. 4.30251 10 *TT T T*

2 *C* 4733.58633 3.99122 0.00677 *T T*

1600 *Cph* (87)

Where, *C*pg and *C*pw are the specific heat capacities of methane and water, J/Kg/K, which are related to temperature; *C*ph and *C*ps are the specific heat capacities of hydrates and rock

The decomposition heat of methane hydrate can be calculated by Masuda (1999)[15] model as

*m c dT H QH MH*

Where, *Q*H is the decomposition of hydrates; *M*H is the molar mass of hydrates; *c* and *d* is the experiment coefficients, the values are proposed 56599J/mol and -16.744J/mol/K for

> ( ) ( )( )

3 2 2 32 *Z B Z A B B Z B B AB* ( 1) ( 2 3 ) ( ) 0 (91)

The density of methane is calculated by Peng-Robinson state equation as follows:

*RT T P*

*V b VV b bV b*

PR state equation is a mostly used cubic equation of state, which can calculate the gas volume and gas compression factor. PR equation can be expressed with compressibility

*pg* (85)

*pw* (86)

*ps* (88)

(89)

(90)

**<IP_ADDRESS> Specific heat capacity**

skeleton, J/Kg/K.

methane hydrates[16].

factor *Z*, as follows:

**<IP_ADDRESS> Density of methane**

follows:

relationship (American National Standard)[14]:

835 *C*

**<IP_ADDRESS> Decomposition heat of methane hydrate**

**2.4.3 Fugacity coefficient and density of methane**

$$\begin{aligned} A &= \frac{\alpha(\Gamma)P}{R^2\Gamma^2} \\ B &= \frac{bP}{RT} \\ \alpha(\Gamma) &= \frac{0.457235R^2T\_\circ^2\alpha(T\_{r'}\alpha)}{P\_\circ} \\ b &= \frac{0.07779607RT\_\circ}{P\_\circ} \\ \alpha(T\_{r'}\alpha) &= \left[1 + k(1 - T\_r^{0.5})\right]^2 \\ k &= 0.37464 + 1.54226\alpha - 0.26992\alpha^2 \end{aligned}$$

Where, *α*(T) and *b* are the parameters of PR equation; *T*c is the critical temperature; *P*c is the critical pressure, and *ω* is the acentric factor.

Eq. (91) has three real roots, in which the maximum one is the gas compressibility factor, the minimum one is the liquid compressibility factor.

After obtaining the methane compressibility factor under the conditions of temperature *T*i and pressure *P*g, the density of methane *ρ*g can be calculated by the following equation:

$$
\rho\_{\text{g}} = \frac{P\_{\text{g}} M\_{\text{g}}}{ZRT\_{\text{l}}} \tag{92}
$$

#### **<IP_ADDRESS> Fugacity coefficient of methane**

The fugacity coefficient of methane is calculated by PR state equation, as follows:

$$\varphi = \frac{f}{P} = \exp\left[ (Z - 1) - \ln(Z - B) - \frac{A}{2\sqrt{2}B} \ln\left(\frac{Z + (1 + \sqrt{2})B}{Z + (1 - \sqrt{2})B}\right) \right] \tag{93}$$

#### **2.4.4 Phase equilibrium equation**

The phase equilibrium of methane hydrates, water and methane is calculated by Makogon (1997) model[17-19], and the expression is:

$$A \log\_{10} \left( P\_{\mathcal{C}} \right) = A \left( T - T\_O \right) + B \left( T - T\_O \right)^2 + \mathcal{C} \tag{94}$$

Where, *P*e is the equilibrium pressure, Pa; the value of *T*0 is 27.15K; *A*, *B*, *C* are experiment coefficients, and the value is 0.0342K-1, 0.0005K-2, 6.4804, respectively.

What's more, the viscosities of water and gas are assumed as constant, which don't change with temperature and pressure.

Fluid-Solid Coupling Numerical Simulation

on Natural Gas Production from Hydrate Reservoirs by Depressurization 167

(1 ) 1 0 1

decomposition effect and change of reservoirs stress state on permeability.

influence of temperature, so the following relationship can be obtained.

**2.5.3 Comprehensive dynamic model of elastic modulus**

modulus and effective stress accords with power function form, as follows:

**2.5.2 Comprehensive dynamic model of porosity**

experiment coefficients; σ is the effective stress, MPa.

Where,

Φ

saturation of hydrates.

state change on the porosity.

this study is:

2 ( )(1 ) 0 *N K Kc d e SH*

The model built up by this study roundly reflects the influence of gas hydrate

JianJun Liu, Chinese academy of sciences Seepage institute, and JishunQin, China University of Petroleum, who conduct a lot of experimental research on the relationship between the porosity of low permeable porous medium and the effective stress, proposed that the index relationship could be used to describe the changing regularity between them.

*m n* exp *e*

Palmer suggests that the relationship between porosity of unconsolidated sandstone and the effective stress can be described by index relationship, whose form is similar with the formula (99). When Yizhong Zhao studied the reservoir of unconsolidated sandstone, they optimize the index relationship which is similar with (99) as dynamic models of porosity. Supposing the distribution of porosity is uniform in gas hydrate reservoirs and ignoring the

(1 ) *S e H*

Where, *Φ* is the absolute porosity when the saturation of hydrates is zero; *S*H is the

In conclusion, the comprehensive dynamic model of porosity for hydrate reservoirs built for

This model above reflects the effect of hydrate decomposition effect and the influence stress

The research of Guangquan Li, Jianjun Liu etc shows that the relation between elastic

/ *b E Ea*

(1 ) exp *Sm n H*

(101)

(102)

σ is the porosity at some stress state;*Φ*e is the initial effective porosity; *m* and *n* are

(97)

(99)

(100)

(98)

2

*V V N f j l*

*K K SH V e e V e*

#### **2.5 Comprehensive dynamic models of property parameters for gas hydrate reservoirs**

To resolve fluid-solid coupling numerical model for gas hydrate reservoirs not only needs equations of seepage fluid and deformation field but also needs relevant supplementary equations. The letter mainly include dynamic models of physical and mechanical parameters such as permeability, porosity and elastic modulus ,which reflect the real time change of the property at the process of exploitation for gas hydrate reservoirs.

Factors affecting physical property parameters of conventional reservoirs include volumetric strain of rock, effective stress, temperature, and so on. Usually one or more factors and property parameters including permeability,porosity and elastic modulus are used to build up a certain relation, that is to build up the dynamic models of physical property parameters.

For gas hydrate reservoirs, hydrate decomposition is the most direct and notable factor to affect physical properties. So in order to build up comprehensive dynamic models of physical properties for gas hydrate reservoirs, two factors must be considered: firstly, it must take example from dynamic models of conventional reservoirs to reflect the relationship between physical property parameters including permeability, porosity and elastic modulus and physical quantity including volumetric strain, effective stress and temperature; secondly, it need to highlight the influence of gas hydrate decomposition effect on related physical parameters.

#### **2.5.1 Comprehensive dynamic model of permeability**

Experiments of permeability stress sensitivity for loose sandstone done by Yizhong Zhao[20] indicate that where far away from borehole, the permeability is mainly controlled by principal stress and the fit accuracy of permeability and effective stress is high. The model built by them is indicated as Eq. (95); since sensitivity of nearly well reservoirs is mainly controlled by stress difference, relativity between permeability and volumetric strain is better. So for nearly well reservoir we could adopt the model of permeability and volumetric strain, as the following equation (96) shows.

$$K\_{\sigma} \;/\; K = c \cdot \sigma^2 + d \cdot \sigma + e \tag{95}$$

Where, *K*σ is permeability of some stress state, 10-3 μm2; *K* is the initial permeability, 10-3 μm2; σ is the effective stress, MPa; *c*, *d* and *e* are regression coefficients of experiment.

$$\frac{K\_{\mathcal{D}}}{K} = \frac{1}{1 + \varepsilon\_V} \cdot \left(1 + \frac{\varepsilon\_V}{\rho\_{\mathcal{E}}}\right)^{f} \left(\frac{\varepsilon\_V}{\rho\_{\mathcal{E}}}\right)^2 + j\frac{\varepsilon\_V}{\rho\_{\mathcal{E}}} + l \tag{96}$$

Where, *ε*v is the volumetric strain; *φ*e is the initial effective porosity; *f*, *j* and *l* are the regression coefficients of experiment.

Combined with the permeability computation model Eq. (77) for gas hydrate reservoirs and considering the difference of permeability effect mechanism for each part of the reservoir, this study builds up two comprehensive dynamic models of permeability for gas hydrate reservoirs. The formula (97) is fit in reservoirs which are far away borehole; however, the formula (98) is fit in the relative analysis of nearly well reservoirs.

