Scaling Tilted Anisotropic Systems
Authors: Dalton R.L. 
Published: December 2007

A Note on an Elementary Iterative Problem

The iterative problem
(1)k+1 k x = f(x)
frequently occurs, in one form or another, in reservoir engineering calculations. The convergence problem is that of obtaining from Eq. 1 a sequence problem is that of obtaining from Eq. 1 a sequence of iterates xk that approach a finite solution x* = lim k-&gt;oo (xk) assumed to exist. Obviously, x* = f(x*). The function /may be an explicit algebraic function or a long and complex algorithm. Since the former case is easily handled by the Newton-Raphson technique, this discussion is pertinent to the latter case.
Convergence of this iterative process is easily analyzed and is discussed by a number of authors, including Wegstein. Denoting this error in the kth iterate by k we have by definition xk = x* + k. Substituting accordingly for xk and xk+1 in Eq. 1 and employing a Taylor expansion, we have for small
(2) k + 1---- = f ′ ( x* ) ,
Since x* = f(x*), this equation gives
(3)k k + 1---- = f ′ (x*),
and the condition for convergence is simply f;(x*) less than 1.
In cases where f'(x*) is close to or greater than 1, convergence often can be accelerated or obtained by weighting i.e., Eq. 1 is replaced by
(4)k+1 k k x = w f(x) + (1 − w) x
Repeating this above error analysis gives
(5)k k + 1----- = w f ′ + 1 − w,
(6)1 - f′ k + 1 and an optimum weight factor is given by ------- = 0 , or k1w = --------,
Eq. 3 and 6 lead to some qualitative observations. Eq. 3 shows that if 0 less than f' less than 1 then convergence should be monotonic in the sense that the successive iterates should approach the value x* from one side. The error k will be constant in sign. Further, Eq. 6 shows that in this case convergence might be accelerated by use of weight factor w greater than unity. If f' greater than 1 then divergence should be expected in use of Eq. 1 but Eq. 6 shows that use of a negative weight factor might result in convergence. If - 1 less than f' less than 0 then Eq. 3 indicates an oscillatory convergence i.e., overshoot followed by alternating sign on the successive errors k. Eq. 6 indicates that in this case a weight factor between 0.5 and 1 might speed convergence. Finally if f' less than - 1 then Eq. 1 will give divergence; but convergence might be obtained by using Eq. 4 with weight factor between 0 and 0.5.
We recently encountered an example of the iterative problem, Eq. 1, in a senior class in natural gas engineering. In a two-dimensional, single-phase computer calculation, the total production rate, Q, from a heterogeneous gas production rate, Q, from a heterogeneous gas reservoir was calculated from the equation
(7)day i=l NwMcfQ ------ = Ci (pi − pw),
where Nw is the number of wells in the reservoir; Ci is a constant reflecting an assumption of quasi-steady-state only within the small grid block containing the well; pi is average pressure in the grid block; and pw is flowing well pressure. The program actually solves for and uses potentials program actually solves for and uses potentials rather than pressures but the present form will suffice for illustration. The desired field deliverability rate Q was specified for each of a sequence of time steps covering the producing period of interest. During each time step, the period of interest. During each time step, the program performed a small number of iterations, in program performed a small number of iterations, in part to determine the value of wellbore producing part to determine the value of wellbore producing pressure pw (same for all wells) necessary to give pressure pw (same for all wells) necessary to give total field production rate equal to the desired, specified value Q.