An integrated approach for relative permeability estimation of fractured porous media: laboratory and numerical simulation studies

Abstract

Most carbonate reservoirs in Middle East are characterized as porous fractured reservoirs. Estimation of relative permeability of these highly heterogeneous reservoirs is challenging due to the existence of discontinuity in the fluid flow fractured porous media. Although relative permeability is an essential data for simulation of flow in fractured media, few attempts have so far been made to estimate the relative permeability curves. Most notable are the studies by Akin (J Pet Sci Eng 30(1)1–14, 2001), Al-sumaiti and Kazemi (2012), and Fahad (2013). This paper presents an integrated approach to history matching the oil drainage tests, which were carried out by unsteady state, on glass bed models with a single fracture at different orientations and to estimate the relative permeability curve. The integrated approach includes an inversion algorithm coupled with forward numerical modeling of fluid flow. The history matching of the displacement test data was obtained by using the Levenberg–Marquardt algorithm to minimize the error between the simulated and experimental data. In this algorithm, Corey-type power law is used to create relative permeability curves during the optimization procedures. The forward modeling is a 3D multiphase fluid simulator for flow through discrete fractures. Numerical results of fluid flow profiles and the optimized relative permeability curves for single fracture with different orientations and experimental validation with oil drainage tests are presented. The results of the optimized relative permeability data for single fracture are in a good agreement with the data derived by the correlation of Fahad (2013). These results prove that the presented approach can be used to upscale the relative permeability curve from laboratory scale to reservoir grid scale. The work on the upscaling of the estimated relative permeability curve of fractured porous media is under preparation and will be published soon.

Appendix: Derivation of multiphase flow equations in a poroelastic framework

The mathematical equations of multiphase fluid flow through matrix and fracture system are derived and then reformulated using finite element technique. The combining of mass and momentum balance equations is used in the derivation process with appropriate boundary conditions and a set of assumptions. These assumptions include: (1) The system is only for water and oil fluid flow, and (2) the fluid is isothermal and fluid phases are in a thermodynamic equilibrium state.

Mass conversation equation

The conversation equations of masses of fluid and solid for a representative element volume (V) in a porous medium will be derived separately. The mass of solid component of the medium can be expressed as a volume fraction of the porous solid as the following equation shows,

$$M_{\text{s}} = (1 - \phi )V\rho {}_{\text{s}}$$

(17)

where $\phi$ is the sum of fracture and matrix porosities and $\rho_{\text{s}}$ is the solid density

The mass conversation of solid constituent can be written as;

$$\frac{{{\text{D}}M_{\text{s}} }}{{{\text{D}}t}} = \frac{\text{D}}{{{\text{D}}t}}\int\limits_{v} {(1 - \phi )V\rho_{\text{s}} = \int\limits_{v} {\left[ {\frac{{\partial (1 - \phi )\rho_{\text{s}} }}{\partial t} + \frac{{\partial (1 - \phi )\rho_{\text{s}} u_{\text{s}} }}{\partial x}} \right]{\text{d}}v} } = 0$$

(18)

Equation (18) can be simplified if the continuum mechanism is assumed to be revealed;

$$\frac{{\partial (1 - \phi )\rho_{\text{s}} }}{\partial t} + \frac{{\partial (1 - \phi )\rho_{\text{s}} u_{\text{s}} }}{\partial x} = 0.0$$

(19)

Mass conversation for oil and water phases in rock and fracture systems can be derived similarly as follows:

The mass of solid component of the medium is expressed as:

$$M_{\psi } = (\phi )V\rho_{\psi }$$

(20)

The mass conversation of fluid constituent can be written as;

$$\frac{{{\text{D}}M_{\psi } }}{{{\text{D}}t}} = \frac{\text{D}}{{{\text{D}}t}}\int\limits_{v} {\phi S_{\psi } \rho_{\psi } = \int\limits_{v} {\left[ {\frac{{\partial \phi S_{\psi } \rho_{\psi } }}{\partial t} + \frac{{\partial \phi S_{\psi } \rho_{\psi } u_{\psi } }}{\partial x} - q_{\psi } } \right]{\text{d}}v} } = 0$$

(21)

where $\psi$ stands for oil and water phases, $S_{\psi }$ is the saturation of fluid phase (oil and water), $\rho_{\psi }$ is the fluid density, and $q_{\psi }$ is the rate of fluid exchange between matrix and fracture system. Equation (21) can be written separately for oil and water fluid phases separately by assuming that the continuum mechanism is prevailing as follows:

$$\frac{{\partial \phi S_{\text{w}} \rho_{\text{w}} }}{\partial t} + \frac{{\partial \phi S_{\text{w}} \rho_{\text{w}} U_{\text{w}} }}{\partial x} - q_{\text{w}} = 0$$

