Gas Injection for Pressure Maintenance in Fractured Reservoirs

Abstract

Gas injection into the gas cap which is known as pressure maintenance or crestal gas injection is done to increase the reservoir pressure. Different types of gas may be injected in this method including producing gas, N₂, CO₂ etc. The injected gas is chosen base on the field development studies. Each of these gases has some advantage and disadvantages. Gas injection in naturally fractured reservoirs is a challenge which needs more investigation on this subject. This chapter summarizes the basic concepts of pressure maintenance and active mechanisms during pressure maintenance in naturally fractured reservoirs. Also, this chapter provides the essential concepts in simulation of pressure maintenance in fractured reservoirs.

Appendices

Appendices

5.1.1 Appendix A: Derivation of the Multiphase Flow Equations in Compositional Simulation

The multiphase flow equations govern compositional simulation will be derived. Basically, these equations are continuity equations for each component cover a volume element ΔxΔyΔz fixed in the space (Fig. 5.30) as following:

$$\begin{aligned} & \left( {Molar\;rate\;of\;component\;c\;in} \right) - \left( {Molar\;rate\;of\;component\;c\;out} \right) \\ &\quad +\left( {Molar\;injection\;rate\;of\;component\;c} \right) \\&\quad- \left( {Molar\;production\;rate\;of\;component\;c} \right) \\ &\quad = \left( {Molar\;accumulation\;rate\;of\;component\;c} \right) \\ \end{aligned}$$

(5.136)

Component c can be transported across the volume element boundary by two mechanisms: diffusion and convection. These mechanisms, injection rate, production rate, and accumulation rate will be discussed in details next.

Fig. 5.30

Volume element for deriving the multiphase flow equations

5.1.1.1 Convection Mechanism

Convective transport is the amount of material carried along by the bulk movement of the fluid. The driving force in convective transport is potential gradient. The molar rate in minus molar rate out of component c (mole/time) for x, y, and z directions due to convective transport in oil and gas phases are:

$$\begin{aligned} & \left( {\rho_{o} x_{c} v_{o,x} } \right)_{x} \Delta y\Delta z + \left( {\rho_{g} x_{c} v_{g,x} } \right)_{x} \Delta y\Delta z \\ & \;\; - \left( {\rho_{o} x_{c} v_{o,x} } \right)_{x + \Delta x} \Delta y\Delta z - \left( {\rho_{g} x_{c} v_{g,x} } \right)_{x + \Delta x} \Delta y\Delta z \\ & + \left( {\rho_{o} x_{c} v_{o,y} } \right)_{y} \Delta x\Delta z + \left( {\rho_{g} x_{c} v_{g,y} } \right)_{y} \Delta x\Delta z \\ & \;\; - \left( {\rho_{o} x_{c} v_{o,y} } \right)_{y + \Delta y} \Delta x\Delta z - \left( {\rho_{g} x_{c} v_{g,y} } \right)_{y + \Delta y} \Delta x\Delta z \\ & + \left( {\rho_{o} x_{c} v_{o,z} } \right)_{z} \Delta x\Delta y + \left( {\rho_{g} x_{c} v_{g,y} } \right)_{z} \Delta x\Delta y \\ & \;\; - \left( {\rho_{o} x_{c} v_{o,z} } \right)_{z + \Delta z} \Delta x\Delta y - \left( {\rho_{g} x_{c} v_{g,z} } \right)_{z + \Delta z} \Delta x\Delta y = 0 \\ \end{aligned}$$

(5.137)

where

$$\vec{v}_{p} = - \frac{{kk_{rp} }}{{\mu_{p} }}\left( {\overrightarrow {\nabla } p_{p} - \gamma_{p} \overrightarrow {\nabla } D} \right)\quad \quad \quad p = oil\;and\;gas$$

(5.138)

• $\rho_{o}$ and $\rho_{g}$ are the molar densities of oil and gas.

• ϕ is the porosity of the volume element.

• x_c is the mole fraction of component c in the oil phase.

• y_c is the mole fraction of component c in the gas phase.

5.1.1.2 Diffusion (Molecular) Transport

Molecular transport, or diffusion, can also add material across the faces of the volume element. If N_o,c and N_g,c are the diffusion molar fluxes of component c (mole c per time per area) in oil and gas phases, these quantities have units of mole per area per time and represent the amount of transport by diffusion. Therefore, following the previous approach, the molar rate in minus molar rate out of component c for x, y, and z directions by diffusion are:

$$\begin{aligned} & \left( {N_{c,o,x} } \right)_{x} \Delta y\Delta z + \left( {N_{c,g,x} } \right)_{x} \Delta y\Delta z - \left( {N_{c,o,x} } \right)_{x + \Delta x} \Delta y\Delta z\\&\quad - \left( {N_{c,g,x} } \right)_{x + \Delta x} \Delta y\Delta z + \left( {N_{c,o,y} } \right)_{y} \Delta x\Delta z + \left( {N_{c,g,y} } \right)_{y} \Delta x\Delta z\\&\quad - \left( {N_{c,o,y} } \right)_{y + \Delta y} \Delta x\Delta z - \left( {N_{c,g,y} } \right)_{y + \Delta y} \Delta x\Delta z \\ & \quad+ \left( {N_{c,o,z} } \right)_{z} \Delta x\Delta y + \left( {N_{c,g,z} } \right)_{z} \Delta x\Delta y \\&\quad- \left( {N_{c,o,z} } \right)_{z + \Delta z} \Delta x\Delta y - \left( {N_{c,g,z} } \right)_{z + \Delta z} \Delta x\Delta y = 0 \\ \end{aligned}$$

(5.139)

where

$$N_{c,o} = \phi \rho_{o} S_{o} \left( {D_{c,o} \nabla x_{c} } \right)$$

(5.140)

$$N_{c,g} = \phi \rho_{g} S_{g} \left( {D_{c,g} \nabla y_{c} } \right)$$

(5.141)

5.1.1.3 Production or Injection

Finally, production and/or injection of component c into the volume element are given by:

$$q_{D,fm,c} + q_{C,fm,c}$$

(5.142)

where $q_{D,fm}$ and $q_{C,fm}$ are diffusion and convection mass transfer between matrix and fracture at the matrix-fracture boundary.