To resolve fluid-solid coupling numerical model for gas hydrate reservoirs not only needs equations of seepage fluid and deformation field but also needs relevant supplementary equations. The letter mainly include dynamic models of physical and mechanical parameters such as permeability, porosity and elastic modulus ,which reflect the real time

Factors affecting physical property parameters of conventional reservoirs include volumetric strain of rock, effective stress, temperature, and so on. Usually one or more factors and property parameters including permeability,porosity and elastic modulus are used to build up a certain relation, that is to build up the dynamic models of physical

For gas hydrate reservoirs, hydrate decomposition is the most direct and notable factor to affect physical properties. So in order to build up comprehensive dynamic models of physical properties for gas hydrate reservoirs, two factors must be considered: firstly, it must take example from dynamic models of conventional reservoirs to reflect the relationship between physical property parameters including permeability, porosity and elastic modulus and physical quantity including volumetric strain, effective stress and temperature; secondly, it need to highlight the influence of gas hydrate decomposition effect

Experiments of permeability stress sensitivity for loose sandstone done by Yizhong Zhao[20] indicate that where far away from borehole, the permeability is mainly controlled by principal stress and the fit accuracy of permeability and effective stress is high. The model built by them is indicated as Eq. (95); since sensitivity of nearly well reservoirs is mainly controlled by stress difference, relativity between permeability and volumetric strain is better. So for nearly well reservoir we could adopt the model of permeability and volumetric

> 2 *K Kc d e* /

Where, *K*σ is permeability of some stress state, 10-3 μm2; *K* is the initial permeability, 10-3 μm2; σ is the effective stress, MPa; *c*, *d* and *e* are regression coefficients of experiment.

*V V f j l K V e e*

Where, *ε*v is the volumetric strain; *φ*e is the initial effective porosity; *f*, *j* and *l* are the

Combined with the permeability computation model Eq. (77) for gas hydrate reservoirs and considering the difference of permeability effect mechanism for each part of the reservoir, this study builds up two comprehensive dynamic models of permeability for gas hydrate reservoirs. The formula (97) is fit in reservoirs which are far away borehole; however, the

1 1

*K V e*

1

formula (98) is fit in the relative analysis of nearly well reservoirs.

2

(95)

(96)

**2.5 Comprehensive dynamic models of property parameters for gas hydrate**

change of the property at the process of exploitation for gas hydrate reservoirs.

**reservoirs**

property parameters.

on related physical parameters.

**2.5.1 Comprehensive dynamic model of permeability**

strain, as the following equation (96) shows.

regression coefficients of experiment.

$$K\_{\sigma} = K\_0 (c \cdot \sigma^2 + d \cdot \sigma + e)(1 - S\_H)^N \tag{97}$$

$$\mathbf{K}\_{\sigma} = \mathbf{K}\_{0} \frac{(1 - S\_{H})^{N}}{1 + \varepsilon\_{V}} \cdot \left(1 + \frac{\varepsilon\_{V}}{\rho\_{\mathcal{E}}}\right)^{f} \left(\frac{\varepsilon\_{V}}{\rho\_{\mathcal{E}}}\right)^{2} + j\frac{\varepsilon\_{V}}{\rho\_{\mathcal{E}}} + l \tag{98}$$

The model built up by this study roundly reflects the influence of gas hydrate decomposition effect and change of reservoirs stress state on permeability.

#### **2.5.2 Comprehensive dynamic model of porosity**

JianJun Liu, Chinese academy of sciences Seepage institute, and JishunQin, China University of Petroleum, who conduct a lot of experimental research on the relationship between the porosity of low permeable porous medium and the effective stress, proposed that the index relationship could be used to describe the changing regularity between them.

$$\left\langle \begin{matrix} \varrho\_{\sigma} \\ \varrho\_{\mathcal{C}} \end{matrix} \right\rangle\_{\mathcal{O}\_{\mathcal{C}}} = m \cdot \exp\left(n \cdot \sigma\right) \tag{99}$$

Where, Φσ is the porosity at some stress state;*Φ*e is the initial effective porosity; *m* and *n* are experiment coefficients; σ is the effective stress, MPa.

Palmer suggests that the relationship between porosity of unconsolidated sandstone and the effective stress can be described by index relationship, whose form is similar with the formula (99). When Yizhong Zhao studied the reservoir of unconsolidated sandstone, they optimize the index relationship which is similar with (99) as dynamic models of porosity.

Supposing the distribution of porosity is uniform in gas hydrate reservoirs and ignoring the influence of temperature, so the following relationship can be obtained.

$$
\sigma\_{\mathcal{C}} = \varphi (1 - S\_{H}) \tag{100}
$$

Where, *Φ* is the absolute porosity when the saturation of hydrates is zero; *S*H is the saturation of hydrates.

In conclusion, the comprehensive dynamic model of porosity for hydrate reservoirs built for this study is:

$$
\varphi\_{\sigma} = \varphi (1 - S\_H) \times m \cdot \exp(n \cdot \sigma) \tag{101}
$$

This model above reflects the effect of hydrate decomposition effect and the influence stress state change on the porosity.

#### **2.5.3 Comprehensive dynamic model of elastic modulus**

The research of Guangquan Li, Jianjun Liu etc shows that the relation between elastic modulus and effective stress accords with power function form, as follows:

$$E\_{\sigma} \;/ \; E = a \cdot \sigma^{b} \tag{102}$$

Fluid-Solid Coupling Numerical Simulation

must be taken into consideration.

mathematical expression is[21-23]:

result of hydrates decomposition.

reservoirs by depressurization.

productivity are also researched.

**hydrate reservoirs by depressurization**

**3.1.1 Basic parameters used in the simulation**

relative permeability capillary pressure curve, respectively.

hydrate reservoirs by depressurization.

**from hydrate reservoirs by depressurization**

fluid-solid coupling numerical simulation.

**2.5.4 Comprehensive dynamic model of cohesion**

on Natural Gas Production from Hydrate Reservoirs by Depressurization 169

Because the change of Poisson ratio is little, the Poisson ratio is seen as constant during the

During the exploitation of gas hydrate reservoirs, hydrates decomposition can cause the cementation of the reservoirs loosened and the remarkable change of the cohesive force

In 2005, Clennell, from the petroleum research institute of Commonwealth Scientific and Industrial Research Organisation's (CSIRO), and Tohidi, from Heriot-Watt University hydrate research center, proposed that as the decomposition of hydrates, the reservoirs cementation will become weaker, and the cohesion will decrease continuously. So they built the model of cohesion decrease for reservoirs, which was widely adopted by scholars. The

1 1.2 0 *C C*

Where, C is the cohesive force of the reservoirs after hydrates decomposition, MPa; C0 is the cohesive force before hydrates decomposition, MPa; Δ*Φ* is the increment of porosity as a

Eq. (109) is also taken as the dynamic model of cohesive force for the production from

**3. Fluid-solid coupling simulation studies on productivity of gas production**

Gas production from hydrate reservoirs by depressurization is a complex fluid-solid coupling process with hydrates decomposition, which is affected by the hydrates decomposition effect, fluid-solid coupling and borehole effect. In this process, the reservoir physical parameters shows the complex characteristics, reservoir porosity characteristics change and fluid-solid coupling have the dynamic effect for gas production from hydrate

Based on the gas and water fluid-solid coupling mathematical model and the physical and mechanical models build above, this section will solve these models. Taking a gas hydrate reservoir of Gulf of Mexico as an example, we simulate the productivity of gas production from hydrate reservoirs by depressurization. At the same time, the main influence factors to

**3.1 Fluid-solid coupling finite element model for productivity of gas production from**

The parameters used in fluid-solid coupling numerical simulation include basic property parameters, reservoir fluid and the basic properties of solid hydrate. All of the parameters input in the simulation of this study are from a natural gas hydrate reservoir in the Gulf of Mexico, as is shown in Table 3-1 and Table 3-2. And Figs. 3-1 and 3-2 show the gas/water

(109)

Where, *E*σ is elastic modulus at some stress state, GPa; *E* is the initial elastic modulus, GPa; *a* and *b* are the fitting coefficients of experiment; *σ* is the effective stress, MPa.

By making sensitivity experiments of elastic modulus for unconsolidated sandstone, Yizhong Zhao points that the regression relation of quadratic polynomial between elastic modulus and effective stress is more accurately, the expression of which is as follows:

$$E\_{\sigma} \;/\; E = A \cdot \sigma^2 + B \cdot \sigma + C \tag{103}$$

Where, *A*, *B*, *C* is the fitting coefficients by experiment.

In recent years,through experimental research,the petroleum research institute of Commonwealth Scientific and Industrial Research Organization(CSIRO)and Tohidi, from hydrate research center of Heriot-Watt University point that the decomposition of hydrates will result in the increase of porosity and the decline in the elastic modulus correspondingly, based on which they built the relation between elastic modulus change and the variation of porosity, as follows:

$$
\Delta E = -\underline{\mathfrak{F}} \cdot \Delta \underline{\mathfrak{\varphi}} \tag{104}
$$

Where, *ξ*is the fitting coefficient by experiment, the value of which is -9.5; Δ*E* is the decrease of elastic modulus, GPa; Δ*Φ* is the increment of porosity.

Based on the comprehensive dynamic model of property for gas hydrate reservoirs which we built for this research, the mathematical relation between the hydrate decomposition effect and the increase of porosity resulting from the change of stress condition is as follows:

$$
\Delta \phi = \phi (1 - S\_{H^{\cdot}}) \times m \cdot \exp(m \cdot \sigma) - \phi (1 - S\_{H \dot{\imath}}) \tag{105}
$$

Where, *Φ* is the absolute porosity when the hydrate saturation is zero; *S*Hi is the initial saturation of hydrates; *S*H is the current hydrates saturation; *σ* is the effective stress, MPa; *m* and *n* is the experimental matched coefficients.