(22)

$$\frac{{\partial \phi S_{\text{o}} \rho_{\text{o}} }}{\partial t} + \frac{{\partial \phi S_{\text{o}} \rho_{\text{o}} U_{\text{o}} }}{\partial x} - q_{\text{o}} = 0$$

(23)

where $U_{\text{w}}$ and $U_{\text{o}}$ are the intrinsic velocities for water and oil phases in matrix and fractured system, respectively. The assumption of quasi-steady-state flux between the fracture network and matrix is used; therefore, flux rate can be expressed as:

$$q_{\psi } = \frac{{k_{1} \rho_{\psi } }}{{\mu_{\psi } }}k_{r\psi } \left( {p_{1\psi } - p_{2\psi } } \right)$$

(24)

where $k_{1}$ is referring to the permeability of the matrix and $p_{1\psi } ,p_{2\psi }$ are the pressure in the matrix and fracture system, respectively.

Darcy velocities for oil and water phases in matrix and fracture system are defined as:

$$\begin{aligned} u_{\text{w}} = \phi S_{\text{w}} \left( {U_{\text{w}} - u_{\text{s}} } \right) \hfill \\ u_{\text{o}} = \phi S_{\text{o}} \left( {U_{\text{o}} - u_{\text{s}} } \right) \hfill \\ \end{aligned}$$

(25)

From Eq. (25) intrinsic phase velocities can be expressed as:

$$\begin{aligned} U_{\text{w}} = \frac{{u_{\text{w}} }}{{\phi S_{\text{w}} }} + u_{\text{s}} \hfill \\ U_{\text{o}} = \frac{{u_{\text{o}} }}{{\phi S_{\text{o}} }} + u_{\text{s}} \hfill \\ \end{aligned}$$

(26)

Substituting Eq. (26) into Eq. (23) and then the equation reformulated as follows:

$$\frac{\partial }{\partial t}\left( {S_{\text{w}} \rho_{\text{w}} } \right) + \frac{\partial }{\partial x}\left( {\phi S_{\text{w}} \rho_{\text{w}} } \right)\left( {\frac{{u_{\text{w}} }}{{\phi S_{\text{w}} }} + u_{\text{s}} } \right) - q_{\text{w}} = 0.0$$

(27)

$$\frac{\partial }{\partial t}\left( {\phi S_{\text{o}} \rho_{\text{o}} } \right) + \frac{\partial }{\partial x}\left( {\phi S_{\text{o}} \rho_{\text{o}} } \right)\left( {\frac{{u_{\text{o}} }}{{\phi S_{\text{o}} }} + u_{\text{s}} } \right) - q_{\text{o}} = 0.0$$

(28)

Expanding the derivatives of Eqs. (19) and (27)

$$\frac{{\partial \rho_{\text{s}} }}{\partial t} - \phi \frac{{\partial \rho_{\text{s}} }}{\partial t} - \rho_{\text{s}} \frac{\partial \phi }{\partial t} - \phi \rho_{\text{s}} \frac{{\partial u_{\text{s}} }}{\partial x} - u_{\text{s}} \rho_{\text{s}} \frac{\partial \phi }{\partial x} - u_{\text{s}} \phi \frac{{\partial \rho_{\text{s}} }}{\partial x} + \rho_{\text{s}} \frac{{\partial u_{\text{s}} }}{\partial x} + u_{\text{s}} \frac{{\partial \rho_{\text{s}} }}{\partial x} = 0$$

(29)

$$\begin{aligned} \phi S_{\text{w}} \frac{{\partial \rho_{\text{w}} }}{\partial t} + \rho_{\text{w}} \phi \frac{{\partial S_{\text{w}} }}{\partial t} + S_{\text{w}} \rho_{\text{w}} \frac{\partial \phi }{\partial t} + \phi S_{\text{w}} \rho_{\text{w}} \frac{{\partial u_{\text{s}} }}{\partial x} + \phi S_{\text{w}} u_{\text{s}} \frac{{\partial \rho_{\text{w}} }}{\partial t} + \phi u_{\text{s}} \rho_{\text{w}} \frac{{\partial S_{\text{w}} }}{\partial x} \hfill \\ \quad + u_{\text{s}} S_{\text{w}} \rho_{\text{w}} \frac{\partial \phi }{\partial x} + u_{\text{w}} \frac{{\partial \rho_{\text{w}} }}{\partial x} + \rho_{\text{w}} \frac{{\partial u_{\text{w}} }}{\partial x} - q_{\text{w}} = 0 \hfill \\ \end{aligned}$$

