Caracterización molecular de las cepas

Theory. We consider Experimental Design (ED) or, as it also known, Design of Ex- periments (DoE) and Response Surface Methodology (RSM) as an excellent example of

reduced physics methods (Section. 2.1).

Response Surface Methodology ( Box and Wilson (1951)) operates through the recognition of a relationship between the system’s input and response variables, as- suming that this relationship can be described by a mathematical equation.

This relationship, once identified by means of some type of Experimental Design, is then used to construct a so called response surface, an emulator to replace the

2.5. Experimental Design and Response Surface Methodology 75

t = 0

Def ine an initial design DN₍₀₎

Evaluate DN_{(0) and identif y M heavy − hitters (M ≤ N)}

while (T ermination criteria are met)

do                               

Def ine design DM_{(t) and perf orm experiments}

Generate response surf ace RS(t)

Optimize(here any of the optimisation techniques can be used f or parameter space exploration)

Def ine new ref ined design DM_{(t + 1)}

t = t + 1

Figure 2.18: Pseudo-code of Response Surface methodology coupled with Experimental

Design

actual simulated system. This response surface is also known as a surrogate model or a proxy. Therefore, after a sufficient amount of forward simulations, performed according to the selected design, the proxy model is fitted to the produced output data (misfit functions) and further forward model evaluations can be carried out purely by interpolating the proxy. The proxy can then be verified at various time intervals, by the performance of check runs.

As it was noted by Eide et al. (1994), HM is done by minimising the distance between the observed values of the response variables, and the response predicted by the proxy. The process is iterative and is presented in Fig. 2.18.

If one assumes a linear correlation between the inputs (I1, I2, . . .) and output O

of the system, a polynomial can be fitted to the solution space to represent/reflect the system’s performance.

O = α0+ α1I1 + α2I2+ α3I1I2+ α4I12+ α5I22 (2.6)

In order to derive the coefficients of the polynomial, one should perform a range of experiments Ej _{= (I}j

1, I2j, . . .), j ∈ 1 . . . N predefined by a selected Experimental

1 5 2 3 4 6 7 8 a) V1 V3 V2 ₁ 5 2 3 4 6 7 8 b) V1 V3 V2 1 5 2 3 4 6 7 8 d) V1 V3 V2 1 5 2 3 4 6 7 8 c) V1 V3 V2

Figure 2.19: Various types of experiment designs for response surface methodology. 2-level

designs including full factorial design 23 _{(a), fractional factorial design 2}3−1 _{(b) and Box-}

Behnken design with middle point (c). 3-level full factorial design 33_(d)

regression then enables us to calculate the coefficients of the polynomial.

In Eq. (2.6), α0 is an intercept term, α1 and α2 are known as main effects, which

show the sensitivity of the proxy to the changes in I1 and I2 factors. α3 defines an

effect of a two-factor interaction, and α4and α5 are known as quadratic terms.

The general notation for a 2-level design is to use +1 and −1 in place of the high

level (i.e. maximum value) and the low level (i.e. minimum value) respectively, for

each factor (optimisation variable). 0 is used as a notation for centre or middle point of the design.

In the case of reservoir history-matching, the output of the system O is the misfit function value MF which quantifies the difference between the historical and sim- ulated production quantities, and the inputs I are reservoir properties (parameters) which require alteration.

The reader is advised to refer to e-Handbook of Statistical Methods (2003) for a very good and basic description of RSM methodology and most widely applied experi- mental designs, some of which we chose to cover in this overview:

2.5. Experimental Design and Response Surface Methodology 77

• Full factorial design, where every setting of every factor appears with every set-

ting of every other factor (Fig. 2.19, a). By factors one should understand opti- misation variables. If there are k optimisation variables, with each allowed to take two values (minimum and maximum), a full factorial design will consist of 2k_{runs. Such designs are only applicable to cases with less than 5 optimisa-} tion variables;

• Fractional factorial design is applied when one deals with high numbers of op-

timisation variables. In this case if one deals with a 2-level design, only 2k−1 experiments will be carried out (Fig. 2.19, b). Broadly speaking, with designs of resolution three, and sometimes four, we seek to screen out the few impor- tant main effects from the many less important others. For this reason, these designs are often termed main effects designs, or screening designs;

• Plackett-Burman design (PB), a very economical design with the number of ex-

periments being a multiple of four (rather than a power of 2). PB designs are very efficient screening designs when only main effects are of interest, since these main effects are, in general, heavily confined within two-factor interac- tions. The PB design in 12 runs, for example, may be used for an experiment containing up to 11 factors. A folded PB (FPB) design is also amongst the most widely used screening designs. It is obtained from an original design with its appended copy in which signs in all the columns have been reversed.

• Box-Behnken design creates experiments which are formed by the variable val-

ues that represent the middle of the sampling intervals plus a mid (center) point (Fig. 2.19, c). Center points are typically added to check for the curva- ture in the response surface;

• Higher level designs, where variables are not only allowed to take minimum

Therefore a total number of performed experiments will be Nk_{. A case of 3-} level 3 variable design is illustrated in Fig. 2.19, d.

Low-resolution designs are typically used for screening the main effects and higher-resolution designs are applied when interaction effects need to be estimated and response surfaces need to be constructed.

Application. Some of the earliest publications on the subject of RSM application in

history-matching were by Egeland et al. (1992); Damsleth et al. (1992) and Aanonsen et al. (1995).

Results of a comparative study of different design schemes can be found in Yeten et al. (2005). Reis (2006) applied RSM for the optimisation of both synthetic and real field reservoirs.

RSM can be coupled with a wide variety of optimisation techniques. Examples include deterministic techniques such as simplex method, presented in the work by Hoffman and Caers (2000) and stochastic techniques such as genetic algorithms, presented in the works by Castellini et al. (2006); Yu et al. (2007, 2008).