Combining Eq. (104) into Eq. (105), the change of elastic modulus is as follows:

$$
\Delta E = -\xi \cdot \left[ \sigma (1 - S\_H) \times m \cdot \exp(n \cdot \sigma) - \varphi (1 - S\_{H\bar{i}}) \right] \tag{106}
$$

At the same time, there is the following relationship:

$$E = E\_{\text{O}} + \Delta E \tag{107}$$

Where, *E*0 is the elastic modulus before hydrates decomposition, GPa.

Allying Eqs. (106) and (107) with Eq. (103), the comprehensive dynamic model of elastic modulus is built for this study, as follows:

$$E\_{\sigma} = \left\{ E0 - \xi \left[ \varphi(1 - S\_{H}) \times m \cdot \exp(n \cdot \sigma) - \varphi(1 - S\_{H\bar{1}}) \right] \right\} \cdot \left( A \cdot \sigma^{2} + B \cdot \sigma + C \right) \tag{108}$$

This model reflects the influence of hydrate decomposition and stress state change on the elastic modulus.

Because the change of Poisson ratio is little, the Poisson ratio is seen as constant during the fluid-solid coupling numerical simulation.

#### **2.5.4 Comprehensive dynamic model of cohesion**

168 Advances in Natural Gas Technology

Where, *E*σ is elastic modulus at some stress state, GPa; *E* is the initial elastic modulus, GPa; *a*

By making sensitivity experiments of elastic modulus for unconsolidated sandstone, Yizhong Zhao points that the regression relation of quadratic polynomial between elastic modulus and effective stress is more accurately, the expression of which is as follows:

> 2 *E EA B C* /

In recent years,through experimental research,the petroleum research institute of Commonwealth Scientific and Industrial Research Organization(CSIRO)and Tohidi, from hydrate research center of Heriot-Watt University point that the decomposition of hydrates will result in the increase of porosity and the decline in the elastic modulus correspondingly, based on which they built the relation between elastic modulus change and the variation of

> *E*

Where, *ξ*is the fitting coefficient by experiment, the value of which is -9.5; Δ*E* is the decrease

Based on the comprehensive dynamic model of property for gas hydrate reservoirs which we built for this research, the mathematical relation between the hydrate decomposition effect and the increase of porosity resulting from the change of stress condition is as follows:

(1 ) exp (1 ) *Sm n S H Hi*

Where, *Φ* is the absolute porosity when the hydrate saturation is zero; *S*Hi is the initial saturation of hydrates; *S*H is the current hydrates saturation; *σ* is the effective stress, MPa; *m*

*E Sm n S* (1 ) exp (1 ) *H Hi*

Allying Eqs. (106) and (107) with Eq. (103), the comprehensive dynamic model of elastic

2 (1 ) exp (1 ) *E E S m n S A BC* 0 *H Hi*

This model reflects the influence of hydrate decomposition and stress state change on the

Combining Eq. (104) into Eq. (105), the change of elastic modulus is as follows:

>

(108)

(106)

0 *EE E* (107)

(103)

(104)

(105)

and *b* are the fitting coefficients of experiment; *σ* is the effective stress, MPa.

Where, *A*, *B*, *C* is the fitting coefficients by experiment.

of elastic modulus, GPa; Δ*Φ* is the increment of porosity.

>

Where, *E*0 is the elastic modulus before hydrates decomposition, GPa.

and *n* is the experimental matched coefficients.

At the same time, there is the following relationship:

modulus is built for this study, as follows:

elastic modulus.

porosity, as follows:

During the exploitation of gas hydrate reservoirs, hydrates decomposition can cause the cementation of the reservoirs loosened and the remarkable change of the cohesive force must be taken into consideration.

In 2005, Clennell, from the petroleum research institute of Commonwealth Scientific and Industrial Research Organisation's (CSIRO), and Tohidi, from Heriot-Watt University hydrate research center, proposed that as the decomposition of hydrates, the reservoirs cementation will become weaker, and the cohesion will decrease continuously. So they built the model of cohesion decrease for reservoirs, which was widely adopted by scholars. The mathematical expression is[21-23]:

$$C = C\_0 \left[ 1 \text{--} 1.2 \left( \Lambda \varphi \right) \right] \tag{109}$$

Where, C is the cohesive force of the reservoirs after hydrates decomposition, MPa; C0 is the cohesive force before hydrates decomposition, MPa; Δ*Φ* is the increment of porosity as a result of hydrates decomposition.

Eq. (109) is also taken as the dynamic model of cohesive force for the production from hydrate reservoirs by depressurization.

#### **3. Fluid-solid coupling simulation studies on productivity of gas production from hydrate reservoirs by depressurization**

Gas production from hydrate reservoirs by depressurization is a complex fluid-solid coupling process with hydrates decomposition, which is affected by the hydrates decomposition effect, fluid-solid coupling and borehole effect. In this process, the reservoir physical parameters shows the complex characteristics, reservoir porosity characteristics change and fluid-solid coupling have the dynamic effect for gas production from hydrate reservoirs by depressurization.

Based on the gas and water fluid-solid coupling mathematical model and the physical and mechanical models build above, this section will solve these models. Taking a gas hydrate reservoir of Gulf of Mexico as an example, we simulate the productivity of gas production from hydrate reservoirs by depressurization. At the same time, the main influence factors to productivity are also researched.

#### **3.1 Fluid-solid coupling finite element model for productivity of gas production from hydrate reservoirs by depressurization**

#### **3.1.1 Basic parameters used in the simulation**

The parameters used in fluid-solid coupling numerical simulation include basic property parameters, reservoir fluid and the basic properties of solid hydrate. All of the parameters input in the simulation of this study are from a natural gas hydrate reservoir in the Gulf of Mexico, as is shown in Table 3-1 and Table 3-2. And Figs. 3-1 and 3-2 show the gas/water relative permeability capillary pressure curve, respectively.

Fluid-Solid Coupling Numerical Simulation

Fig. 3-2. Capillary pressure curve

bottomhole pressure.

fixed to both X-axis and Y-axis.

Fig. 3-3. Schematic diagram of mechanical model

**Z X**

**Y**

**3.1.2 Geometrical model and boundary conditions**

the finite element model are shown in Figs. 3-3 and 3-4, respectively.

on Natural Gas Production from Hydrate Reservoirs by Depressurization 171

To simplify the problem, the plane strain model is adopted considering the symmetry of the hydrate reservoir. Set the overall dimensions as 100m×100m for this finite element model. With respect to the significance of the near borehole formation to the whole capacity simulation, the model meshing is obtained with gradient grid technology, so the mesh density is higher in formation near borehole. The diagram of mechanical model and grid of

In the process of simulation the boundary conditions for the seepage field are as follows: Line *BC* and *CD* keep constant pressure boundary, Line *AB* and *DE* keep free pressure boundary, and at the position of the wellbore, namely Point *A*, the boundary keeps the

The boundary conditions of the deformation field are listed as follows: The maximum effective horizontal principle stress (*σ*H) is applied on the boundary line BC, and the minimum effective horizontal principle stress (*σ*h) is applied on Line *CD*. On the Line AB the displacement is constant as zero along X-axis and free along Y-axis; And on the Line DE the displacement is zero along X-axis and free along Y-axis. At the Point A the displacement is

σh

100m

D C

A B

100m

σ

H

Table 3-1. Basic parameters for reservoir fluid, rock and gas hydrate

Table 3-2. Basic parameters of finite element simulation for gas hydrate reservoir

Fig. 3-1. Gas/water relative permeability curves

Fig. 3-2. Capillary pressure curve

Methane Thermal Conductivity /W.m-1.K-1

temperature 0.00335 Rely on

Hydrate Decomposition Rate/mol. (m2.Pa.s)-1

Specific Heat of Methane /J.Kg-1.K-1

temperature 1.5

Absolute Porosity to X-axis /μm2

Angle of Internal Friction

Sensitiveness Coefficient of Permeability

*e*

Hydrate Decomposition Activating Energy /J.mol-1

Thermal Conductivity of Rock Skeleton /W.m-1.K-1

Methane Hydrate Number

> Absolute Porosity to Y-axis /μm2

Biot Factor

Sensitiveness Coefficient of Porosity *m*

Specific Heat of Water /J.Kg-1.K-1

Specific Heat of Solid Hydrate /J.Kg-1.K-1

Table 3-1. Basic parameters for reservoir fluid, rock and gas hydrate

Initial Hydrate Saturation

Initial Elastic Modulus/MPa

Minimum Horizontal Principal Stress

Sensitiveness Coefficient of Elastic Modulus

/MPa

*B*

Fig. 3-1. Gas/water relative permeability curves

835 910 0.393 1600 8060 77330 6.0

Initial Water Saturation

1310 365 0.5 0.4 0.4 0.2 0.2

Poission's Ratio

290 16.9 4845 0.40 2.0 30º 1.0

Sensitiveness Coefficient of Permeability *c*

21.80 20.45 19.70 0.0038 -0.1096 1.3086 1.1455

Sensitiveness Coefficient of Elastic Modulus *C*

Absolute Porosity (Without Hydrate)

Cohesion /MPa

Sensitiveness Coefficient of Permeability

Borehole Radius/m

*d*

Water Density /Kg.m-3

Specific Heat of Rock /J.Kg-1.K-1

Depth of Sea Water/m

Initial Temperature

Vertical Stress/MPa

Sensitiveness Coefficient of Porosity *n*

/K

Water viscosity /Pa.s

Hydrate Density /Kg.m-3

> Distance to Sea Bed/m

Initial Pressure/ MPa

Maximum Horizontal Principal Stress /MPa

Sensitiveness Coefficient of Elastic Modulus *A*

Water Thermal Conductivity /W.m-1.K-1

1000 0.001 0.6 Rely on

Hydrate Thermal Conductivity /W.m-1.K-1

#### **3.1.2 Geometrical model and boundary conditions**

To simplify the problem, the plane strain model is adopted considering the symmetry of the hydrate reservoir. Set the overall dimensions as 100m×100m for this finite element model. With respect to the significance of the near borehole formation to the whole capacity simulation, the model meshing is obtained with gradient grid technology, so the mesh density is higher in formation near borehole. The diagram of mechanical model and grid of the finite element model are shown in Figs. 3-3 and 3-4, respectively.