5.1.1.4 Accumulation

The total mole of fluid in the volume element at any time is $\phi \left( {\rho_{o} S_{o} + \rho_{g} S_{g} } \right)\Delta x\Delta y\Delta z$, and the mole of component c is $\phi \left( {\rho_{o} S_{o} x_{c} + \rho_{g} S_{g} y_{c} } \right)\Delta x\Delta y\Delta z$. Therefore, the rate of accumulation of mole of component c is:

$$\frac{\partial }{\partial t}\left[ {\phi \left( {\rho_{o} S_{o} x_{c} + \rho_{g} S_{g} y_{c} } \right)} \right]\Delta x\Delta y\Delta z$$

(5.143)

5.1.1.5 Flow Equations

The flow equations can be obtained by substituting Eqs. (5.137) to (5.143) into Eq. (5.136). If the resulting equations are divided by volume element $\Delta x\Delta y\Delta z$ and applying limit when the volume element goes to zero, it becomes:

$$\begin{aligned} & - \left[ {\frac{{\partial \left( {\rho_{o} x_{c} v_{o,x} + \rho_{g} y_{c} v_{g,x} } \right)}}{\partial x} + \frac{{\partial \left( {\rho_{o} x_{c} v_{o,y} + \rho_{g} y_{c} v_{g,y} } \right)}}{\partial y} + \frac{{\partial \left( {\rho_{o} x_{c} v_{o,z} + \rho_{g} y_{c} v_{g,z} } \right)}}{\partial z}} \right] \\ &\,\,\,\, - \left[ {\frac{{\partial \left( {N_{c,o,x} + N_{c,g,x} } \right)}}{\partial x} + \frac{{\partial \left( {N_{c,o,y} + N_{c,g,y} } \right)}}{\partial y} + \frac{{\partial \left( {N_{c,o,z} + N_{c,g,z} } \right)}}{\partial z}} \right] \\ &\,\,\,\,+ q_{D,fm,c} + q_{C,fm,c} = \frac{\partial }{\partial t}\left[ {\phi \left( {\rho_{o} S_{o} x_{c} + \rho_{g} S_{g} y_{c} } \right)} \right] \\ \end{aligned}$$

(5.144)

Or, in vector notation,

$$\begin{aligned} & - \nabla . \left( {\rho_{o} x_{c} v_{o} + \rho_{g} y_{c} v_{g} } \right) - \nabla . \left( {N_{c,o} + N_{c,g} } \right) + q_{D,fm,c} + q_{C,fm,c} \\ & = \frac{\partial }{\partial t}\left[ {\phi \left( {\rho_{o} S_{o} x_{c} + \rho_{g} S_{g} y_{c} } \right)} \right] \\ \end{aligned}$$

(5.145)

By substituting v_o and v_g from Eq. (5.138) and N_c,o and N_c,g from Eqs. (5.140) and (5.141) into Eq. (5.145), it becomes:

$$\begin{aligned} & \nabla . \left( {\rho_{o} x_{c} \frac{{kk_{ro} }}{{\mu_{o} }}\left( {\overrightarrow {\nabla } p_{o} - \gamma_{o} \overrightarrow {\nabla } D} \right) + \rho_{g} y_{c} \frac{{kk_{rg} }}{{\mu_{g} }}\left( {\overrightarrow {\nabla } p_{g} - \gamma_{g} \overrightarrow {\nabla } D} \right)} \right) \\ & + \nabla . \left( {\phi \rho_{o} S_{o} \left( {D_{c,o} \nabla x_{c} } \right) + \phi \rho_{g} S_{g} \left( {D_{c,g} \nabla y_{c} } \right)} \right) + q_{D,fm,c} + q_{C,fm,c} \\ & \; = \frac{\partial }{\partial t}\left[ {\phi \left( {\rho_{o} S_{o} x_{c} + \rho_{g} S_{g} y_{c} } \right)} \right] \\ \end{aligned}$$

(5.146)

Equation (5.146) is a general case of compositional multiphase flow through porous media for each component in oil and gas phases. For the water phase, considering that hydrocarbon phases are immiscible in water, we have,

$$\nabla . \left( {\rho_{w} \frac{{kk_{rw} }}{{\mu_{w} }}\left( {\overrightarrow {\nabla } p_{w} - \gamma_{w} \overrightarrow {\nabla } D} \right)} \right) = \frac{\partial }{\partial t}\left[ {\phi \left( {\rho_{w} S_{w} } \right)} \right].$$

(5.147)

5.1.2 Appendix B: Discretizing the Flow Equations

The final form of the general hydrocarbon flow equations, as obtained in Appendix A (Eq. (5.147)), is as follows:

$$\begin{array}{*{20}l} &{ \nabla \cdot \left( {\rho _{o} x_{c} \frac{{kk_{{ro}} }}{{\mu _{o} }}\left( {\vec{\nabla }p_{o} - \gamma _{o} \vec{\nabla }D} \right) + \rho _{g} y_{c} \frac{{kk_{{rg}} }}{{\mu _{g} }}\left( {\vec{\nabla }p_{g} - \gamma _{g} \vec{\nabla }D} \right)} \right)_{{i,j,k}}^{{n + 1}} } \hfill \\ &{ \; + \nabla \cdot \left( {\phi \rho _{o} S_{o} \left( {D_{{c,o}} \nabla x_{c} } \right) + \phi \rho _{g} S_{g} \left( {D_{{c,g}} \nabla y_{c} } \right)} \right)_{{i,j,k}}^{{n + 1}} } \hfill \\ &{ \; + \nabla \cdot \left( {q_{{D,fm,c}} + q_{{C,fm,c}} } \right)_{{i,j,k}}^{{n + 1}} \frac{\partial }{{\partial t}}\left[ {\phi \left( {\rho _{o} S_{o} x_{c} + \rho _{g} S_{g} y_{c} } \right)} \right]_{{i,j,k}}^{{n + 1}} } \hfill \\ \end{array} \;c = 1,2, \ldots ,n_{c}$$