(30)

Equations (29) and (30) can be reformulated and written as follows:

$$\left( {\frac{{\partial \rho_{\text{s}} }}{\partial t} + u_{\text{s}} \frac{{\partial \rho_{\text{s}} }}{\partial x}} \right) - \phi \left( {\frac{{\partial \rho_{\text{s}} }}{\partial t} + u_{\text{s}} \frac{{\partial \rho_{\text{s}} }}{\partial x}} \right) - \rho_{\text{s}} \left( {\frac{\partial \phi }{\partial t} + u_{\text{s}} \frac{\partial \phi }{\partial x}} \right) + \left( {1 - \phi } \right)\rho_{\text{s}} \frac{{\partial u_{\text{s}} }}{\partial x} = 0$$

(31)

$$\begin{aligned} \phi S_{\text{w}} \left( {\frac{{\partial \rho_{\text{w}} }}{\partial t} + u_{\text{s}} \frac{{\partial \rho_{\text{w}} }}{\partial x}} \right) + \rho_{\text{w}} \phi \left( {\frac{{\partial S_{\text{w}} }}{\partial t} + u_{\text{s}} \frac{{\partial S_{\text{w}} }}{\partial x}} \right) + S_{\text{w}} \rho_{\text{w}} \left( {\frac{\partial \phi }{\partial t} + u_{\text{s}} \frac{\partial \phi }{\partial x}} \right) + \phi S_{\text{w}} \rho_{\text{w}} \frac{{\partial u_{\text{s}} }}{\partial x} \hfill \\ \quad + u_{\text{w}} \frac{{\partial \rho_{\text{w}} }}{\partial x} - q_{\text{w}} = 0 \hfill \\ \end{aligned}$$

(32)

By considering the total derivative as:

$$\frac{{{\text{D}}(*)}}{{{\text{D}}t}} = \frac{\partial (*)}{\partial t} + u\frac{\partial (*)}{\partial x}$$

(33)

Equations (31) and (32) can be written by considering the total derivative form as:

$$\left( {\frac{{{\text{D}}\rho_{s} }}{{{\text{D}}t}}} \right) - \phi \left( {\frac{{{\text{D}}\rho_{\text{s}} }}{{{\text{D}}t}}} \right) - \rho_{\text{s}} \left( {\frac{{{\text{D}}\phi }}{{{\text{D}}t}}} \right) + \left( {1 - \phi } \right)\rho_{\text{s}} \frac{{\partial u_{\text{s}} }}{\partial x} = 0$$

(34)

$$\phi S_{\text{w}} \left( {\frac{{{\text{D}}\rho_{\text{w}} }}{{{\text{D}}t}}} \right) + \rho_{\text{w}} \phi \left( {\frac{{{\text{D}}S_{w} }}{{{\text{D}}t}}} \right) + S_{\text{w}} \rho_{\text{w}} \left( {\frac{{{\text{D}}\phi }}{{{\text{D}}t}}} \right) + \phi S_{\text{w}} \rho_{\text{w}} \frac{{\partial u_{\text{s}} }}{\partial x} + u_{\text{w}} \frac{{\partial \rho_{\text{w}} }}{\partial x} - q_{\text{w}} = 0$$

(35)

From Eq. (34), the expression of change of porosity with time can be written as:

$$\frac{{{\text{D}}\phi }}{{{\text{D}}t}} = \frac{1}{{\rho_{\text{s}} }}\left( {\frac{{{\text{D}}\rho_{\text{s}} }}{\partial D}} \right) - \frac{\phi }{{\rho_{\text{s}} }}\left( {\frac{{{\text{D}}\rho_{\text{s}} }}{{{\text{D}}t}}} \right) + \left( {1 - \phi } \right)\frac{{\partial u_{\text{s}} }}{\partial x}$$

(36)

Substituting Eq. (36) into Eq. (35), the following equation can be got:

$$\begin{aligned} \phi S_{\text{w}} \left( {\frac{{{\text{D}}\rho_{\text{w}} }}{{{\text{D}}t}}} \right) + \rho_{\text{w}} \phi \left( {\frac{{{\text{D}}S_{\text{w}} }}{{{\text{D}}t}}} \right) + S_{\text{w}} \rho_{\text{w}} \left( {\frac{1}{{\rho_{\text{s}} }}\left( {\frac{{{\text{D}}\rho_{\text{s}} }}{\partial Dt}} \right) - \frac{\phi }{{\rho_{\text{s}} }}\left( {\frac{{{\text{D}}\rho_{\text{s}} }}{{{\text{D}}t}}} \right) + \left( {1 - \phi } \right)\frac{{\partial u_{\text{s}} }}{\partial x}} \right) \hfill \\ \quad + \phi S_{\text{w}} \rho_{\text{w}} \frac{{\partial u_{\text{s}} }}{\partial x} + u_{\text{w}} \frac{{\partial \rho_{\text{w}} }}{\partial x} - q_{\text{w}} = 0 \hfill \\ \end{aligned}$$