In the process of simulation the boundary conditions for the seepage field are as follows: Line *BC* and *CD* keep constant pressure boundary, Line *AB* and *DE* keep free pressure boundary, and at the position of the wellbore, namely Point *A*, the boundary keeps the bottomhole pressure.

The boundary conditions of the deformation field are listed as follows: The maximum effective horizontal principle stress (*σ*H) is applied on the boundary line BC, and the minimum effective horizontal principle stress (*σ*h) is applied on Line *CD*. On the Line AB the displacement is constant as zero along X-axis and free along Y-axis; And on the Line DE the displacement is zero along X-axis and free along Y-axis. At the Point A the displacement is fixed to both X-axis and Y-axis.

Fig. 3-3. Schematic diagram of mechanical model

Fluid-Solid Coupling Numerical Simulation

1

2

3

Noncoupling normal model

Simplified fluid-solid coupling model

Comprehens ive fluidsolid coupling model

Fig. 3-5. Gas production rate and time

Fig. 3-6. Water production rate and time

on Natural Gas Production from Hydrate Reservoirs by Depressurization 173

Num. Model name Factors in consideration Contrast analysis

Based on the seepage model of phases transition,

temperature variation, but neglect the fluid-solid coupling effect(bottomhole pressure is 13.9 MPa)

Based on the seepage model of phases transition, consider hydrates decomposition, temperature variation, the influence of rock skeleton volumetric strain on the seepage, but ignore reservoir properties and mechanical parameters variation (bottomhole pressure is13.9 MPa)

This is the gas-water two-phase non-isothermal fluid-solid coupling model for natural gas hydrate reservoir built in Section Two, which considers hydrates decomposition, reservoir physical and mechanical properties variation

Fig. 3-5 shows that the gas production rate curve fluctuates greatly because of the instability of gas-liquid two-fluid phase seepage. The variation tendency of gas production rate curve indicates that the gas production rate increases sharply at the early stage, and become stable lately. Only 43 days is set to exploit which is too short to finish hydrates decompose, but it

(bottomhole pressure is 13.9 MPa) Table 3-3. Essential characteristics of the three models for deliverability analysis

consider hydrates decomposition and

method

(1)By comparing the results of Models 1 and 2, we analyze the effect of matrix volumetric strain on

production performance; (2)By comparing the results of Models 2 and 3, we analyze the effect of physical parameters on production performance.

Fig. 3-4. Grid of the FEM model

#### **3.2 Fluid-solid coupling numerical simulation on productivity of gas production from hydrate reservoirs by depressurization**

For the further studying the influential mechanism of fluid-solid coupling effects on the gas production from hydrate reservoirs by depressurization, this section also analyses the other simulation results of the two simplified models which is based on the gas-water two-phase non-isothermal fluid-solid coupling model established above.

The first model ignores physical parameters change caused by the fluid-solid coupling effect, but it considers the coupling effect between seepage field and deformation field, and it is named the simplified fluid-solid coupling model. Based on the first model, the second model further ignores the coupling effects of seepage field and solid field, so it is named the non-coupling normal model.

For the convenience of analysis, here the gas-water two-phase non-isothermal fluid-solid coupling model established above is renamed as the comprehensive fluid-solid coupling model. The basic characteristics of the three models are shown in Table 3-3.

#### **3.2.1 Numerical simulation on productivity of non-coupling normal model for production by depressurization**

Taking into account the non-coupling normal model with phase transition in Table 3-3, which considers the petrophysical properties change as a result of hydrates decomposition and neglects the coupling effect between the seepage field and the deformation field, the gas and water production rate are simulated. The results are shown Figs. 3-5 and 3-6, the unit of which is STCMD.

**3.2 Fluid-solid coupling numerical simulation on productivity of gas production from**

For the further studying the influential mechanism of fluid-solid coupling effects on the gas production from hydrate reservoirs by depressurization, this section also analyses the other simulation results of the two simplified models which is based on the gas-water two-phase

The first model ignores physical parameters change caused by the fluid-solid coupling effect, but it considers the coupling effect between seepage field and deformation field, and it is named the simplified fluid-solid coupling model. Based on the first model, the second model further ignores the coupling effects of seepage field and solid field, so it is named the

For the convenience of analysis, here the gas-water two-phase non-isothermal fluid-solid coupling model established above is renamed as the comprehensive fluid-solid coupling

Taking into account the non-coupling normal model with phase transition in Table 3-3, which considers the petrophysical properties change as a result of hydrates decomposition and neglects the coupling effect between the seepage field and the deformation field, the gas and water production rate are simulated. The results are shown Figs. 3-5 and 3-6, the unit of

model. The basic characteristics of the three models are shown in Table 3-3.

**3.2.1 Numerical simulation on productivity of non-coupling normal model for**

Fig. 3-4. Grid of the FEM model

non-coupling normal model.

**production by depressurization**

which is STCMD.

**hydrate reservoirs by depressurization**

non-isothermal fluid-solid coupling model established above.

Table 3-3. Essential characteristics of the three models for deliverability analysis

Fig. 3-5. Gas production rate and time

Fig. 3-6. Water production rate and time

Fig. 3-5 shows that the gas production rate curve fluctuates greatly because of the instability of gas-liquid two-fluid phase seepage. The variation tendency of gas production rate curve indicates that the gas production rate increases sharply at the early stage, and become stable lately. Only 43 days is set to exploit which is too short to finish hydrates decompose, but it

Fluid-Solid Coupling Numerical Simulation

Fig. 3-7. Gas production rate from experiment

Fig. 3-8. Water production rate from experiment

cumulative gas production finally stays at a certain level.

corresponding with the water production rate.

of this simulation results.

on Natural Gas Production from Hydrate Reservoirs by Depressurization 175

The size of the pipe adopted in the simulation experiment by ShuXia Lee is *Φ*38×500mm and *Φ*38×800mm, which belongs to small one-dimensional model. However, in the numerical simulation of this study a plane strain model is used. The difference of the two models leads to no relative property of the gas and water production rate value. But it is significant to analyze the comparison of the change regulation. Because the model dimension is large and the time is limited in this research, the dropping stage of gas and water production rates have not been happened. Comparing Fig. 3-5 with Fig. 3-7, and Fig. 3-6 with Fig. 3-8, in general the characteristic of the gas and water production rate and the regulation variation match with results of physical simulation experiment. This shows the validity of the results

The cumulative gas production is shown as Fig. 3-9. It can be seen that the cumulative gas production curve is well corresponding with the gas production rate curve. At the early stage of depressurization the gas production rate is bigger, and correspondingly the cumulative gas production rises quickly. When the gas production rate reaches the relatively steady stage, the cumulative gas production increases steadily and slowly. It can be inferred that, with gas production rate gradually decreasing at the later stage of depressurization production, correspondingly the increasing range of cumulative gas production will gradually decrease. Until the hydrates decomposition totally, the

The cumulative water production is shown as Fig. 3-10. It is easy to see that, the cumulative water production curve is vary similar with the one of gas production, and it is also well

can be indicated that gas production rate will decrease as hydrate saturation reducing, and finally no gas is produced. In general, the gas production rate curve can be divided into three stages, rapidly rising stage, stable stage and dropping stage. We can summarize the characteristic as follows:

From Fig. 3-6 it can be seen that the water production rate curve is similar with the one of gas production. It can also be divided into three stages as follows:

Shuxia Lee and Yongmao Hao, who are from the Hydrate Research Center of China University of Petroleum, established hydrates synthesizing and exploiting equipment. They obtained the water and gas production rate characteristic curve through small scale hydrate exploiting experiment by depressurization[24,25]. Figs. 3-7 and 3-9 show the results.