(5.148)

For simplicity, flow equations are discretized in x-direction only. The same procedure can be applied to y and z directions. Equation (5.148) in x-direction becomes:

$$\begin{aligned} & \frac{\partial }{\partial x}\left( {\rho_{o} x_{c} \frac{{kk_{ro} }}{{\mu_{o} }}\left( {\frac{{\partial p_{o} }}{\partial t} - \gamma_{o} \frac{\partial D}{{\partial t}}} \right) + \rho_{g} y_{c} \frac{{kk_{rg} }}{{\mu_{g} }}\left( {\frac{{\partial p_{g} }}{\partial t} - \gamma_{g} \frac{\partial D}{{\partial t}}} \right)} \right)_{i,j,k}^{n + 1} \\ & + \frac{\partial }{\partial x}\left( {\phi \rho_{o} S_{o} \left( {D_{c,o} \frac{{\partial x_{c} }}{\partial x}} \right) + \phi \rho_{g} S_{g} \left( {D_{c,g} \frac{{\partial y_{c} }}{\partial x}} \right)} \right)_{i,j,k}^{n + 1} \\ & + \left( {q_{D,fm,c} + q_{C,fm,c} } \right)_{i,j,k}^{n + 1} \frac{\partial }{\partial t}\left[ {\phi \left( {\rho_{o} S_{o} x_{c} + \rho_{g} S_{g} y_{c} } \right)} \right]_{i,j,k}^{n + 1} \\ \end{aligned}$$

(5.149)

5.1.2.1 Discretization Oil and Gas Convective Terms in x-direction

Defining oil and gas convective terms as:

$$v_{ox} = \rho_{o} x_{c} \frac{{k_{x} k_{ro} }}{{\mu_{o} }}\left( {\frac{{\partial p_{o} }}{\partial x} - \gamma_{o} \frac{\partial D}{{\partial x}}} \right)$$

(5.150)

$$v_{gx} = \rho_{g} y_{c} \frac{{k_{x} k_{rg} }}{{\mu_{g} }}\left( {\frac{{\partial p_{g} }}{\partial x} - \gamma_{g} \frac{\partial D}{{\partial x}}} \right)$$

(5.151)

Oil convective flux term will be discretized first. By substituting Eq. (5.150) into Eq. (5.149), the oil convective flux term in Eq. (5.149) in x-direction becomes:

$$\frac{\partial }{\partial x}\left[ {\rho_{o} x_{c} \frac{{k_{x} k_{ro} }}{{\mu_{o} }}\left( {\frac{{\partial p_{o} }}{\partial x} - \gamma_{o} \frac{\partial D}{{\partial x}}} \right)} \right]_{i,j,k}^{n + 1} = \frac{{\partial v_{{o, x_{i,j,k} }}^{n + 1} }}{\partial x}$$

(5.152)

Using central finite differences into Eq. (5.152):

$$\frac{{\partial v_{{o, x_{i,j,k} }}^{n + 1} }}{\partial x} = \frac{{v_{{o, x_{{i + \frac{1}{2},j,k}} }}^{n + 1} - v_{{o, x_{{i - \frac{1}{2},j,k}} }}^{n + 1} }}{{\Delta x_{i,j,k} }}\quad \quad \quad \Delta x_{i,j,k} = x_{{i + \frac{1}{2},j,k}} - x_{{i - \frac{1}{2},j,k}}$$

(5.153)

where,

$$\begin{array}{*{20}c} {v_{{o,x_{{i + \frac{1}{2},j,k}}^{n + 1} }} = \left( {\rho_{o} x_{c} \frac{{k_{x} k_{ro} }}{{\mu_{o} }}} \right)_{{i + \frac{1}{2},j,k}}^{n + 1} \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ {\left[ {\frac{{p_{{o_{i + 1,j,k} }} - p_{{o_{i,j,k} }} }}{{\Delta x_{{i + \frac{1}{2},j,k}} }} - \gamma_{{o_{{i + \frac{1}{2},j,k}} }} \frac{{D_{i + 1,j,k} - D_{i,j,k} }}{{\Delta x_{{i + \frac{1}{2},j,k}} }}} \right]^{n + 1} } \\ \end{array} \Delta x_{{i + \frac{1}{2},j,k}} = x_{i + 1,j,k} - x_{i,j,k}$$

(5.154)

and,

$$\begin{aligned} v_{{o,x_{{i - \frac{1}{2},j,k}}^{n + 1} }}& = \left( {\rho_{o} x_{c} \frac{{k_{x} k_{ro} }}{{\mu_{o} }}} \right)_{{i - \frac{1}{2},j,k}}^{n + 1} \\& \quad\left[ {\frac{{p_{{o_{i,j,k} }} - p_{{o_{i - 1,j,k} }} }}{{\Delta x_{{i - \frac{1}{2},j,k}} }} - \gamma_{{o_{{i - \frac{1}{2},j,k}} }} \frac{{D_{i,j,k} - D_{i - 1,j,k} }}{{\Delta x_{{i - \frac{1}{2},j,k}} }}} \right]^{n + 1}\end{aligned}$$

(5.155)

