|Publisher||Society of Petroleum Engineers||Language||English|
|Content Type||Journal Paper|
|Title||The Laplace Transform Finite Difference (LTFD) Numerical Method for the Simulation of Compressible Liquid Flow in Reservoirs|
G.J. Moridis, SPE, Lawrence Berkeley Laboratory, D.A. McVay, SPE, S.A. Holditch & Assoc., D.L. Reddell and T.A. Blasingame, SPE, Texas A&M U.
|Journal||SPE Advanced Technology Series|
|Volume||Volume 2, Number 2||Pages||122-131|
1994. Society of Petroleum Engineers
A new numerical method, the Laplace Transform Finite Difference (LTFD) method, was developed for the simulation of single-phase compressible liquid flow through porous media. The major advantage of LTFD is that it eliminates time dependency, the need for time discretization, and the problems stemming from the treatment of the time derivative in the flow equation by employing a Laplace transform formulation. The LTFD method yields a solution semi-analytical in time and numerical in space. and renders the effects of the time derivative on accuracy and stability irrelevant because time is no longer a consideration. The method was tested against results from three test cases obtained from a standard Finite Difference (FD) simulator, as well as from analytical models. For a single timestep, LTFD requires an execution time 6 to 10 times longer than the analogous FD requirement without an increase in storage. However, this disadvantage is outweighed by the fact that LTFD allows an unlimited timestep size. Execution times may be reduced by orders of magnitude because calculations are necessary only at the desired observation times, while in a standard FD method calculations are needed at all the intermediate times of the discretized time domain. Thus, FD may require several hundred timesteps and matrix inversions to reach the desired solution time, but LTFD requires only one timestep and no more than 6 to 10 matrix inversions. Moreover, LTFD yields a stable, non-increasing material balance error in addition to a more accurate solution than FD.
In transient flow through porous media, the general Partial Differential Equation (PDE) to be solved is obtained by combining appropriate forms of Darcy's Law and the equation of mass conservation, yielding:
Eq. 1 is generally nonlinear and contains the time derivative ¶p/¶t, of which the numerical approximation is consistently the most important source of instability and error in numerical models. The treatment of ¶p/¶t in a traditional Finite Difference (FD) approximation scheme involves the discretization of the continuous time coordinate into a large number of small timesteps Dt. Numerical solutions at a number of points k. in the domain are then sought at the discrete times
where the dependent variable pk(t) is approximated by a set of values =1,2, . . . n.
The PDE problem with a continuous smooth solution surface is thus reduced to a set of algebraic equations relating the discrete approximate values pk to each point k. A Taylor series approximation of the time derivative yields
Equation 3 and 4
is the truncation error, and n+1 and n denote the current time and previous times of the discretized time domain. For v=n, Eq. 3 represents a forward difference approximation and results in the explicit formulation of the FD, which is not unconditionally stable; for v=n+1, Eq. 3 represents a backward difference approximation and results in the unconditionally stable implicit formulation of the FD. The above approximation introduces an error of order O(Dt). Accuracy (and, in the case of the explicit formulation, stability) considerations preclude the use of a large Dt. Minimization of Ek,n often dictates the use of a large number of small timesteps Dt between desired observation times, requiring longer execution times and resulting in potentially larger roundoff errors. The problem of restriction on the size of Dt is further exacerbated by the nonlinearity of the PDE, which is caused by the pressure dependence of the liquid-density and the formation porosity. This necessitates even shorter timesteps, dictates internal iterations within each timestep until a convergence criterion is met, and adds significantly to the computational load.
The Laplace Transform Finite Difference (LTFD) method belongs to a family of new, Laplace transform-based numerical methods recently introduced by Moridis and Reddell1-5. It was first applied to the solution of the diffusion-type (parabolic) PDE of incompressible flow through porous media1,2, and was extended to the solution of the advection-diffusion (hyperbolic) PDE of solute transport (miscible displacement) in porous media3. The major advantage of LTFD is that it eliminates the time dependency of the problem because of the Laplace transform formulation employed, and thus the need for time discretization for an accurate, stable solution. In essence, LTFD yields a solution semi-analytical in time and numerical in space. An unlimited Dt size is possible without loss of accuracy or stability, and the need for a large number of intermediate steps between desired observation times is eliminated.
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