Fig. 3-7. Gas production rate from experiment

can be indicated that gas production rate will decrease as hydrate saturation reducing, and finally no gas is produced. In general, the gas production rate curve can be divided into three stages, rapidly rising stage, stable stage and dropping stage. We can summarize the

1. The rising stage of gas production rate. As the pressure releases the gas production rate rises rapidly. This is because: a. the initial saturation of free gas is 0.1 and when the pressure decreases, the free gas could swell out quickly; b. the pressure of the reservoir prior to production is close to hydrate decomposition pressure at the initial temperature, so the pressure around wellbore will be below the equilibrium pressure of

2. The stable stage of gas production rate. Most of the gas produced after free gas output is from the hydrate decomposition, which is controlled by the rate of hydrate

3. The dropping stage of gas production rate. The hydrate saturation become lower and lower as the decomposition front going ahead further, and the gas production becomes less correspondingly, and finally no gas could be produced. As only 43 days is set to simulate the gas production in Fig. 3-5, which is not long enough to lead to hydrate decomposition totally, so the obvious dropping stage of gas production is not reached

From Fig. 3-6 it can be seen that the water production rate curve is similar with the one of

1. Instead of gas, water is the only production in the original depressurization stage, and the time of water producing falls behind the gas producing time. This is because the existence of free gas in the reservoir has better flowing ability than the water when the

2. After the output of free gas, the water saturation of reservoirs increases gradually as hydrates decompose. The rate of water production decreases rapidly after reaching a peak under the influence of the relative permeability of water and gas. The reason for this phenomenon is as follows, the free water and produced water accumulate towards the wellbore as a result of the propelling of gas, and they come out rapidly when reach to some extent. Consequently, a water production peak takes place. However, due to the fact that both the gross of the free water and the water comes from the hydrate decomposition in the original stage are relatively limited, the rate of water production

3. The water producing rate remains steady level after falling. In this period, as the free water output is almost completely, the production is mainly the decomposing water from hydrates, which is controlled by the decomposition rate of hydrates. And the water production exhibits the similar character to that of gas. The decrease of the hydrate saturation and decomposition rate leads to the reduction of the water

Shuxia Lee and Yongmao Hao, who are from the Hydrate Research Center of China University of Petroleum, established hydrates synthesizing and exploiting equipment. They obtained the water and gas production rate characteristic curve through small scale hydrate

exploiting experiment by depressurization[24,25]. Figs. 3-7 and 3-9 show the results.

hydrates rapidly and they begin to decompose immediately.

gas production. It can also be divided into three stages as follows:

decreases speedily after reaching the peak.

producing rate, and finally no water is generated.

characteristic as follows:

dissociation.

yet in fact.

pressure decreasing.

Fig. 3-8. Water production rate from experiment

The size of the pipe adopted in the simulation experiment by ShuXia Lee is *Φ*38×500mm and *Φ*38×800mm, which belongs to small one-dimensional model. However, in the numerical simulation of this study a plane strain model is used. The difference of the two models leads to no relative property of the gas and water production rate value. But it is significant to analyze the comparison of the change regulation. Because the model dimension is large and the time is limited in this research, the dropping stage of gas and water production rates have not been happened. Comparing Fig. 3-5 with Fig. 3-7, and Fig. 3-6 with Fig. 3-8, in general the characteristic of the gas and water production rate and the regulation variation match with results of physical simulation experiment. This shows the validity of the results of this simulation results.

The cumulative gas production is shown as Fig. 3-9. It can be seen that the cumulative gas production curve is well corresponding with the gas production rate curve. At the early stage of depressurization the gas production rate is bigger, and correspondingly the cumulative gas production rises quickly. When the gas production rate reaches the relatively steady stage, the cumulative gas production increases steadily and slowly. It can be inferred that, with gas production rate gradually decreasing at the later stage of depressurization production, correspondingly the increasing range of cumulative gas production will gradually decrease. Until the hydrates decomposition totally, the cumulative gas production finally stays at a certain level.

The cumulative water production is shown as Fig. 3-10. It is easy to see that, the cumulative water production curve is vary similar with the one of gas production, and it is also well corresponding with the water production rate.

Fluid-Solid Coupling Numerical Simulation

Fig. 3-11. Comparison of gas production rate

Fig. 3-12. Comparison of cumulative gas

Fig. 3-13. Comparison of water production rate

Fig. 3-14. Comparison of cumulative water

on Natural Gas Production from Hydrate Reservoirs by Depressurization 177

Fig. 3-9. Cumulative gas production

Fig. 3-10. Cumulative water production

#### **3.2.2 Numerical simulation on productivity of simplified fluid-solid coupling model for production by depressurization**

During the fluid-solid coupling effect of production by depressurization, the influence of reservoir deformation field to seepage field mainly reflects in two aspects. On the one hand, the deformation of reservoir skeleton will shrink the pore space, which will increase elastic drive energy. On the other hand, the change of effective stress state will cause petrophysical variations of stress sensitive reservoirs, such as porosity, permeability and so on. All of these will affect the production performance of hydrate reservoirs.

In this chapter, the simplified fluid-solid coupling model in Table 3-3 is adopted to simulate gas production from hydrate reservoirs by depressurization, in which the physical parameters alternation caused by fluid-solid coupling is neglected. By comparing with the simulation results of the non-coupling normal model above, we discuss the influence of fluid-solid coupling effects on the production behaviors of hydrate reservoirs.

Fig. 3-11 gives the gas production rate curves of both the non-coupling normal model and simplified fluid-solid coupling model. Fig. 3-13 gives the water production rate curves of the two models. The cumulative gas and water production curves of the two models are shown in Figs. 3-12 and 3-14.

From the comparison above, it can be seen that the results of the simplified fluid-solid coupling model and non-coupling normal model are similar in some aspects, but there is also some obvious difference, as follows:

1. Similar to the non-coupling normal model, the gas and water production curves of the simplified fluid-solid coupling model can be divided into three stages generally: rapidly rising stage, stable stage and gradually reduce stage. This is consistent with the results of experiments by Shuxia Lee and Yongmao Hao in China University of Petroleum.

Fig. 3-11. Comparison of gas production rate

176 Advances in Natural Gas Technology

**3.2.2 Numerical simulation on productivity of simplified fluid-solid coupling model for**

During the fluid-solid coupling effect of production by depressurization, the influence of reservoir deformation field to seepage field mainly reflects in two aspects. On the one hand, the deformation of reservoir skeleton will shrink the pore space, which will increase elastic drive energy. On the other hand, the change of effective stress state will cause petrophysical variations of stress sensitive reservoirs, such as porosity, permeability and so on. All of these

In this chapter, the simplified fluid-solid coupling model in Table 3-3 is adopted to simulate gas production from hydrate reservoirs by depressurization, in which the physical parameters alternation caused by fluid-solid coupling is neglected. By comparing with the simulation results of the non-coupling normal model above, we discuss the influence of

Fig. 3-11 gives the gas production rate curves of both the non-coupling normal model and simplified fluid-solid coupling model. Fig. 3-13 gives the water production rate curves of the two models. The cumulative gas and water production curves of the two models are shown

From the comparison above, it can be seen that the results of the simplified fluid-solid coupling model and non-coupling normal model are similar in some aspects, but there is

1. Similar to the non-coupling normal model, the gas and water production curves of the simplified fluid-solid coupling model can be divided into three stages generally: rapidly rising stage, stable stage and gradually reduce stage. This is consistent with the results of experiments by Shuxia Lee and Yongmao Hao in China University of Petroleum.

fluid-solid coupling effects on the production behaviors of hydrate reservoirs.

Fig. 3-9. Cumulative gas production

Fig. 3-10. Cumulative water production

will affect the production performance of hydrate reservoirs.

**production by depressurization**

in Figs. 3-12 and 3-14.

also some obvious difference, as follows:

Fig. 3-12. Comparison of cumulative gas

Fig. 3-13. Comparison of water production rate

Fig. 3-14. Comparison of cumulative water

Fluid-Solid Coupling Numerical Simulation

Fig. 3-16. Comparison of cumulative gas production

Fig. 3-17. Comparison of water production rate

Fig. 3-18. Comparison of cumulative water production

difference, as follows:

The comparisons above show that the gas and water production rate curves of the comprehensive and simplified coupling models can also be divided into three stages, namely the rapidly rising stage, stable stage and gradually reduce stage, which is similar to that of the non-coupling normal model, and matches with the experiment results of Shuxia Lee. Further analysis shows that the numerical size of simulative result has significantly

1. From Figs. 3-15 and 3-16 it can seen that the average gas production rate and accumulative production which are obtained by the simplified fluid-solid coupling model are obviously higher than that of the comprehensive fluid-solid coupling model. The average gas production rate in stable stage of the former model is about 1.144 times

on Natural Gas Production from Hydrate Reservoirs by Depressurization 179

2. The major difference between two models lies in that both the average production rate and the accumulative production of gas obtained by the simplified fluid-solid coupling model are 3.45% higher than that of the non-coupling normal model. Similarly, both the average water production rate and the water accumulative production are 4.28% higher than that of non-coupling normal model. This is because reservoir skeleton volume strain can lead to reservoir porosity reducing and elastic drive energy increasing to some extent. This is flavor to raise gas and water production rate, accumulative gas and water production of hydrate reservoirs. The increasing of elastic drive energy has more effect to water production than gas production as a result of the poor liquidity of the gas with respect to the liquid.

#### **3.2.3 Numerical simulation on productivity of comprehensive fluid-solid coupling model for production by depressurization**

The analysis above shows that shrinkage of reservoir pore space can raise the elastic drive energy because of the fluid-solid coupling effect, which does help to increase the gas production rate and accumulative gas production. This is one side that the fluid-solid coupling affecting production performance. And on the other side, the change of the reservoir effective stress can lead to physical parameters, such as porosity and permeability, which can also influence production behaviors.

In this chapter, we adopt the comprehensive fluid-solid coupling model in Table 3-3 to simulate gas production from hydrate reservoirs by depressurization. Based on the simplified model, the comprehensive fluid-solid coupling model considers the change of physical parameters caused by the stress state variation. Thus, by comparing with the two models, we can study the influence of the physical parameters change of sensitive stress reservoirs, which is caused by the fluid-solid coupling effect, on the production performance of hydrate reservoirs.