(37)

Or.

$$\begin{aligned} \phi S_{\text{w}} \left( {\frac{{{\text{D}}\rho_{\text{w}} }}{{{\text{D}}t}}} \right) + \rho_{\text{w}} \phi \left( {\frac{{{\text{D}}S_{\text{w}} }}{{{\text{D}}t}}} \right) + \left( {1 - \phi } \right)\frac{{S_{\text{w}} \rho_{\text{w}} }}{{\rho_{\text{s}} }}\frac{{{\text{D}}\rho_{\text{s}} }}{\partial Dt} + \left( {1 - \phi } \right)S_{\text{w}} \rho_{\text{w}} \frac{{\partial u_{\text{s}} }}{\partial x} \hfill \\ \quad + \rho_{\text{w}} \frac{{\partial u_{\text{w}} }}{\partial x} + u_{\text{w}} \frac{{\partial \rho_{\text{w}} }}{\partial x} - q_{\text{w}} = 0 \hfill \\ \end{aligned}$$

(38)

The relationships between the change of fluid and rock densities with fluid and rock bulk modulus are given as:

$$\left\{ \begin{array}{lll}& \frac{{{\text{D}}\rho_{\text{w}} }}{{{\text{D}}t}} = \frac{{\rho_{\text{w}} }}{{K_{\text{w}} }}\frac{{{\text{D}}P_{\text{w}} }}{{{\text{D}}t}} \hfill \\ &\frac{{{\text{D}}\rho_{\text{o}} }}{{{\text{D}}t}} = \frac{{\rho_{\text{w}} }}{{K_{\text{o}} }}\frac{{{\text{D}}P_{\text{o}} }}{{{\text{D}}t}} \hfill \\ &\frac{{{\text{D}}\rho_{\text{s}} }}{{{\text{D}}t}} = \frac{{\rho_{\text{s}} \left[ {\frac{{\left( {1 - \phi } \right)}}{{K_{\text{s}} }}\frac{{{\text{D}}P}}{{{\text{D}}t}} + \frac{{\left( {1 - \phi } \right)}}{{K_{\text{ns}} }}\frac{{{\text{D}}P}}{{{\text{D}}t}} - \left( {1 - \alpha } \right)\frac{{\partial u_{\text{s}} }}{\partial x}} \right]}}{{\left( {1 - \phi } \right)}} \hfill \\ \end{array} \right.$$

(39)

where K_w and K_o are the bulk modulus of water and oil, respectively. K_s and K_ns are the bulk modulus of solid rock.

Substituting Eq. (39) into Eq. (38) gets the following equation:

$$\begin{aligned} \phi S_{\text{w}} \frac{{\rho_{\text{w}} }}{{K_{\text{w}} }}\frac{{{\text{D}}P_{\text{w}} }}{{{\text{D}}t}} + \rho_{\text{w}} \phi \left( {\frac{{{\text{D}}S_{\text{w}} }}{{{\text{D}}t}}} \right) + \left( {1 - \phi } \right)\frac{{S_{\text{w}} \rho_{\text{w}} }}{{\rho_{\text{s}} }}\left( {\frac{{\rho_{\text{s}} \left[ {\frac{{\left( {1 - \phi } \right)}}{{K_{\text{s}} }}\frac{{{\text{D}}P}}{{{\text{D}}t}} + \frac{{\left( {1 - \phi } \right)}}{{K_{\text{ns}} }}\frac{{{\text{D}}P}}{{{\text{D}}t}} - \left( {1 - \alpha } \right)\frac{{\partial u_{\text{s}} }}{\partial x}} \right]}}{{\left( {1 - \phi } \right)}}} \right) \hfill \\ \quad + \left( {1 - \phi } \right)S_{\text{w}} \rho_{\text{w}} \frac{{\partial u_{\text{s}} }}{\partial x} + \rho_{\text{w}} \frac{{\partial u_{\text{w}} }}{\partial x} + u_{\text{w}} \frac{{\partial \rho_{\text{w}} }}{\partial x} - q_{\text{w}} = 0 \hfill \\ \end{aligned}$$

(40)

In order to simplify Eq. (40) and get the final form of the two-phase fluid flow equation, the following assumptions are used as:

