Temporal Discretization

After spatial discretization, performing a reservoir simulation requires discretization of the time derivative x( ).t Since the initial condition value is known, the time derivative x( )t is

approximated by a first-order Euler scheme, viz:

x( )t x(t t) x( )t t

   

 (3.47)

In so doing, it is important to select appropriate instances at which the time-dependent inputs and parameters are evaluated, as the choice can significantly influence the computational efficiency as well as the stability. Numerical temporal discretization schemes may be fully explicit, fully implicit, IMplicit Pressure Explicit Saturation (IMPES) or Adaptive Implicit Method (AIM).

If the inputs and parameters are evaluated at the current time instance k, the discretization is referred to as fully explicit; thus, the state vector at the next time-step x_k1 can be obtained as

an explicit expression in terms of state vector at current time-step x_k. Because there is no

need to solve any system of equations, this discretization approach often results into fast computation of variables. However, stability issues arising from time-step restrictions can be problematic; and this can considerably increase the computational requirements. The explicit temporal discretization leads to the nonlinear discrete-time state-space equation, viz:

x_k1 A x x( _k) _k B x u( _k) _k (3.48) y_k C x x( _k) _k D x u( _k) _k (3.49) For a reservoir model of N grid-blocks, with m inputs, p outputs and n state variables, the vectors x, u and y are defined as follows:

x: p 2 , u: q p s well n N m well       _{ }  _{ }     and , , : . well p well w well o      _{ }     p y q q

Note that the state vector x_k n2Nrepresents the state variables i.e. pressure and water saturation values in all the grid-blocks, the input vector uk m _{contains the designated flow}

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rates and bottom-hole pressure in the wells, and the output vector y_k p _{include measured}

flow rates and bottom-hole pressure in each well.

The state-space equation presented in (3.48) and (3.49) can be represented schematically as shown in Figure 3.5 u k y Input uk Outputyk State xk System Model ABCD xk+1 =A(xk)xk + B(xk)uk yk =C(xk)xk + D(xk)uk k

Figure 3.5: A schematic representation of reservoir model state space equation

In systems and control literature, matrix A n n is often referred to as the ‗system matrix‘ or ‗dynamics matrix‘ as it contains the dynamic properties of the system; matrix B n m is the ‗input matrix‘ as it maps the inputs to the states; matrix C p n _{is the ‗}output matrix‘ since it maps the states to the output, and matrix D p m _{is the ‗}feedthrough matrix‘.

Under the assumption of no capillary pressure (as we have earlier underlined), Eq. 3.48 can be represented in the form:

 

 

  

1 1 11 1 1 21 2 ( ) x ( ) x A x B x p A s 0 p B s u s A s 0 s B s k k k k k k k k k k k k k             _{ } _{ }            (3.50)

showing that the reservoir dynamic states (grid-blocks pressures and saturations) are driven by the pressures of the previous time step. Again, because of the saturation dependency of relative permeability in the system matrix A21, the saturation part of the grid-block states are also driven by the saturations of the previous time step.

For the fully implicit scheme, the dependent variables and inputs are evaluated at the next time-step i.e. at time k+1. The computation of the variablesat the next time-step requires that a system of N nonlinear equations (where N denotes the total number of grid-blocks) is solve simultaneously; and this is often carried out through some iterative procedure such as the Newton–Raphson method. Fully implicit discretization scheme are usually robust and

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unconditionally stable. In other words, they yield a stable solution for large time-steps; the only restrictions on time-step size are those necessary to ensure convergence of the iteration scheme employed. However, because of the huge number of nonlinear system of equations (equal to the number of reservoir grid-blocks) involved in this approach, they are usually computationally expensive. The fully implicit scheme results to a coupled system of nonlinear equations of the format:

gk



u xk, k1,xk



0 (3.51) h_k



u x y_k, _k, _k



0 (3.52) where g and h are nonlinear vector-valued functions.

The IMplicit Pressure Explicit Saturation (IMPES) scheme employs a splitting approach which capitalizes on the physics or nature of the coupled systems of flow equations. Under this scheme, the coupled system of partial differential equations is decoupled into the two fundamentally different equations – pressure and saturation equations; and each of the equation is solved using different discretization methods. The pressures are subsequently determined by solving the pressure equations implicitly, while the saturations are determined explicitly by solving the material balance equations. In other words, after solving for pressures, the two-phase saturations are updated explicitly by computing Darcy‘s velocity from the pressure distribution obtained earlier. Now, it is important to note that since the time scales of the dynamic behavior of the two-phase flow model differ remarkably (the changes in pressure are less rapid than the changes in saturation); it is therefore, logical to assign different time-sizes to the decoupled equations. While the implicit pressure update can deal with large time-step sizes, there is a time-step size restriction on the explicit saturation updates so as to guarantee stability. It is noted that the instability of the IMPES approach is as a result of the explicit treatment of the capillary pressure and the decoupling between the pressure equation and the saturation equation, Kou and Sun (2010). This means that the IMPES approach is conditionally stable; hence, its application in highly heterogeneous permeable media (where capillary pressure effects play a significant role in fluid flow path) must be restricted to very small time-step sizes. Because saturation dependent parameters such as relative permeabilities and capillary pressures are often assumed to be constants at each time-step, the formulation and implementation of IMPES are relatively easy; its low memory and CPU time requirements also makes it quite attractive, Markovinovic (2009).

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Another popular discretization scheme is the approach known as adaptive implicit method (AIM). Under this scheme, the time-dependent variables in some grid blocks are solved fully implicitly while the rest are solved using IMPES. Thus, this scheme assigns different levels of implicitness to the grid-blocks, and these levels are appropriately adjusted (as required) in space and time to maintain stability. The method therefore gives robustness in problematic areas with large changes in pressure and saturations (like near a wellbore), while at the same time giving high computational efficiency away from problem regions, Aarnes et al. (2007).

In document A unified metaheuristic and system-theoretic framework for petroleum reservoir management (Page 79-82)