Figs. 3-15 and 3-17 show the gas and water production rate curves respectively obtained from the comprehensive fluid-solid coupling model and the simplified model. The accumulative gas and water production curves of the two models are shown as Figs. 3-16 and 3-18.

Fig. 3-15. Comparison of gas production rate

2. The major difference between two models lies in that both the average production rate and the accumulative production of gas obtained by the simplified fluid-solid coupling model are 3.45% higher than that of the non-coupling normal model. Similarly, both the average water production rate and the water accumulative production are 4.28% higher than that of non-coupling normal model. This is because reservoir skeleton volume strain can lead to reservoir porosity reducing and elastic drive energy increasing to some extent. This is flavor to raise gas and water production rate, accumulative gas and water production of hydrate reservoirs. The increasing of elastic drive energy has more effect to water production than gas production as a result of the poor liquidity of the

**3.2.3 Numerical simulation on productivity of comprehensive fluid-solid coupling**

The analysis above shows that shrinkage of reservoir pore space can raise the elastic drive energy because of the fluid-solid coupling effect, which does help to increase the gas production rate and accumulative gas production. This is one side that the fluid-solid coupling affecting production performance. And on the other side, the change of the reservoir effective stress can lead to physical parameters, such as porosity and permeability,

In this chapter, we adopt the comprehensive fluid-solid coupling model in Table 3-3 to simulate gas production from hydrate reservoirs by depressurization. Based on the simplified model, the comprehensive fluid-solid coupling model considers the change of physical parameters caused by the stress state variation. Thus, by comparing with the two models, we can study the influence of the physical parameters change of sensitive stress reservoirs, which is caused by the fluid-solid coupling effect, on the production performance

Figs. 3-15 and 3-17 show the gas and water production rate curves respectively obtained from the comprehensive fluid-solid coupling model and the simplified model. The accumulative gas and water production curves of the two models are shown as Figs. 3-16

gas with respect to the liquid.

**model for production by depressurization**

which can also influence production behaviors.

Fig. 3-15. Comparison of gas production rate

of hydrate reservoirs.

and 3-18.

Fig. 3-16. Comparison of cumulative gas production

Fig. 3-17. Comparison of water production rate

Fig. 3-18. Comparison of cumulative water production

The comparisons above show that the gas and water production rate curves of the comprehensive and simplified coupling models can also be divided into three stages, namely the rapidly rising stage, stable stage and gradually reduce stage, which is similar to that of the non-coupling normal model, and matches with the experiment results of Shuxia Lee. Further analysis shows that the numerical size of simulative result has significantly difference, as follows:

1. From Figs. 3-15 and 3-16 it can seen that the average gas production rate and accumulative production which are obtained by the simplified fluid-solid coupling model are obviously higher than that of the comprehensive fluid-solid coupling model. The average gas production rate in stable stage of the former model is about 1.144 times

Fluid-Solid Coupling Numerical Simulation

higher than that of the later.

to production performance.

coupling model and non-coupling normal model.

Fig. 3-22. Comparison of cumulative water production

on Natural Gas Production from Hydrate Reservoirs by Depressurization 181

The comparisons of Figs. 3-19 and 3-20 show that the average gas production rate and accumulative gas production of the non-coupling normal model are obviously higher than that of the comprehensive fluid-solid coupling model. The average gas production rate in the stable stage of the former model is about 1.106 times higher than that of the later. And accordingly, the accumulative gas production of the former is about 15.9% higher than that of the later. Similarly, from Fig. 3-21 and Fig. 3-22 it can be seen that the average water production rate and accumulative water production of non-coupling normal model is also obviously greater than that of comprehensive model. The average water production rate during the steady stage of the former model is about 1.111 times higher than that of the later. Correspondingly, the accumulative water production of the former is about 11.1%

In conclusion, by analyzing the production performances of the non-coupling normal model, the simplified coupling model and the comprehensive, we can find that the development index, namely the gas production rate, the accumulative gas production and so on, obtained from the simplified model is the highest in the three models. The results of the non- coupling normal model are slightly lower. But the development index simulated by the comprehensive coupling model is significantly lower than the two former models. The reason for this phenomenon is that fluid-solid coupling has different influencing mechanism

During the production by depressurization, the fluid-solid coupling effect has two aspects. On one side, the shrinkage of rock matrix caused by fluid-solid coupling effect can raise the elastic drive energy, which is helpful for gas output. So the development index of simplified fluid-solid coupling model which only considers the skeleton volume strain is higher than that of the non-coupling normal model. This is the good aspect that fluid-solid coupling effect contributes to gas production. On the other side, the shrinkage of rock pore space can also cause the reduction of permeability and porosity, and accordingly the seepage captivity also decreases which causes the adverse effect for production. Relatively, the change of porosity and permeability which is caused by fluid-solid coupling plays the dominated role on affecting the production performance. But the increasing of elastic drive energy is limited. Therefore, the predictive value of production index simulated by the comprehensive fluid-solid coupling model which considers the skeleton volume strain and physical parameters change is obviously lower than the results of the simplified fluid-solid

greater than that of the later. Accordingly, the gas accumulative production is about 4.4% higher than that of the later.

2. Similarly, from Figs. 3-17 and 3-18 we can also see that the average water production rate and accumulative production simulated by the simplified fluid-solid coupling model is obviously higher than that of the comprehensive fluid-solid coupling model. The average water production rate in the stable stage of the former model is about 1.159 times greater than that of the later. Correspondingly, the accumulative water production is about 15.9% higher than that of the later.

Fig. 3-19 and Fig. 3-21 show the gas production rate and water production rate curve respectively, which are simulated by the comprehensive fluid-solid coupling model and the non-coupling normal model. The accumulative gas and water production curves respectively in two cases are shown in Figs. 3-20 and 3-22.

Fig. 3-19. Comparison of gas production rate

Fig. 3-20. Comparison of cumulative gas production

Fig. 3-21. Comparison of water production rate

2. Similarly, from Figs. 3-17 and 3-18 we can also see that the average water production rate and accumulative production simulated by the simplified fluid-solid coupling model is obviously higher than that of the comprehensive fluid-solid coupling model. The average water production rate in the stable stage of the former model is about 1.159 times greater than that of the later. Correspondingly, the accumulative water

Fig. 3-19 and Fig. 3-21 show the gas production rate and water production rate curve respectively, which are simulated by the comprehensive fluid-solid coupling model and the non-coupling normal model. The accumulative gas and water production curves

4.4% higher than that of the later.

production is about 15.9% higher than that of the later.

respectively in two cases are shown in Figs. 3-20 and 3-22.

Fig. 3-19. Comparison of gas production rate

Fig. 3-20. Comparison of cumulative gas production

Fig. 3-21. Comparison of water production rate

greater than that of the later. Accordingly, the gas accumulative production is about

Fig. 3-22. Comparison of cumulative water production

The comparisons of Figs. 3-19 and 3-20 show that the average gas production rate and accumulative gas production of the non-coupling normal model are obviously higher than that of the comprehensive fluid-solid coupling model. The average gas production rate in the stable stage of the former model is about 1.106 times higher than that of the later. And accordingly, the accumulative gas production of the former is about 15.9% higher than that of the later. Similarly, from Fig. 3-21 and Fig. 3-22 it can be seen that the average water production rate and accumulative water production of non-coupling normal model is also obviously greater than that of comprehensive model. The average water production rate during the steady stage of the former model is about 1.111 times higher than that of the later. Correspondingly, the accumulative water production of the former is about 11.1% higher than that of the later.

In conclusion, by analyzing the production performances of the non-coupling normal model, the simplified coupling model and the comprehensive, we can find that the development index, namely the gas production rate, the accumulative gas production and so on, obtained from the simplified model is the highest in the three models. The results of the non- coupling normal model are slightly lower. But the development index simulated by the comprehensive coupling model is significantly lower than the two former models. The reason for this phenomenon is that fluid-solid coupling has different influencing mechanism to production performance.

During the production by depressurization, the fluid-solid coupling effect has two aspects. On one side, the shrinkage of rock matrix caused by fluid-solid coupling effect can raise the elastic drive energy, which is helpful for gas output. So the development index of simplified fluid-solid coupling model which only considers the skeleton volume strain is higher than that of the non-coupling normal model. This is the good aspect that fluid-solid coupling effect contributes to gas production. On the other side, the shrinkage of rock pore space can also cause the reduction of permeability and porosity, and accordingly the seepage captivity also decreases which causes the adverse effect for production. Relatively, the change of porosity and permeability which is caused by fluid-solid coupling plays the dominated role on affecting the production performance. But the increasing of elastic drive energy is limited. Therefore, the predictive value of production index simulated by the comprehensive fluid-solid coupling model which considers the skeleton volume strain and physical parameters change is obviously lower than the results of the simplified fluid-solid coupling model and non-coupling normal model.

Fluid-Solid Coupling Numerical Simulation

Fig. 3-24. SH for different permeability

Fig. 3-25. Effect of permeability on gas rate

days.