• The total derivative is considered as:

  $$\frac{{{\text{D}}(*)}}{{{\text{D}}t}} = \frac{\partial (*)}{\partial t} + u\frac{\partial (*)}{\partial x}$$

• Combine Eq. (C.24) with Darcy’s law:

  $$u_{\psi } = - \frac{{k_{\psi } }}{{\mu_{\psi } }}k_{r\psi } \left( {P_{\psi } + \rho_{\psi } gh} \right)$$

  (41)

• Solid velocity is small and can be neglected

where $\psi$ is standing for oil and water phase, $k_{r\psi }$ is phase relative permeability, $\rho_{\psi }$ is phase fluid density, g is the gravitational acceleration, and $h$ is the height above reference level.

The water-phase flow governing equation will be as follows:

$$\begin{aligned} \phi S_{\text{w}} \frac{{\rho_{\text{w}} }}{{K_{\text{w}} }}\frac{{\partial P_{\text{w}} }}{\partial t} + \rho_{\text{w}} \phi \left( {\frac{{\partial S_{\text{w}} }}{\partial t}} \right) + S_{\text{w}} \rho_{\text{w}} \left( \begin{aligned} \frac{{\left( {1 - \phi } \right)}}{{K_{\text{s}} }}\left( {S_{\text{o}} \frac{{\partial P_{\text{o}} }}{\partial t} + P_{\text{o}} \frac{{\partial S_{\text{o}} }}{\partial t} + S_{\text{w}} \frac{{\partial P_{\text{w}} }}{\partial t} + P_{\text{w}} \frac{{\partial S_{\text{w}} }}{\partial t}} \right) \hfill \\ + \frac{{\left( {1 - \phi } \right)}}{{K_{\text{ns}} }}\left( {S_{\text{o}} \frac{{\partial P_{\text{o}} }}{\partial t} + P_{\text{o}} \frac{{\partial S_{\text{o}} }}{\partial t} + S_{\text{w}} \frac{{\partial P_{\text{w}} }}{\partial t} + P_{\text{w}} \frac{{\partial S_{\text{w}} }}{\partial t}} \right) \hfill \\ \end{aligned} \right) \hfill \\ \quad - \rho_{\text{w}} \frac{{k_{\text{w}} }}{{\mu_{\text{w}} }}k_{\text{rw}} \frac{{\partial \left( {P_{\text{w}} + \rho_{\text{w}} gh} \right)}}{\partial x} - \rho_{\text{w}} \frac{{k_{\text{w}} }}{{\mu_{\text{w}} }}k_{\text{rw}} \left( {P_{\text{wm}} - P_{\text{wf}} } \right) + \left( {1 - \phi } \right)S_{\text{w}} \rho_{\text{w}} \frac{{\partial u_{\text{s}} }}{\partial x} - q_{\text{w}} = 0 \hfill \\ \end{aligned}$$

(42)

The oil-phase fluid flow governing equation can be written as:

$$\begin{aligned} \phi S_{\text{o}} \frac{{\rho_{\text{o}} }}{{K_{\text{o}} }}\frac{{\partial P_{\text{o}} }}{\partial t} + \rho_{\text{o}} \phi \left( {\frac{{\partial S_{\text{o}} }}{\partial t}} \right) + S_{\text{o}} \rho_{\text{o}} \left( \begin{aligned} \frac{{\left( {1 - \phi } \right)}}{{K_{\text{s}} }}\left( {S_{\text{o}} \frac{{\partial P_{\text{o}} }}{\partial t} + P_{\text{o}} \frac{{\partial S_{\text{o}} }}{\partial t} + S_{\text{w}} \frac{{\partial P_{\text{w}} }}{\partial t} + P_{\text{w}} \frac{{\partial S_{\text{w}} }}{\partial t}} \right) \hfill \\ + \frac{{\left( {1 - \phi } \right)}}{{K_{\text{ns}} }}\left( {S_{\text{o}} \frac{{\partial P_{\text{o}} }}{\partial t} + P_{\text{o}} \frac{{\partial S_{\text{o}} }}{\partial t} + S_{\text{w}} \frac{{\partial P_{\text{w}} }}{\partial t} + P_{\text{w}} \frac{{\partial S_{\text{w}} }}{\partial t}} \right) \hfill \\ \end{aligned} \right) \hfill \\ \quad - \rho_{\text{o}} \frac{{k_{\text{o}} }}{{\mu_{\text{o}} }}k_{\text{ro}} \frac{{\partial \left( {P_{\text{o}} + \rho_{\text{o}} gh} \right)}}{\partial x} - \rho_{\text{o}} \frac{{k_{\text{o}} }}{{\mu_{\text{o}} }}k_{\text{ro}} \left( {P_{\text{om}} - P_{\text{of}} } \right) + \left( {1 - \phi } \right)S_{\text{w}} \rho_{\text{w}} \frac{{\partial u_{\text{s}} }}{\partial x} - q_{\text{o}} = 0 \hfill \\ \end{aligned}$$

(43)

where $P_{\text{wm}}$ and $P_{\text{wf}}$ are the water pressures inside matrix and fractures, respectively.