By substituting Eqs. (5.154) and (5.155) into Eq. (5.152):

$$\begin{aligned} & \frac{{\partial v_{{g,~x_{{i,j,k}} }}^{{n + 1}} }}{{\partial x}} \\ & \;\; = \frac{1}{{\Delta x_{{i,j,k}} }}\left[ {\left( {\frac{{\rho _{g} x_{c} \frac{{k_{x} k_{{rg}} }}{{\mu _{g} }}}}{{\Delta x}}} \right)_{{i + \frac{1}{2},j,k}} \left( {(p_{{g_{{i + 1,j,k}} }} - p_{{g_{{i,j,k}} }} ) - \gamma _{{g_{{i + \frac{1}{2},j,k}} }} \left( {D_{{i + 1,j,k}} - D_{{i,j,k}} } \right)} \right)} \right]^{{n + 1}} \\ & \;\; - \frac{1}{{\Delta x_{{i,j,k}} }}\left[ {\left( {\frac{{\rho _{g} x_{c} \frac{{k_{x} k_{{rg}} }}{{\mu _{g} }}}}{{\Delta x}}} \right)_{{i - \frac{1}{2},j,k}} \left( {(p_{{g_{{i,j,k}} }} - p_{{g_{{i - 1,j,k}} }} ) - \gamma _{{g_{{i - \frac{1}{2},j,k}} }} \left( {D_{{i,j,k}} - D_{{i - 1,j,k}} } \right)} \right)} \right]^{{n + 1}} \\ \end{aligned}$$

(5.156)

Gas convective flux term can be discretized by the same manner as:

$$\begin{aligned} &\frac{{\partial v_{{g, x_{i,j,k} }}^{n + 1} }}{\partial x}\\&\;\; = \frac{1}{{\Delta x_{i,j,k} }}\left[ {\left( {\frac{{\rho_{g} x_{c} \frac{{k_{x} k_{rg} }}{{\mu_{g} }}}}{\Delta x}} \right)_{{i + \frac{1}{2},j,k}} \left( {(p_{{g_{i + 1,j,k} }} - p_{{g_{i,j,k} }} ) - \gamma_{{g_{{i + \frac{1}{2},j,k}} }} \left( {D_{i + 1,j,k} - D_{i,j,k} } \right)} \right)} \right]^{n + 1} \\ & \;\; - \frac{1}{{\Delta x_{i,j,k} }}\left[ {\left( {\frac{{\rho_{g} x_{c} \frac{{k_{x} k_{rg} }}{{\mu_{g} }}}}{\Delta x}} \right)_{{i - \frac{1}{2},j,k}} \left( {(p_{{g_{i,j,k} }} - p_{{g_{i - 1,j,k} }} ) - \gamma_{{g_{{i - \frac{1}{2},j,k}} }} \left( {D_{i,j,k} - D_{i - 1,j,k} } \right)} \right)} \right]^{n + 1} \\ \end{aligned}$$

(5.157)

5.1.2.2 Discretization the Oil and Gas Diffusive Flux Term in x-Direction

Oil diffusive flux term will be discretized first as:

$$\begin{aligned} & \frac{\partial }{\partial x}\left( {\phi \rho_{o} S_{o} \left( {D_{c,o} \frac{{\partial x_{c} }}{\partial x}} \right)} \right)_{i,j,k}^{n + 1} \\ & = \frac{1}{{\Delta x_{i,j,k} }}\left[ {\left( {\phi \rho_{o} S_{o} \left( {D_{c,o} \frac{{\partial x_{c} }}{\partial x}} \right)} \right)_{{i + \frac{1}{2},j,k}}^{n + 1} - \left( {\phi \rho_{o} S_{o} \left( {D_{c,o} \frac{{\partial x_{c} }}{\partial x}} \right)} \right)_{{i - \frac{1}{2},j,k}}^{n + 1} } \right] \\ \end{aligned}$$

(5.158)

Expanding the (i+1/2) and (i−1/2) term in Eq. (5.158):

$$\left( {\phi \rho_{o} S_{o} \left( {D_{c,o} \frac{{\partial x_{c} }}{\partial x}} \right)} \right)_{{i + \frac{1}{2},j,k}}^{n + 1} = \left( {\frac{{\phi \rho_{o} S_{o} D_{c,o} }}{\Delta x}} \right)_{{i + \frac{1}{2},j,k}}^{n + 1} \left( {x_{{c_{i + 1,j,k} }} - x_{{c_{i,j,k} }} } \right)^{n + 1}$$

(5.159)

$$\left( {\phi \rho_{o} S_{o} \left( {D_{c,o} \frac{{\partial x_{c} }}{\partial x}} \right)} \right)_{{i - \frac{1}{2},j,k}}^{n + 1} = \left( {\frac{{\phi \rho_{o} S_{o} D_{c,o} }}{\Delta x}} \right)_{{i - \frac{1}{2},j,k}}^{n + 1} \left( {x_{{c_{i,j,k} }} - x_{{c_{i - 1,j,k} }} } \right)^{n + 1}$$

(5.160)

Substituting Eqs. (5.159) and (5.160) in Eq. (5.158):

$$\begin{aligned} & \frac{\partial }{{\partial x}}\left( {\phi \rho _{o} S_{o} \left( {D_{{c,o}} \nabla x_{c} } \right)} \right)_{{i,j,k}}^{{n + 1}} \\ & \;\; = \frac{1}{{\Delta x_{{i,j,k}} }}\left[ {\left( {\frac{{\phi \rho _{o} S_{o} D_{{c,o}} }}{{\Delta x}}} \right)_{{i + \frac{1}{2},j,k}}^{{n + 1}} \left( {x_{{c_{{i + 1,j,k}} }} - x_{{c_{{i,j,k}} }} } \right)^{{n + 1}} } \right. \\ & \left. {\;\; - \left( {\frac{{\phi \rho _{o} S_{o} D_{{c,o}} }}{{\Delta x}}} \right)_{{i - \frac{1}{2},j,k}}^{{n + 1}} \left( {x_{{c_{{i,j,k}} }} - x_{{c_{{i - 1,j,k}} }} } \right)^{{n + 1}} } \right] \\ \end{aligned}$$

(5.161)