Fig. 3-23. Decomposition front for different permeability

on Natural Gas Production from Hydrate Reservoirs by Depressurization 183

stronger the flow ability of reservoir fluid, the faster hydrate break down, and the faster decomposition front position promote. From Fig. 3-23 it can be seen that the distances between the decomposition front and the wellbore are 36m, 45.08m and 55.07m with respect to the formation permeability being 100mD, 200mD, 300mD after the well producing for 43

Figs. 3-25 and 3-27 show the gas and water production rate of hydrate reservoir respectively under the condition of different permeability. The accumulative gas and water production

curves with different formation permeability are shown as Figs. 3-26 and 3-28.

The variation of skeleton volume strain and petrophysical properties caused by fluid-solid coupling in the reservoir is the objective existence. So, the overall effect of fluid-solid coupling causes the development index, namely the gas production rate, the accumulative gas production and so on, is obviously lower than that of the non-coupling normal model. Therefore, from the productivity point of view, the coupling effect between the seepage field and the deformation field must not be neglected in the exploitation of hydrate reservoirs. Although there are many numerical models for production from hydrate reservoir in previous research, they failed to totally consider the coupling effect between the seepage filed and rock deformation field.

Compared with the current numerical models, the fluid-solid coupling model, which this study established for the production from hydrate reservoirs production by depressurization, has the superiority as follows: fully consideration of hydrate decomposition effect, the coupling effect of seepage field and deformation field, the change of the temperature, and the dynamic variation of the physical and mechanical properties of formations under the influence of many factors.

In the next portion, based on the gas-water two-phase non-isothermal fluid-solid coupling model established in this research, we will carry out a sensitive analysis on the main factors which affect the productivity from hydrate reservoirs.

#### **3.3 Influences analysis on productivity of gas production from hydrate reservoirs by depressurization**

The factors that affect the production performance for hydrate reservoirs by depressurization are very more and the extent of them is various. In order to comprehensively evaluate the influence degree of all factors on the development index and provide the theoretical basis for the production optimization of hydrate reservoirs, we primarily study the influence of the reservoir absolute permeability, the bottomhole pressure and the reservoir temperature on hydrate reservoir production index, such as the accumulative gas production.

#### **3.3.1 The effect of absolute permeability of formation**

The absolute permeability of formation is a parameter reflecting the seepage ability of porous medium and it is an important factor affecting the production rate after gas hydrates decomposing. In this portion, we use the non-isothermal fluid-solid coupling model developed in this article to simulate the production behaviors with the condition of three reservoir absolute permeability, namely 100mD, 200mD and 300mD. In the simulation the other conditions of hydrate reservoir keep the same. The emphasis is to analyze the influence of formation absolute permeability on production index, such as the hydrate decomposing rate, the gas production rate and accumulative gas production.

Figs. 3-23 and 3-24 show the hydrate decomposition front and the distribution of hydrate saturation SH under different permeability condition after the well producing for 43 days.

It is not difficult to find that the influence of absolute formation permeability is obvious to production performance of hydrate reservoir. The greater the formation absolute permeability is, the faster the propagate speed of the pressure drop is in the reservoir, the

The variation of skeleton volume strain and petrophysical properties caused by fluid-solid coupling in the reservoir is the objective existence. So, the overall effect of fluid-solid coupling causes the development index, namely the gas production rate, the accumulative gas production and so on, is obviously lower than that of the non-coupling normal model. Therefore, from the productivity point of view, the coupling effect between the seepage field and the deformation field must not be neglected in the exploitation of hydrate reservoirs. Although there are many numerical models for production from hydrate reservoir in previous research, they failed to totally consider the coupling effect between the seepage

Compared with the current numerical models, the fluid-solid coupling model, which this study established for the production from hydrate reservoirs production by depressurization, has the superiority as follows: fully consideration of hydrate decomposition effect, the coupling effect of seepage field and deformation field, the change of the temperature, and the dynamic variation of the physical and mechanical properties of

In the next portion, based on the gas-water two-phase non-isothermal fluid-solid coupling model established in this research, we will carry out a sensitive analysis on the main factors

**3.3 Influences analysis on productivity of gas production from hydrate reservoirs by**

The factors that affect the production performance for hydrate reservoirs by depressurization are very more and the extent of them is various. In order to comprehensively evaluate the influence degree of all factors on the development index and provide the theoretical basis for the production optimization of hydrate reservoirs, we primarily study the influence of the reservoir absolute permeability, the bottomhole pressure and the reservoir temperature on hydrate reservoir production index, such as the

The absolute permeability of formation is a parameter reflecting the seepage ability of porous medium and it is an important factor affecting the production rate after gas hydrates decomposing. In this portion, we use the non-isothermal fluid-solid coupling model developed in this article to simulate the production behaviors with the condition of three reservoir absolute permeability, namely 100mD, 200mD and 300mD. In the simulation the other conditions of hydrate reservoir keep the same. The emphasis is to analyze the influence of formation absolute permeability on production index, such as the hydrate

Figs. 3-23 and 3-24 show the hydrate decomposition front and the distribution of hydrate saturation SH under different permeability condition after the well producing for 43 days. It is not difficult to find that the influence of absolute formation permeability is obvious to production performance of hydrate reservoir. The greater the formation absolute permeability is, the faster the propagate speed of the pressure drop is in the reservoir, the

decomposing rate, the gas production rate and accumulative gas production.

filed and rock deformation field.

**depressurization**

accumulative gas production.

formations under the influence of many factors.

which affect the productivity from hydrate reservoirs.

**3.3.1 The effect of absolute permeability of formation**

Fig. 3-23. Decomposition front for different permeability

Fig. 3-24. SH for different permeability

stronger the flow ability of reservoir fluid, the faster hydrate break down, and the faster decomposition front position promote. From Fig. 3-23 it can be seen that the distances between the decomposition front and the wellbore are 36m, 45.08m and 55.07m with respect to the formation permeability being 100mD, 200mD, 300mD after the well producing for 43 days.

Figs. 3-25 and 3-27 show the gas and water production rate of hydrate reservoir respectively under the condition of different permeability. The accumulative gas and water production curves with different formation permeability are shown as Figs. 3-26 and 3-28.

Fig. 3-25. Effect of permeability on gas rate

Fluid-Solid Coupling Numerical Simulation

**3.3.2 The effect of bottomhole pressure**

production rate, the accumulative gas production, and so on.

and 29.11m respectively far from the wellbore.

Fig. 3-29. Dissociation front with different pressure

Fig. 3-30. SH for different pressure

on Natural Gas Production from Hydrate Reservoirs by Depressurization 185

The value of bottomhole pressure is an important parameter to control the gas production from hydrate reservoirs reasonably, and it not only affects the cumulative gas and water production of hydrate reservoirs, but also determines the propagate speed of hydrate dissociation front. Based on the non-isothermal fluid-solid coupling model above, this portion simulate the production performance under the conditions of the initial formation pressure 16.9MPa and three bottomhole pressure, namely 12.9MPa, 13.9MPa and 14.9MPa respectively. Other conditions in the progress keep the same. Emphatically the influence of bottomhole pressure on production index, such as the hydrate decomposing rate, the gas

Figs. 3-29 and 3-30 show the hydrates dissociation front position and the hydrate saturation SH distribution after 43 days production under the condition of different bottomhole pressures. Comparative analysis shows the lower bottomhole pressure leads to the faster pressure drop spread in reservoir, the greater pressure gradient, and the faster hydrate dissociation and decomposition front promoting rate. From Fig. 3-29, we can see that after 43 days production under the pressure being 12.9MPa, 13.9MPa and 14.9MPa respectively, the hydrate dissociation fronts move to the positions where the distances are 60.06m, 45.08m

Figs. 3-31 and 3-33 show of the gas and water production rate of hydrate reservoirs respectively under different bottomhole pressures. And Figs. 3-32 and 3-34 give the

accumulation gas and water production from hydrate reservoirs respectively.

Fig. 3-26. Effect of permeability on cumulative gas

Fig. 3-27. Effect of permeability on water rate

Fig. 3-28. Effect of permeability on cumulative water

From the results of the figures above, it is easy to find that despite of the different permeability of the formation, the gas and water production rate curves are similar, which can be divided into three stages on the whole, namely the rapidly rising stage, the stable stage and the gradually reducing stage. The greater the absolute permeability of reservoir is, the faster the propagate speed of pressure drop is, the faster hydrates break down, the greater the gas and water production rates are, and accordingly the bigger the accumulative gas and water productions are. After the well produces for about 200days, the accumulative gas production with the absolute permeability being 300mD is 7.05% higher than that of the formation with permeability being 200mD. The accumulative water production is 7.4% higher than that of the formation of which the permeability is 200mD. The accumulative gas and water production of the formation with the absolute permeability being 200mD is 5.9% and 6.34% respectively higher than that of formation whose absolute permeability is 100mD.

#### **3.3.2 The effect of bottomhole pressure**

184 Advances in Natural Gas Technology

Fig. 3-26. Effect of permeability on cumulative gas

Fig. 3-27. Effect of permeability on water rate

Fig. 3-28. Effect of permeability on cumulative water

From the results of the figures above, it is easy to find that despite of the different permeability of the formation, the gas and water production rate curves are similar, which can be divided into three stages on the whole, namely the rapidly rising stage, the stable stage and the gradually reducing stage. The greater the absolute permeability of reservoir is, the faster the propagate speed of pressure drop is, the faster hydrates break down, the greater the gas and water production rates are, and accordingly the bigger the accumulative gas and water productions are. After the well produces for about 200days, the accumulative gas production with the absolute permeability being 300mD is 7.05% higher than that of the formation with permeability being 200mD. The accumulative water production is 7.4% higher than that of the formation of which the permeability is 200mD. The accumulative gas and water production of the formation with the absolute permeability being 200mD is 5.9% and 6.34% respectively higher than that of formation whose absolute permeability is 100mD.

