General Theory of Casing Optimization
The combination casing string design is considered a multistage decision-making procedure in which the next step decision depends upon the previous decisions. The general concept of the discrete version of dynamic programming is applied (Roberts. 1964; Phillips et al.. 1976). Dynamic programming terminology is defined by the following five attributes.
1. A stage is a unit section of casing string (length At) or a step in the recurrent design procedure. At each stage, the set of the optimal casing variants is selected (Fig. 5.3).
2. Stage variables, Fn. are loads supported by the nth casing unit section:
Fn — Fn (Арь, Apc. FAn) (5.14)
In general, there are (Ax Л’ц-) combinations of the loads at stage n. where:
Nsn_1 = number of possible different variants of casing string below section n.
Nw — number of different casing unit weights.
The axial loads, Тдп, for the nth unit section are calculated using Eq. 5.5. These loads can also be computed using Eq. 5.6 for vertical wells and Eqs. 5.39 — 5.45 for directional wells.
5. Accumulated total return, CPn, is the minimal cost of n sections of casing for each load, Fn. As the load is dependent only on the unit length’s weight, each of which is represented here by m, cost optimization is carried out at each stage by selecting the cheapest casing within each of the possible casing weights and by identifying the casings that are lighter than the cheapest one. The procedure is described as follows.
|
Wmn = mill (iu;l(s)) s€(a, o) |
(5.18) |
|
Pwn = PwJWmn) |
(5.19) |
|
WreJn = min Wmn(u) u€(c, d) |
(5.20) |
|
Wrn < wT’fn |
(5.21) |
|
PPn = PPAWPn) |
(5.22) |
|
c;n = A (xPP„(v) + CkPn_t |
(5723) |
where:
a = the smallest value of Pn within m, US8/100 ft.
b = the largest value of Pn within m. US8/100 ft.
с — the smallest value of Pwn, USS/100 ft.
d = the largest value of Pwn — USS/100 ft.
к = varies from 1 to Asn_, ■
г = varies from 1 to (rPn x k).
PPn = distributed price of the И’rP/n of casing. USS/100 ft.
Pwn = distributed price of the cheapest casing
within m, US8/100 ft. rPn = number of WPn weights.
Wm,, = distributed weight of the cheapest casing
within m, lb/ft.
WPn = distributed weight of casing lighter or
equal to WTf, jn, lb/ft.
VUre/n = distributed weight of the cheapest casing
at stage n, lb/ft.
6. Absolute minimal cost, Cmmn, at stage n is given by:
Cmmn = min (Af x P (w) + ) (5.24)
ve(ej) 4 и
|
e = the smallest value of PPn, USS/100 ft. |
where: e =
/ = the largest value of PPn. USS/100 ft.
Inasmuch as the transition of the cost and transition of the axial load from step n — 1 to step n is achieved by simple addition, the principle of optimality can be
applied and Eq. 5.24 becomes:
|
С. _ ^ mmn — |
min (Д£ x />,»)
For n — A, Eq. 5.25 gives the minimal cost of the combination casing string desired. This cost corresponds to the optimum configuration of the casing string stored in the computer memory.
Simplification of the Theory. In some practical computations, the lack of the price/weight conflict has been observed. Mathematically, this means that r (WPn) has only one value and this is equal to r (lFrfyn). For the particular cases where this happens the optimization procedure can be simplified. Namely, at any unit section of the casing string, there is only one set of loads supported by the n — 1 casing section, meaning that the above formulation will equal both formulations for the minimum weight method and the minimum cost method presented earlier.