The same procedure can be used to discretize gas molecular diffusion term as:

$$\begin{aligned} & \frac{\partial }{{\partial x}}\left( {\phi \rho _{g} S_{g} \left( {D_{{c,g}} \nabla y_{c} } \right)} \right)_{{i,j,k}}^{{n + 1}} \\ & \;\; = \frac{1}{{\Delta x_{{i,j,k}} }}\left[ {\left( {\frac{{\phi \rho _{g} S_{g} D_{{c,g}} }}{{\Delta x}}} \right)_{{i + \frac{1}{2},j,k}}^{{n + 1}} \left( {y_{{c_{{i + 1,j,k}} }} - y_{{c_{{i,j,k}} }} } \right)^{{n + 1}} } \right. \\ &\left. { \;\; - \left( {\frac{{\phi \rho _{g} S_{g} D_{{c,g}} }}{{\Delta x}}} \right)_{{i - \frac{1}{2},j,k}}^{{n + 1}} \left( {y_{{c_{{i,j,k}} }} - y_{{c_{{i - 1,j,k}} }} } \right)^{{n + 1}} } \right] \\ \end{aligned}$$

(5.162)

5.1.2.3 Discretization the Accumulation Term

The accumulation term is discretized by using regressive finite differences in time as:

$$\begin{aligned} & \frac{\partial }{\partial t}\left[ {\phi \left( {x_{c} \rho_{o} S_{o} + y_{c} \rho_{g} S_{g} } \right)} \right]_{i,j,k} \\ & \;\; = \frac{1}{\Delta t}\left[ {\left( {\phi x_{c} \rho_{o} S_{o} + \phi y_{c} \rho_{g} S_{g} } \right)^{n + 1} - \left( {\phi x_{c} \rho_{o} S_{o} + \phi y_{c} \rho_{g} S_{g} } \right)^{n} } \right]_{i,j,k} \\ \end{aligned}$$

(5.163)

5.1.2.4 Final Form of Discretized Flow Equations

Substituting Eqs. (5.156), (5.157), (5.161), (5.162), and (5.163) into Eq. (5.149):

$$\begin{aligned} & \frac{1}{{\Delta x_{{i,j,k}} }}\left[ {\left( {\frac{{\rho _{o} x_{c} \frac{{k_{x} k_{{ro}} }}{{\mu _{o} }}}}{{\Delta x}}} \right)_{{i + \frac{1}{2},j,k}} \left( {(p_{{o_{{i + 1,j,k}} }} - p_{{o_{{i,j,k}} }} ) - \gamma _{{o_{{i + \frac{1}{2},j,k}} }} \left( {D_{{i + 1,j,k}} - D_{{i,j,k}} } \right)} \right)} \right]^{{n + 1}} \\ & \; - \frac{1}{{\Delta x_{{i,j,k}} }}\left[ {\left( {\frac{{\rho _{o} x_{c} \frac{{k_{x} k_{{ro}} }}{{\mu _{o} }}}}{{\Delta x}}} \right)_{{i - \frac{1}{2},j,k}} \left( {(p_{{o_{{i,j,k}} }} - p_{{o_{{i - 1,j,k}} }} ) - \gamma _{{o_{{i - \frac{1}{2},j,k}} }} \left( {D_{{i,j,k}} - D_{{i - 1,j,k}} } \right)} \right)} \right]^{{n + 1}} \\ & \; + ~\frac{1}{{\Delta x_{{i,j,k}} }}\left[ {\left( {\frac{{\rho _{g} y_{c} \frac{{k_{x} k_{{rg}} }}{{\mu _{g} }}}}{{\Delta x}}} \right)_{{i + \frac{1}{2},j,k}} \left( {(p_{{g_{{i + 1,j,k}} }} - p_{{g_{{i,j,k}} }} ) - \gamma _{{g_{{i + \frac{1}{2},j,k}} }} \left( {D_{{i + 1,j,k}} - D_{{i,j,k}} } \right)} \right)} \right]^{{n + 1}} \\ & \; - \frac{1}{{\Delta x_{{i,j,k}} }}\left[ {\left( {\frac{{\rho _{g} y_{c} \frac{{k_{x} k_{{rg}} }}{{\mu _{g} }}}}{{\Delta x}}} \right)_{{i - \frac{1}{2},j,k}} \left( {(p_{{g_{{i,j,k}} }} - p_{{g_{{i - 1,j,k}} }} ) - \gamma _{{g_{{i - \frac{1}{2},j,k}} }} \left( {D_{{i,j,k}} - D_{{i - 1,j,k}} } \right)} \right)} \right]^{{n + 1}} \\ & \; + \frac{1}{{\Delta x_{{i,j,k}} }}\left[ {\left( {\frac{{\phi \rho _{o} S_{o} D_{{c,o}} }}{{\Delta x}}} \right)_{{i + \frac{1}{2},j,k}}^{{n + 1}} \left( {x_{{c_{{i + 1,j,k}} }} - x_{{c_{{i,j,k}} }} } \right)^{{n + 1}} } \right. \\ &\left. {\;\; - \left( {\frac{{\phi \rho _{o} S_{o} D_{{c,o}} }}{{\Delta x}}} \right)_{{i - \frac{1}{2},j,k}}^{{n + 1}} \left( {x_{{c_{{i,j,k}} }} - x_{{c_{{i - 1,j,k}} }} } \right)^{{n + 1}} } \right] \\ & \; + \frac{1}{{\Delta x_{{i,j,k}} }}\left[ {\left( {\frac{{\phi \rho _{g} S_{g} D_{{c,g}} }}{{\Delta x}}} \right)_{{i + \frac{1}{2},j,k}}^{{n + 1}} \left( {y_{{c_{{i + 1,j,k}} }} - y_{{c_{{i,j,k}} }} } \right)^{{n + 1}} } \right. \\ &\left. { \;\; - \left( {\frac{{\phi \rho _{g} S_{g} D_{{c,g}} }}{{\Delta x}}} \right)_{{i - \frac{1}{2},j,k}}^{{n + 1}} \left( {y_{{c_{{i,j,k}} }} - y_{{c_{{i - 1,j,k}} }} } \right)^{{n + 1}} } \right] \\ & \; + \left( {q_{{D,fm,c}} + q_{{C,fm,c}} } \right)_{{i,j,k}}^{{n + 1}} \\ & \; = \frac{1}{{\Delta t}}\left[ {\left( {\phi x_{c} \rho _{o} S_{o} + \phi y_{c} \rho _{g} S_{g} } \right)^{{n + 1}} - \left( {\phi x_{c} \rho _{o} S_{o} + \phi y_{c} \rho _{g} S_{g} } \right)^{n} } \right]_{{i,j,k}} \\ \end{aligned}$$