The value of bottomhole pressure is an important parameter to control the gas production from hydrate reservoirs reasonably, and it not only affects the cumulative gas and water production of hydrate reservoirs, but also determines the propagate speed of hydrate dissociation front. Based on the non-isothermal fluid-solid coupling model above, this portion simulate the production performance under the conditions of the initial formation pressure 16.9MPa and three bottomhole pressure, namely 12.9MPa, 13.9MPa and 14.9MPa respectively. Other conditions in the progress keep the same. Emphatically the influence of bottomhole pressure on production index, such as the hydrate decomposing rate, the gas production rate, the accumulative gas production, and so on.

Figs. 3-29 and 3-30 show the hydrates dissociation front position and the hydrate saturation SH distribution after 43 days production under the condition of different bottomhole pressures. Comparative analysis shows the lower bottomhole pressure leads to the faster pressure drop spread in reservoir, the greater pressure gradient, and the faster hydrate dissociation and decomposition front promoting rate. From Fig. 3-29, we can see that after 43 days production under the pressure being 12.9MPa, 13.9MPa and 14.9MPa respectively, the hydrate dissociation fronts move to the positions where the distances are 60.06m, 45.08m and 29.11m respectively far from the wellbore.

Fig. 3-29. Dissociation front with different pressure

Fig. 3-30. SH for different pressure

Figs. 3-31 and 3-33 show of the gas and water production rate of hydrate reservoirs respectively under different bottomhole pressures. And Figs. 3-32 and 3-34 give the accumulation gas and water production from hydrate reservoirs respectively.

Fluid-Solid Coupling Numerical Simulation

**3.3.3 The influence of formation temperature**

Fig. 3-35. Dissociation front with different temperature

cumulative gas production, etc.

being 14.9MPa.

restraint.

on Natural Gas Production from Hydrate Reservoirs by Depressurization 187

Comparative analysis show that, similarly to the former, the gas and water production rate curves with different bottomhole pressures generally can be divided into three stages: rapidly rising stage, stable stage and gradually reducing stage. And the lower bottomhole pressure, correspondingly, leads to the faster hydrate dissociation, the bigger gas and water production rate and the greater gas and water accumulative production. After 200 days production, when bottomhole pressure is 12.9MPa, gas accumulative production is 7.9% higher than that with the bottomhole pressure being 13.9MPa, and accordingly, the water accumulative production increase 9.6%. When bottomhole pressure is 13.9MPa, the gas and water accumulative production increase 6.9% and 8.7% respectively compared with that

It should be pointed out that low of the bottomhole pressure is contributive to improve the gas accumulation production and other development indexes. But at the same time, the lower bottomhole pressure can lead to the greater producing pressure difference and the worse stability in near well reservoir. Therefore, the stability of the reservoir near well bore should be considered in optimizing design of bottomhole pressure. To improve the production rate of hydrate reservoirs, the bottomhole pressure can not be reduced without

The formation temperature is a key parameter to affect the hydrate dissociation rate. Using the non-isothermal fluid-solid coupling model of the hydrate reservoir, this portion will simulate the production performance when formation temperatures are 289.5K, 290K and 290.5K respectively and other conditions are the same. We analyze emphatically the influence of formation temperature to the hydrate dissociation rate, gas production rate and

Figs. 3-35 and 3-36 show the location of hydrate dissociation front and the distribution of the saturation of hydrate (SH) after 43 days production under different formation temperatures. Comparisons indicate that the higher formation temperature leads to the faster hydrate dissociation, the greater dissociation front promoting rate and the further forward distance during the same time. As Fig. 3-35 shows, when the formation temperatures are 289.5K, 290K and 290.5K respectively, the hydrate dissociation fronts are 34.105m, 45.08m and

56.08m respectively far away from borehole after 43 days production similarly.

Fig. 3-31. Effect of pressure on gas production rate

Fig. 3-32. Effect of pressure on cumulative gas

Fig. 3-33. Effect of pressure on water production rate

Fig. 3-34. Effect of pressure on cumulative water

Fig. 3-31. Effect of pressure on gas production rate

Fig. 3-32. Effect of pressure on cumulative gas

Fig. 3-33. Effect of pressure on water production rate

Fig. 3-34. Effect of pressure on cumulative water

Comparative analysis show that, similarly to the former, the gas and water production rate curves with different bottomhole pressures generally can be divided into three stages: rapidly rising stage, stable stage and gradually reducing stage. And the lower bottomhole pressure, correspondingly, leads to the faster hydrate dissociation, the bigger gas and water production rate and the greater gas and water accumulative production. After 200 days production, when bottomhole pressure is 12.9MPa, gas accumulative production is 7.9% higher than that with the bottomhole pressure being 13.9MPa, and accordingly, the water accumulative production increase 9.6%. When bottomhole pressure is 13.9MPa, the gas and water accumulative production increase 6.9% and 8.7% respectively compared with that being 14.9MPa.

It should be pointed out that low of the bottomhole pressure is contributive to improve the gas accumulation production and other development indexes. But at the same time, the lower bottomhole pressure can lead to the greater producing pressure difference and the worse stability in near well reservoir. Therefore, the stability of the reservoir near well bore should be considered in optimizing design of bottomhole pressure. To improve the production rate of hydrate reservoirs, the bottomhole pressure can not be reduced without restraint.

### **3.3.3 The influence of formation temperature**

The formation temperature is a key parameter to affect the hydrate dissociation rate. Using the non-isothermal fluid-solid coupling model of the hydrate reservoir, this portion will simulate the production performance when formation temperatures are 289.5K, 290K and 290.5K respectively and other conditions are the same. We analyze emphatically the influence of formation temperature to the hydrate dissociation rate, gas production rate and cumulative gas production, etc.

Figs. 3-35 and 3-36 show the location of hydrate dissociation front and the distribution of the saturation of hydrate (SH) after 43 days production under different formation temperatures. Comparisons indicate that the higher formation temperature leads to the faster hydrate dissociation, the greater dissociation front promoting rate and the further forward distance during the same time. As Fig. 3-35 shows, when the formation temperatures are 289.5K, 290K and 290.5K respectively, the hydrate dissociation fronts are 34.105m, 45.08m and 56.08m respectively far away from borehole after 43 days production similarly.

Fig. 3-35. Dissociation front with different temperature

Fluid-Solid Coupling Numerical Simulation

Fig. 3-40. Effect of temperature on cumulative water

**4. Conclusions**

conclusions are obtained as follows:

on Natural Gas Production from Hydrate Reservoirs by Depressurization 189

Comparisons indicate that, as the same with the previous, the gas and water rate curves can be divided into three stages. They are the rapidly increasing stage, the stable stage and the gradually decreasing stage respectively. The higher formation temperature leads to the faster hydrate dissociation, the larger gas and water production rate. After 200 days production, the gas accumulative production with the formation temperature of 290.5 K is 7.0% higher than that of 290 K, and correspondingly, the water accumulative production increases 8.8%. And, the gas accumulative production with the formation temperature of 290 K is 5.9% higher than

Production from gas hydrate reservoirs by depressurization is a non-isothermal chemical and physical fluid-solid coupling process containing phase change. Based on the former study, this research closely held all kinds of the complicated physical and chemical mechanism during depressurization for gas production from hydrate reservoirs, and systematically studied the fluid-solid coupling numerical simulation of the stress state, the rule of physical parameters change and the productivity during the recovery process. Some

1. In view of the shortcomings of current mathematical models for gas production from hydrate reservoirs by depressurization and the comprehensive consideration of the hydrates decomposition thermodynamics, decomposition dynamics, mass and energy conservation in the decomposition process, gas-water two-phase flow in porous media, heat transfer and hydrates decomposition effect etc, we built up new phase change flow

2. Based on phase change seepage models, according to the theory of the fluid-solid coupling seepage and considering the factors of reservoir permeability anisotropy etc, this study established the gas-water two-phase non-isothermal fluid-solid coupling mathematical model for gas production from hydrate reservoirs by depressurization. This model mainly includes the equation of fluid-solid coupling seepage, the equations of fluid-solid coupling solid deformation field, the equations of decomposition dynamics for gas hydrate reservoirs, the temperature field equation, the auxiliary

3. On the base of the previous stress sensitivity experiments for the unconsolidated sandstone, this study built up comprehensive dynamic models of physical property parameters for gas hydrate reservoirs, which include permeability, porosity and elastic modulus etc. This model reflects the relationship between reservoir physical property

models of production by depressurization for gas hydrate reservoirs.

equation and the initial and boundary conditions etc.

that of 289.5 K, correspondingly, the water accumulative production increase 6.9%.

Fig. 3-36. SH with different temperature

Figs. 3-37 and 3-39 show the gas and water production rate of the hydrate reservoir with different formation temperatures. And Figs. 3-38 and 3-40 give the gas and water accumulative production with different formation temperatures.

Fig. 3-37. Effect of temperature on gas rate

Fig. 3-38. Effect of temperature on cumulative gas

Fig. 3-39. Effect of temperature on water rate

Fig. 3-40. Effect of temperature on cumulative water