Las dificultades de una reconstrucción intercultural de derechos humanos

For conventional well history matching the objective function calculates the misfit between the modelled well data mod

k

d (bottom-hole pressures, production rates, water cuts, GORs, etc.) and the corresponding observations obs

k

d . The index k is this notation refers to a specific well, data type, and the time moment. The typical simple definition of the objective function is through the sum of squared differences with appropriate scaling: , 1 2 mod 1



where the sum is taken over all available observed data obs k

d , and N is the total _d

number of these data. The normalising constants _k allow handling data with the different magnitude. The other beneficial side of definition (4.1) comes from its application to uncertainty estimation, because it can be used to find the likelihood function Lexp(0.5f₁) for the situation where the errors mod

k obs

k d

d  are

uncorrelated and have normal distribution with zero mean and standard deviation equal to _k. For a more general case with correlated errors, a full covariance matrix Cw should be introduced, resulting in the definition

). ( ) ( mod 1 mod 1 d d C d d f  obs  T _w obs  (4.2)

The latter approach will be discussed in detail in the subsequent Chapter 5. For the numerical tests considered in this chapter, definition (4.1) will be used.

For seismic history matching I will consider the objective function consisting of two parts which account for the model fit to both observed well data and seismic data:

,

2

1 f

f

f   (4.3)

where, leaping ahead, the value of the seismic part of objective function f is found ₂

either as a sum of squares (4.11) or a more general quadratic form (4.12). The approach for calculation of f I introduce here avoids the full-physics petro-elastic and seismic ₂

modelling, and the motivation behind it will be set forth in section 4.4. Note that definition (4.3) does not include any weights for the two components of the objective function. These weights could be included for the flexibility of the history matching workflow, so that an engineer can quickly put more emphasis on either component. However, for uncertaiunty estimation application no weights should be used, and the relative impact of both components is accounted for by their covariance matrices.

It is worth making a comment on the expected value of the objective function defined by (4.1), (4.2) or (4.3) after optimisation has finished. As discussed in [56], if the forward model – i.e. the link between the model parameters and the model data – was linear, and if the covariance matrix C correctly described the data errors, then f at its minimum would follow a 2 distribution with  degrees of freedom, where

p

d N

N 



 , the quantity N is the number of data, and _d N_p is the number of parameters. This statement is true if the inverse problem is not under-determined and the forward modelling matrix has full rank15. For this 2 the mean value is , and the variance equals 2 , so the following estimate of the minimum of f can be considered,  as suggested in [42]: . 2 5 2 5  min     f   (4.4)

If, in addition, a Gaussian prior model was used, then, as discussed in [56], [42], the total misfit for the maximum a posteriori (MAP) estimate of the model would then follow 2 distribution with N degrees of freedom, independent of the number of _d

model parameters used. However, in this thesis only simple priors are considered, viz. uniform priors defined over some regions in the parameter space, so the previous estimate with  N_d N_p will be more relevant. These considerations are only

15 _{The rank in this case is N}

rigorous for the case of linear forward modelling, and the quasi linear situations are to be treated with caution. However, Oliver et al. [42] report that for synthetic history matching problems the expected minimum value of misfit for MAP estimates is in reasonable agreement with the above theory. If the final value of the objective function after optimisation is not consistent with the estimate (4.4), this may mean failure of the optimisation algorithm to find the global minimum, or incorrectly defined standard deviations of the data errors. Too low values of f_min will mean overestimated std’s, too high values will correspond to underestimated std’s. Since in this work priors with uniform distribution over some region are used, if this region happens to be too small and restrictive, this may also lead to higher values of f at the global optima. It is well known that as the number of degrees of freedom  increases, the 2 distribution converges16 to a normal distribution with mean  and standard deviation ₀  2. Because of this, if the minimum objective function value f_min follows 2 distribution with large  (e.g. there is a large number of data), then one can also consider f_min to be distributed according to N(,₀), and the estimate (4.4) can be rewritten as

. 5 5_{0 min}  ₀

  f   (4.5)