(5.164)

Multiplying Eq. (5.164) by the volume of the grid cell, $V_{r,i,j,k} = \Delta y_{i,j,k} \Delta z_{i,j,k} \Delta x_{i,j,k}$, and rearranging, it becomes:

$$\begin{aligned} & \left[ {\left( {\frac{{\Delta y_{{i,j,k}} \Delta z_{{i,j,k}} \rho _{o} x_{c} \frac{{k_{x} k_{{ro}} }}{{\mu _{o} }}}}{{\Delta x}}} \right)_{{i + \frac{1}{2},j,k}} \left( {(p_{{o_{{i + 1,j,k}} }} - p_{{o_{{i,j,k}} }} ) - \gamma _{{o_{{i + \frac{1}{2},j,k}} }} \left( {D_{{i + 1,j,k}} - D_{{i,j,k}} } \right)} \right)} \right]^{{n + 1}} \\ & \; - \left[ {\left( {\frac{{\Delta y_{{i,j,k}} \Delta z_{{i,j,k}} \rho _{o} x_{c} \frac{{k_{x} k_{{ro}} }}{{\mu _{o} }}}}{{\Delta x}}} \right)_{{i - \frac{1}{2},j,k}} \left( {(p_{{o_{{i,j,k}} }} - p_{{o_{{i - 1,j,k}} }} ) - \gamma _{{o_{{i - \frac{1}{2},j,k}} }} \left( {D_{{i,j,k}} - D_{{i - 1,j,k}} } \right)} \right)} \right]^{{n + 1}} \\ & \; + ~\left[ {\left( {\frac{{\Delta y_{{i,j,k}} \Delta z_{{i,j,k}} \rho _{g} y_{c} \frac{{k_{x} k_{{rg}} }}{{\mu _{g} }}}}{{\Delta x}}} \right)_{{i + \frac{1}{2},j,k}} \left( {(p_{{g_{{i + 1,j,k}} }} - p_{{g_{{i,j,k}} }} ) - \gamma _{{g_{{i + \frac{1}{2},j,k}} }} \left( {D_{{i + 1,j,k}} - D_{{i,j,k}} } \right)} \right)} \right]^{{n + 1}} \\ & \; - \left[ {\left( {\frac{{\Delta y_{{i,j,k}} \Delta z_{{i,j,k}} \rho _{g} y_{c} \frac{{k_{x} k_{{rg}} }}{{\mu _{g} }}}}{{\Delta x}}} \right)_{{i - \frac{1}{2},j,k}} \left( {(p_{{g_{{i,j,k}} }} - p_{{g_{{i - 1,j,k}} }} ) - \gamma _{{g_{{i - \frac{1}{2},j,k}} }} \left( {D_{{i,j,k}} - D_{{i - 1,j,k}} } \right)} \right)} \right]^{{n + 1}} \\ & \; + \left[ {\left( {\frac{{\Delta y_{{i,j,k}} \Delta z_{{i,j,k}} \phi \rho _{o} S_{o} D_{{c,o}} }}{{\Delta x}}} \right)_{{i + \frac{1}{2},j,k}}^{{n + 1}} \left( {x_{{c_{{i + 1,j,k}} }} - x_{{c_{{i,j,k}} }} } \right)^{{n + 1}} } \right. \\ &\left. {\;\; - \left( {\frac{{\Delta y_{{i,j,k}} \Delta z_{{i,j,k}} \phi \rho _{o} S_{o} D_{{c,o}} }}{{\Delta x}}} \right)_{{i - \frac{1}{2},j,k}}^{{n + 1}} \left( {x_{{c_{{i,j,k}} }} - x_{{c_{{i - 1,j,k}} }} } \right)^{{n + 1}} } \right] \\ & \; + \left[ {\left( {\frac{{\phi \Delta y_{{i,j,k}} \Delta z_{{i,j,k}} \rho _{g} S_{g} D_{{c,g}} }}{{\Delta x}}} \right)_{{i + \frac{1}{2},j,k}}^{{n + 1}} \left( {y_{{c_{{i + 1,j,k}} }} - y_{{c_{{i,j,k}} }} } \right)^{{n + 1}} } \right. \\ &\left. {\;\; - \left( {\frac{{\Delta y_{{i,j,k}} \Delta z_{{i,j,k}} \phi \rho _{g} S_{g} D_{{c,g}} }}{{\Delta x}}} \right)_{{i - \frac{1}{2},j,k}}^{{n + 1}} \left( {y_{{c_{{i,j,k}} }} - y_{{c_{{i - 1,j,k}} }} } \right)^{{n + 1}} } \right] \\ & \; + \left( {q_{{D,fm,c}} + q_{{C,fm,c}} } \right)_{{i,j,k}}^{{n + 1}} \\ & \;\; = \frac{{V_{{r,i,j,k}} }}{{\Delta t}}\left[ {\left( {\phi x_{c} \rho _{o} S_{o} + \phi y_{c} \rho _{g} S_{g} } \right)^{{n + 1}} - \left( {\phi x_{c} \rho _{o} S_{o} + \phi y_{c} \rho _{g} S_{g} } \right)^{n} } \right]_{{i,j,k}} \\ \end{aligned}$$

