CHAPTER 2: THE CONTINUUM MECHANICS THEORY APPLIED
2.5 Numerical solution of the governing equations
The commonly used numerical methods in reservoir simulations are finite difference methods (FDM’s), finite volume methods (FVM’s), and finite element methods (FEM’s). Each has its advantages and disadvantages in the implementation.
2.5.1 Finite Difference Method (FDM)
To achieve approximate solutions to partial difference equations (PDEs) in all aspects of engineering, the FDM is well-know numerical method used, especially in fluid dynamics, heat transfer and solid mechanics. The foundation of FDM is to replace the partial derivatives of the governing equations by differences defined over certain spatial intervals in the coordinate directions dx, dy, and dz; which generate a system of algebraic simultaneous equations of the governing equations at a mesh of nodes over the domain of interest. Applying boundary conditions defined at boundary nodes, solution of the simultaneous algebraic system equations will yield the required values of the governing equation at all nodes, which meet with both the governing
Chapter 2: The continuum mechanics theory applied to coupled reservoir engineering particularly in subsidence and compaction research difference (FD) are easy to construct but this method requires the use of a structured grid. In addition, FD code is memory and computational efficient, and can be easily parallelized. However, the requirement of a structured grid makes girding of complicated domains difficult. For example, it may be challenging to treat complicated geology structures such as faults, sills and fractures. If the grid needs to be aligned with these features, adaptive mesh refinement can sometimes alleviate grid geometry problems. Although there are still limitations, FDM is common used in petroleum industry (Eclipse 2005).
2.5.2 Finite Volume Method (FVM)
Another numerical method to obtain approximation of the PDE's is the Finite Volume Method (FVM). In an FVM, the integral form of an equation is discretized, which guarantees conversation. This method overcomes the limitation of FDM as FVM does not require a structured mesh. In addition, an FVM is preferable to other methods as a result of the fact that boundary conditions can be applied noninvasively because the values of the conserved variables are positioned within the volume element, and not at nodes or surfaces. FVM’s are especially powerful on coarse nonuniform grids and in calculations where the mesh moves to track interface or shocks.
2.5.3 Finite Element Method (FEM)
The most useful numerical method is FEM. This method deals with an integral formulation of an equation by using a weak formulation. Because an FEM can be applied to problems of great complexity and unusual geometry, it is a powerful tool in the solution of complexity problems such as fracture in mechanics or fault within
Chapter 2: The continuum mechanics theory applied to coupled reservoir engineering particularly in subsidence and compaction research geology structure. An FEM is very flexible at representing complex geometries.
Consequently, using this method the mesh can be refined near particular features. In addition, the mesh can also be reduced at the far field region. So that large volumes of free space can be included. But for a fine-grid system, the expense involved in solving this large system is usually prohibitive. Considering so many uncertain factors in reservoir parameters and the solving efficiency, this expense may be not justified.
2.5.4 Equation discretization
The formulations of coupled theories between fluid and solid applied to radial flow in reservoir engineering have been presented. Unfortunately, the general analytical solution for this equation has not been resolved except in simplified cases.
Despite this, a numerical method can be applied to achieve the general solution. The Galerkin finite element method is chosen because of its ability to handle anisotropic and heterogeneous regions with complex boundaries (Young and Hyochong 1996).
In the Galerkin method, the unknown variable pressure and displacements can be approximated by a trial solution in space using of the shape function N and nodal values (P, u)
(2-94)
i i n
1
i NP
P=∑=
(2-95)
ri i n
1
r i N u
u =∑=
(2-96)
zi i n
1
z i N u
u =∑=
Therefore, in the coupled simulation, there are three principal degrees of freedoms at each node of the mesh.
We have the generally fully coupled equations as following
Chapter 2: The continuum mechanics theory applied to coupled reservoir engineering particularly in subsidence and compaction research
(2-97)
1 . 2 1
.
1P K P M u F
M + + =
(2-98)
2 2
2d C P F
K + =
Applying the time integration technique with Crank-Nicholson method described we derived the coupled matrix system.
(2-99)
( ) ( )
( ) ( )
+
− +
+
∆
− + +
= ∆
+∆
∆ +
∆ +
∆ +
∆ +
t 2 t 2 t 2 t t 2
t 2 t 1 1 t 1 t t 1
t t t 2
2
1 1 2
P C u K F F
u M 2 P tK M 2 F F t
P . u C K
tK M 2 M 2
Where
∫αβàφ
= CN N dR
K
M1 1 t i j
∫
∫∂∂ ∂∂ + ∂∂ ∂∂
= dR
z N z dR N
r N r
K1 Ni j i j
z dR N r N N K
M2 =∫β1àφ i∂∂ j+ ∂∂ j
B z i B
r i
1 n
z rN P 2 r n
rN P 2
F ∂
π ∂
∂ + π ∂
=
g dR . N D g .
K2 =∫∂∂Ni ∂∂ j
∫ ∂∂ +∂∂
φ
−
= dR
z N r . N N ) 1 (
C2 0 j i i
in which
. T 4 .
3 .
2 .
1
. P,P,P,P
P
=
T 4 . 4 . 3 . 3 . 2 . 2 . 1 . 1 .
. u v u v u v u v
u
=
{u1v1u2v2u3v3u4v4}T
u =
{ ' 'rz}T
z ' '
r, , ,
'= σ σ σ τ
σ θ
Chapter 2: The continuum mechanics theory applied to coupled reservoir engineering particularly in subsidence and compaction research
{ }
T z r z r r
T rz z r
r u z , u z , u r ,u r u
, , ,
∂ +∂
∂
∂
∂
∂
∂
= ∂
τ ε ε ε
=
ε θ
Noting that, porosity and permeability will be updated at each time step on each element. The influence of permeability and porosity on coupled simulation results can be found in Ta and Hunt (2005).
Boundary condition
There are 2 main types of boundary condition for each phase Solid phase:
Dirichlet-type prescribed displacement u~
t) z,
u(r, = with ∀r,z∈Γs1 (2-100)
Neumann-type prescribed surface traction.
~h n . =
σ with ∀r,z∈Γs2 (2-101)
Where Γs1∪Γs2 =Γs Fluid phase:
Dirichlet-type prescribed initial pressure p~
t) z,
p(r, = ∀r,z∈Γf1 (2-102)
Neumann-type prescribed normal flux to boundary
0 (2-103)
n . p =
∇ ∀r,z∈Γf2
Where Γf1∪Γf2 =Γfand unit normal vectorn={nr,nz,nθ}T. Subscript s and f refer to solid and fluid phase, respectively.
Initial condition (t=0)
Chapter 2: The continuum mechanics theory applied to coupled reservoir engineering particularly in subsidence and compaction research
(2-105) P0
z,0) P(r, =
Matrix Equation (2-99) must be completely constrained by initial and boundary conditions described by Equation (2-100) – (2-105) before a solution is obtained.