Momentum balance equation

The relationship between total applied stresses $\sigma_{ij}$ and intergranular (effective) stresses $\sigma_{ij}^{\prime}$ is given by:

$$\sigma_{ij} = \sigma_{ij}^{\prime} - \alpha \delta_{ij} P$$

(44)

where $\alpha$ is pore pressure ratio factor, and $\delta_{ij}$ is the Kronecker delta.The linear constitutive relationships of the system can be expressed as:

$$\sigma_{ij}^{\prime} = D_{ijkl} \varepsilon_{kl}$$

(45)

where $D_{ijkl}$ is the elasticity matrix.

The equilibrium equation of motion for a solid can be defined as:

$$\sigma_{ij}^{{}} + F = 0$$

(46)

where $F$ is the vector of tractions applied on the body. The strain–displacement relationship is defined as:

$$\varepsilon_{ij} = \frac{1}{2}\left( {u_{ij} + u_{ji} } \right)$$

(47)

Rearranging Eq. 42 for water phase and introduce strain, then same procedure for oil phase:

$$- \nabla^{T} \left[ {\frac{{k_{ij} k_{\text{rw}} }}{{\mu_{\text{w}} \beta_{\text{w}} }}\nabla \left( {p_{\text{w}} + \rho_{\text{w}} gh} \right)} \right] + \phi \frac{\partial }{\partial t}\left( {\frac{{\rho_{\text{w}} s_{\text{w}} }}{{\beta_{\text{w}} }}} \right) + \rho_{\text{w}} \frac{{s_{\text{w}} }}{{\beta_{\text{w}} }}\left[ \begin{aligned} \left( {1 - \frac{D}{{3K_{\text{m}} }}} \right)\frac{\partial \varepsilon }{\partial t} + \frac{Dc}{{3K_{\text{m}} }} + \hfill \\ \hfill \\ \left( {\frac{1 - \phi }{{K_{\text{m}} }} - \frac{D}{{\left( {3K_{\text{m}} } \right)^{2} }}} \right)\frac{{\partial \bar{p}}}{\partial t} \hfill \\ \end{aligned} \right] + \rho_{\text{w}} Q_{\text{w}} = 0$$

(48)

Finite element discretization

In this section the finite element technique is used to derive the integral formulation of the derived coupled fluid flow and rock deformation equations through fractured system. In addition, the finite element technique is used to discretize the problem domain into nodes and elements. In this paper, four-node tetrahedral elements were used to represent the rock matrix in a 3D space, while the discrete fractures are represented by triangle elements in a 2D space. The value of the material properties is assumed to be constant within the element and allowed to vary from one element to the next. For the three-dimensional four-node tetrahedral element, the shape functions have the following form:

$$\begin{aligned} N_{1} &= 1 - \xi - \eta - \zeta \hfill \\ N_{2} &= \xi \hfill \\ N_{3} &= \eta \hfill \\ N_{4} &= \zeta \hfill \\ \end{aligned}$$

(49)

where $\left( {\xi ,\eta ,\zeta } \right)$ are the element local coordinates system. In order to transform the element geometry from the global system coordinates (x, y, and z) to the local coordinates $\left( {\xi ,\eta ,\zeta } \right)$, the following equations were used as:

$$\left\{ {\left. {\begin{array}{*{20}c} {\frac{{\partial N_{i} }}{\partial \xi }} \\ {\frac{{\partial N_{i} }}{\partial \eta }} \\ {\frac{{\partial N_{i} }}{\partial \zeta }} \\ \end{array} } \right\} = } \right.\left[ {\begin{array}{*{20}c} {\frac{\partial x}{\partial \xi }} & {\frac{\partial y}{\partial \xi }} & {\frac{\partial z}{\partial \xi }} \\ {\frac{\partial x}{\partial \eta }} & {\frac{\partial y}{\partial \eta }} & {\frac{\partial z}{\partial \eta }} \\ {\frac{\partial x}{\partial \zeta }} & {\frac{\partial y}{\partial \zeta }} & {\frac{\partial z}{\partial \zeta }} \\ \end{array} } \right]\left\{ {\left. {\begin{array}{*{20}c} {\frac{{\partial N_{i} }}{\partial x}} \\ {\frac{{\partial N_{i} }}{\partial y}} \\ {\frac{{\partial N_{i} }}{\partial z}} \\ \end{array} } \right\}} \right. = \left[ J \right]\left\{ {\left. {\begin{array}{*{20}c} {\frac{{\partial N_{i} }}{\partial x}} \\ {\frac{{\partial N_{i} }}{\partial y}} \\ {\frac{{\partial N_{i} }}{\partial z}} \\ \end{array} } \right\}} \right.$$