(5.165)

Now, defining the transmissibility terms for oil and gas phases as:

$$T_{{o_{{i \pm \frac{1}{2},j,k}} }} = \left( {\frac{{\Delta y_{i,j,k} \Delta z_{i,j,k} \rho_{o} \frac{{k_{x} k_{ro} }}{{\mu_{o} }}}}{\Delta x}} \right)_{{i \pm \frac{1}{2},j,k}}$$

(5.166)

$$T_{{o_{{i \pm \frac{1}{2},j,k}} }} = \left( {\frac{{\Delta y_{i,j,k} \Delta z_{i,j,k} \rho_{g} \frac{{k_{x} k_{rg} }}{{\mu_{g} }}}}{\Delta x}} \right)_{{i \pm \frac{1}{2},j,k}}$$

(5.167)

$$T_{{o,c_{{i \pm \frac{1}{2},j,k}} }}^{M} = \left( {\frac{{\Delta y_{i,j,k} \Delta z_{i,j,k} \phi \rho_{o} S_{o} D_{c,o} }}{\Delta x}} \right)_{{i + \frac{1}{2},j,k}}$$

(5.168)

$$T_{{g,c_{{i \pm \frac{1}{2},j,k}} }}^{M} = \left( {\frac{{\Delta y_{i,j,k} \Delta z_{i,j,k} \phi \rho_{g} S_{g} D_{c,g} }}{\Delta x}} \right)_{{i + \frac{1}{2},j,k}}$$

(5.169)

Substituting Eqs. (5.166) through (5.169) into Eq. (5.165), it becomes:

$$\begin{aligned} & \left[ {\left( {T_{o} x_{c} } \right)_{{i + \frac{1}{2},j,k}} \left( {(p_{{o_{i + 1,j,k} }} - p_{{o_{i,j,k} }} ) - \gamma_{{o_{{i + \frac{1}{2},j,k}} }} \left( {D_{i + 1,j,k} - D_{i,j,k} } \right)} \right)} \right]^{n + 1} \\ & \quad - \left[ {\left( {T_{o} x_{c} } \right)_{{i - \frac{1}{2},j,k}} \left( {(p_{{o_{i,j,k} }} - p_{{o_{i - 1,j,k} }} ) - \gamma_{{o_{{i - \frac{1}{2},j,k}} }} \left( {D_{i,j,k} - D_{i - 1,j,k} } \right)} \right)} \right]^{n + 1} \\ & \quad + \left[ {\left( {T_{g} y_{c} } \right)_{{i + \frac{1}{2},j,k}} \left( {(p_{{g_{i + 1,j,k} }} - p_{{g_{i,j,k} }} ) - \gamma_{{g_{{i + \frac{1}{2},j,k}} }} \left( {D_{i + 1,j,k} - D_{i,j,k} } \right)} \right)} \right]^{n + 1} \\ & \quad - \left[ {\left( {T_{g} y_{c} } \right)_{{i - \frac{1}{2},j,k}} \left( {(p_{{g_{i,j,k} }} - p_{{g_{i - 1,j,k} }} ) - \gamma_{{g_{{i - \frac{1}{2},j,k}} }} \left( {D_{i,j,k} - D_{i - 1,j,k} } \right)} \right)} \right]^{n + 1} \\ & \quad + \left[ {\left( {T_{o,c}^{M} } \right)_{{i + \frac{1}{2},j,k}}^{n + 1} \left( {x_{{c_{i + 1,j,k} }} - x_{{c_{i,j,k} }} } \right)^{n + 1} - \left( {T_{o,c}^{M} } \right)_{{i - \frac{1}{2},j,k}}^{n + 1} \left( {x_{{c_{i,j,k} }} - x_{{c_{i - 1,j,k} }} } \right)^{n + 1} } \right] \\ & \quad + \left[ {\left( {T_{g,c}^{M} } \right)_{{i + \frac{1}{2},j,k}}^{n + 1} \left( {y_{{c_{i + 1,j,k} }} - y_{{c_{i,j,k} }} } \right)^{n + 1} - \left( {T_{g,c}^{M} } \right)_{{i - \frac{1}{2},j,k}}^{n + 1} \left( {y_{{c_{i,j,k} }} - y_{{c_{i - 1,j,k} }} } \right)^{n + 1} } \right] \\ & \quad + \left( {q_{D,fm,c} + q_{C,fm,c} } \right)_{i,j,k}^{n + 1} \\&\quad= \frac{{V_{r,i,j,k} }}{\Delta t}\left[ {\left( {\phi x_{c} \rho_{o} S_{o} + \phi y_{c} \rho_{g} S_{g} } \right)^{n + 1} - \left( {\phi x_{c} \rho_{o} S_{o} + \phi y_{c} \rho_{g} S_{g} } \right)^{n} } \right]_{i,j,k} \\ \end{aligned}$$

(5.170)

Equation (5.170) can also be written as:

$$\begin{aligned} & \Delta \left[ {T_{o} x_{c} \left( {\Delta p_{o} - \Delta \left( {\gamma_{o} D} \right)} \right)} \right]_{i,j,k}^{n + 1} + \Delta \left[ {T_{g} y_{c} \left( {\Delta p_{g} - \Delta \left( {\gamma_{g} D} \right)} \right)} \right]_{i,j,k}^{n + 1} \\ & + \Delta \left[ {T_{o,c}^{M} \Delta x_{c} } \right]_{i,j,k}^{n + 1} + \Delta \left[ {T_{g,c}^{M} \Delta y_{c} } \right]_{i,j,k}^{n + 1} + \left( {q_{D,fm,c} + q_{C,fm,c} } \right)_{i,j,k}^{n + 1} \\ & = \frac{{V_{r,i,j,k} }}{\Delta t}\Delta_{t} \left[ {\phi x_{c} \rho_{o} S_{o} + \phi y_{c} \rho_{g} S_{g} } \right]_{i,j,k} \\ \end{aligned}$$