(50)

where J is the Jacobin matrix and can be expressed as:

$$\left[ J \right] = \left[ {\begin{array}{*{20}c} {\frac{\partial x}{\partial \xi }} & {\frac{\partial y}{\partial \xi }} & {\frac{\partial z}{\partial \xi }} \\ {\frac{\partial x}{\partial \eta }} & {\frac{\partial y}{\partial \eta }} & {\frac{\partial z}{\partial \eta }} \\ {\frac{\partial x}{\partial \zeta }} & {\frac{\partial y}{\partial \zeta }} & {\frac{\partial z}{\partial \zeta }} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\sum {\frac{{\partial N_{i} }}{\partial \xi }x_{i} } } & {\sum {\frac{{\partial N_{i} }}{\partial \xi }y_{i} } } & {\sum {\frac{{\partial N_{i} }}{\partial \xi }z_{i} } } \\ {\sum {\frac{{\partial N_{i} }}{\partial \eta }} x_{i} } & {\sum {\frac{{\partial N_{i} }}{\partial \eta }} y_{i} } & {\sum {\frac{{\partial N_{i} }}{\partial \eta }} z_{i} } \\ {\sum {\frac{{\partial N_{i} }}{\partial \zeta }} x_{i} } & {\sum {\frac{{\partial N_{i} }}{\partial \zeta }} x_{i} } & {\sum {\frac{{\partial N_{i} }}{\partial \zeta }} x_{i} } \\ \end{array} } \right]$$

(51)

Shape functions are used to obtain the variation of unknown variables within the element. These variables are approximated by using the interpolation function in finite element space as follows:

$$u = N_{u} \bar{u}$$

(52)

$$P_{\text{w}} = N_{\text{p}} \bar{P}_{\text{w}}$$

(53)

$$P_{\text{o}} = N_{\text{p}} \bar{P}_{\text{o}}$$

(54)

$$\varepsilon = B\bar{u}$$

(55)

where N is the corresponding shape function, $\bar{u}$, $\bar{P}_{\text{w}}$, and $\bar{P}_{\text{o}}$ are the nodal unknown variables. B is the strain displacement matrix and can be defined as:

$$\left[ {B^{T} } \right] = \left[ {\begin{array}{*{20}c} {\frac{{\partial N_{i} }}{\partial x}} & 0 & 0 \\ 0 & {\frac{{\partial N_{i} }}{\partial y}} & 0 \\ 0 & 0 & {\frac{{\partial N_{i} }}{\partial z}} \\ \end{array} \begin{array}{*{20}c} {\frac{{\partial N_{i} }}{\partial y}} & 0 & {\frac{{\partial N_{i} }}{\partial z}} \\ {\frac{{\partial N_{i} }}{\partial x}} & {\frac{{\partial N_{i} }}{\partial z}} & 0 \\ 0 & {\frac{{\partial N_{i} }}{\partial y}} & {\frac{{\partial N_{i} }}{\partial x}} \\ \end{array} } \right]$$

(56)

Equilibrium equation can be written in its general form as follows:

$$\int\limits_{v} {B^{T} \partial \sigma {\text{d}}v - \partial f} = 0$$

(57)

where $\partial f$ is the load vector applied on the boundary and V is the volume of the element.

Equation (57) can be written as:

$$\int\limits_{v} {B^{T} DB{\text{d}}v\frac{\partial u}{\partial t} - \int\limits_{v} {B^{T} \alpha N{\text{d}}v\frac{\partial P}{\partial t}} } = \frac{\partial f}{\partial t}$$

(58)

Average pore pressure (P) can be defined as:

$$P = S_{\text{o}} P_{\text{o}} + S_{\text{w}} P_{\text{w}}$$

(59)

Substituting Eq. (58) into Eq. (59);

$$\int\limits_{v} {B^{T} DB{\text{d}}v\frac{\partial u}{\partial t} - \int\limits_{v} {B^{T} \alpha N{\text{d}}v\left( {S_{\text{o}} \frac{{\partial P_{\text{o}} }}{\partial t} + P_{\text{o}} \frac{{\partial S_{\text{o}} }}{\partial t} + S_{\text{w}} \frac{{\partial P_{\text{w}} }}{\partial t} + P_{\text{w}} \frac{{\partial S_{\text{w}} }}{\partial t}} \right)} } = \frac{\partial f}{\partial t}$$