(5.171)

Equation (5.171) is the final discretized form of the hydrocarbon flow equations in x direction. Following the same procedure, the discretized water flow equation in x direction becomes:

$$\Delta \left[ {T_{w} \left( {\Delta p_{w} - \gamma_{w} \Delta D} \right)} \right]_{i,j,k}^{n + 1} = \frac{{V_{r,i,j,k} }}{\Delta t}\Delta_{t} \left[ {\phi \rho_{w} S_{w} } \right]_{i,j,k}$$

(5.172)

where

$$T_{{w_{{i \pm \frac{1}{2},j,k}} }} = \left( {\frac{{\Delta y_{i,j,k} \Delta z_{i,j,k} \rho_{w} \frac{{k_{x} k_{rw} }}{{\mu_{w} }}}}{\Delta x}} \right)_{{i \pm \frac{1}{2},j,k}}$$

(5.173)

5.1.3 Appendix C: Newton–Raphson Method

The Newton–Raphson method to solve a set of nonlinear equations is described in detail. The problem consists of solving the following set of non-linear equations:

$$\begin{array}{*{20}l} {F_{1} \left( {x_{1} ,x_{2} , \ldots ,x_{n} } \right)} \hfill \\ {F_{2} \left( {x_{1} ,x_{2} , \ldots ,x_{n} } \right)} \hfill \\ \cdot \hfill \\ \cdot \hfill \\ \cdot \hfill \\ {F_{n} \left( {x_{1} ,x_{2} , \ldots ,x_{n} } \right)} \hfill \\ \end{array}$$

(5.174)

or,

$$F_{i} (x) = 0$$

(5.175)

where F_i, i = 1,2,…,n are the equations and x₁, x₂, …, x_n are the unknowns. To develop the Newton–Raphson algorithm, all functions are first expressed as a Taylor series expansion about an arbitrary point (x₁, x₂, …,x_n, F₁, F₂, …,F_n) as:

$$\begin{aligned} & F_{i} \left( {x_{1} + \Delta x_{1} ,x_{2} + \Delta x_{2} , \ldots ,x_{n} + \Delta x_{n} } \right) \\ & = F_{i} \left( {x_{1} ,x_{2} , \ldots ,x_{n} } \right) + \Delta x_{1} \frac{{\partial F_{i} }}{{\partial x_{1} }} + \Delta x_{2} \frac{{\partial F_{i} }}{{\partial x_{2} }} + \cdots + \Delta x_{n} \frac{{\partial F_{i} }}{{\partial x_{n} }}\quad \quad \quad i = 1,2, \ldots ,n \\ \end{aligned}$$

(5.176)

The objective is to find the roots of the equations by setting the left-hand sides of these n equations equal to zero. If initial values of the unknowns are assumed, the n equations of Eq. (5.157) can be solved for Δx₁, Δx₂,…, Δx_n. This system of n equations may also be written as:

$$\begin{array}{*{20}l} {\frac{{\partial F_{1} }}{{\partial x_{1} }}\Delta x_{1} + \frac{{\partial F_{1} }}{{\partial x_{2} }}\Delta x_{2} + \cdots + \frac{{\partial F_{1} }}{{\partial x_{n} }}\Delta x_{n} = - F_{1} } \hfill \\ {\frac{{\partial F_{2} }}{{\partial x_{1} }}\Delta x_{1} + \frac{{\partial F_{2} }}{{\partial x_{2} }}\Delta x_{2} + \cdots + \frac{{\partial F_{2} }}{{\partial x_{n} }}\Delta x_{n} = - F_{2} } \hfill \\ \cdot \hfill \\ \cdot \hfill \\ \cdot \hfill \\ {\frac{{\partial F_{n} }}{{\partial x_{1} }}\Delta x_{1} + \frac{{\partial F_{n} }}{{\partial x_{2} }}\Delta x_{2} + \cdots + \frac{{\partial F_{n} }}{{\partial x_{n} }}\Delta x_{n} = - F_{1} } \hfill \\ \end{array}$$

(5.177)

Equation (5.177) can also be expressed in matrix form as:

$$\left[ {\begin{array}{*{20}c} {\frac{{\partial F_{1} }}{{\partial x_{1} }} \frac{{\partial F_{1} }}{{\partial x_{2} }} \ldots \frac{{\partial F_{1} }}{{\partial x_{n} }}} \\ {\frac{{\partial F_{2} }}{{\partial x_{1} }} \frac{{\partial F_{2} }}{{\partial x_{2} }} \ldots \frac{{\partial F_{2} }}{{\partial x_{n} }}} \\ . \\ . \\ . \\ . \\ {\frac{{\partial F_{n} }}{{\partial x_{1} }} \frac{{\partial F_{n} }}{{\partial x_{2} }} \ldots \frac{{\partial F_{n} }}{{\partial x_{n} }}} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\Delta x_{1} } \\ {\Delta x_{2} } \\ . \\ . \\ . \\ . \\ {\Delta x_{n} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {F_{1} } \\ {F_{2} } \\ . \\ . \\ . \\ . \\ {F_{n} } \\ \end{array} } \right]$$

(5.178)

Equation (5.178) can be written as:

$$\left[ J \right]\Delta x = - F$$

(5.179)

J is called the Jacobian of the n equations system. The system of Eq. (5.179) can be solved either by Gaussian elimination or by any appropriate procedure. The unknowns (x₁, x₂, …,x_n) are updated after each iteration as:

$$x_{i}^{l + 1} = x_{i}^{l} + \Delta x_{i}^{l + 1} \quad \quad \quad i = 1,2, \ldots ,n$$

(5.180)

where l is the iteration level. The iteration is terminated when $\max \left( {\left| {\Delta x_{i} } \right|} \right) < tolerance$.