(60)

Equation (60) can be modified by inserting the capillary pressure as:

$$\int\limits_{v} {B^{T} DB{\text{d}}v\frac{\partial u}{\partial t} - \int\limits_{v} {B^{T} \alpha N{\text{d}}v\left( \begin{aligned} S_{\text{o}} \frac{{\partial P_{\text{o}} }}{\partial t} - P_{\text{o}} \frac{{\partial S_{\text{w}} }}{{\partial P_{\text{c}} }}\frac{{\partial P_{\text{c}} }}{\partial t} \hfill \\ + S_{\text{w}} \frac{{\partial P_{\text{w}} }}{\partial t} + P_{\text{w}} \frac{{\partial S_{\text{w}} }}{{\partial P_{\text{c}} }}\frac{{\partial P_{\text{c}} }}{\partial t} \hfill \\ \end{aligned} \right)} } = \frac{\partial f}{\partial t}$$

(61)

Discretization form for water-phase flow equation through fractured porous media is given below as:

$$\begin{aligned}& - \int\limits_{v} {\nabla N^{T} \rho_{\text{w}} \frac{{k_{\text{w}} }}{{\mu_{\text{w}} }}k_{\text{rw}} \nabla N\left( {P_{\text{w}} + \rho_{\text{w}} gh} \right){\text{d}}v} + \int\limits_{v} {N^{T} \phi S_{w} \frac{{\rho_{\text{w}} }}{{K_{\text{w}} }}N\frac{{\partial P_{\text{w}} }}{\partial t} + \rho_{\text{w}} \phi \left( {\frac{{\partial S_{\text{w}} }}{\partial t}} \right)} \hfill \\ &\quad + \int\limits_{v} {N^{T} S_{\text{w}} \rho_{\text{w}} N\left( \begin{aligned} \frac{{\left( {1 - \phi } \right)}}{{K_{\text{s}} }}\left( {S_{\text{w}} \frac{{\partial P_{\text{w}} }}{\partial t} + \left( {1 - S_{\text{w}} } \right)\frac{{\partial P_{\text{o}} }}{\partial t} - P_{\text{o}} \frac{{\partial S_{\text{w}} }}{\partial t}\frac{{\partial P_{\text{c}} }}{\partial t} + P_{\text{w}} \frac{{\partial S_{\text{w}} }}{\partial t}} \right) \hfill \\ + \frac{{\left( {1 - \phi } \right)}}{{K_{\text{ns}} }}\left( {S_{\text{w}} \frac{{\partial P_{\text{w}} }}{\partial t} + \left( {1 - S_{\text{w}} } \right)\frac{{\partial P_{\text{o}} }}{\partial t} - P_{\text{o}} \frac{{\partial S_{\text{w}} }}{\partial t}\frac{{\partial P_{\text{c}} }}{\partial t} + P_{\text{w}} \frac{{\partial S_{\text{w}} }}{\partial t}} \right) \hfill \\ \end{aligned} \right)} {\text{d}}v \hfill \\& \quad - \int\limits_{v} {N^{T} \rho_{\text{w}} \frac{{k_{\text{w}} }}{{\mu_{\text{w}} }}k_{\text{rw}} N\left( {P_{\text{wm}} - P_{\text{wf}} } \right) + \int\limits_{v} {N^{T} \left( {1 - \phi } \right)S_{\text{w}} \rho_{\text{w}} B\frac{{\partial u_{\text{s}} }}{\partial t}{\text{d}}v} - Q_{\text{w}} } = 0 \hfill \\ \end{aligned}$$

(62)

The process is repeated for oil-phase flow in fractured porous media. All of the previous equations are used for fluid flow through discrete fractures, but in 2D space the final flow equation through fracture network and matrix is given as:

$$\int \int_{\varOmega } {{\text{FEQ}}\,{\text{d}}\varOmega } = \int \int_{{\varOmega_{\text{m}} }} {{\text{FEQ}}\,{\text{d}}\varOmega_{\text{m}} } + b \times \int \int_{{\varOmega_{\text{f}} }} {{\text{FEQ d}}\bar{\varOmega }_{\text{f}} }$$

(63)

where $\bar{\varOmega }_{\text{f}}$ represents the fracture part of the domain as a 2D entity, and $\varOmega_{\text{m}}$ represents matrix domain and $\varOmega$ is the entire domain.