Seismic Data Integration

• Calibration of Data

• Cross spatial variability

• Cokriging

• Simulation Alternatives

• Lecture 11 Quiz Introduction

Data Integration is a fundamental principle of geostatistics / reservoir modeling; the goal is to explicitly account for all of the available data. Seismic data provides abundant data compared to well data that is sparse. Seismic data however does not give any directly useable information and must be calibrated to the limited well data. There are numerous techniques that permit the direct use of seismic data such as cokriging and collocated cokriging. There are also many methods for determining the uncertainty in estimated maps such as cosimulation, colocated cosimulation. Other simulation alternatives include annealing.

Calibration of Data

Usually the most difficult step in the geostatistical study is finding a relationship between the well parameter of interest and some aspect of the seismic data (Wolf et al., 1994). A relationship between seismic data to well data is often quite difficult to infer and once found summarized as a simplistic linear regression; none of the scatter is preserved. Figure 11.1 shows a calibration between sand thickness and a seismic attribute. This is a simplistic approach and is not always appropriate for some petrophysical properties such as permeability.

Figure 11.1

At all stages the user must justify the calibration.

Sometimes there are too few data to infer a relationship and the user must infer a relationship based on previous experience or analogue data. Figure 11.2 shows a scenario where there are too few data to infer a useful relation between porosity and depth. The distribution of depth throughout the reservoir is exhaustively known as a result of seismic interpretation. The depth information is exhaustively known, yet there are only five well data to infer a relation between porosity and depth.

Figure 11.2

Instead of abandoning the calibration a relation has been manually inferred (the grey ellipses) from analogue data retrieved from a reservoir that is similar. The proposed methodology is to sum the inferred distributions of porosity conditional to the depth yielding an inferred global distribution for porosity. In terms of probabilistic notation:

(11.1) where c is a normalizing constant.

In summary the calibrations is performed: (1) map the secondary variable X at all locations, (2) develop a bivariate relationship between X and the Y variable interest, and (3) generate a distribution of Y by combining the conditional distributions. Beware that the user must always justify the calibration routine.

Cross Spatial Variability

In the univariate case (one variable) the model variogram must be a compilation of different variogram models that are known to allow the solution to the system of kriging equations to be unique. As well, the model variogram that defines the spatial correlation in 3-D must be modeled such a way that each variogram structure be the same and have the same contribution in all directions; only the range may change for each structure in each direction. In the multivariate case, where there is a primary variable of interest and a secondary correlated variable, and an equivalent requirement called the linear model of coregionalization. Just as the univariate variogram is constructed as a set of intercorrelated random functions. The linear model of coregionalization. provides a method for modeling the auto- and cross-variogram of two or more variables so that the variance of any possible combination of these variograms is always positive. This ensures a unique solution to the cokriging system of equations.

The criteria are: (1) the determinant must be greater than zero, and (2) all diagonals must be greater than zero.

The linear model of coregionalization is a technique that ensures that estimates derived from cokriging have a positive or zero variance. To ensure this, the sill matrices for each basic structure must be positive definite. As an example, for the two variable case, we can check whether the linear model of coregionalization is honored:

• Gamma_x(h) = 10 + 20 Sph + 40 Exp

• Gamma_y(h) = 20 + 35 Sph + 55 Exp

• Gamma_xy(h) = 7 + 10 Sph + 30 Exp (cross-variogram) For the nugget structure:

10 7 7 20

• determinant = (10 x 20) - (7 x 7) = 151 > zero --- OK

• all diagonals are positive (10 and 20) --- OK For the spherical structure:

20 10 10 35

• determinant = (20 x 35) - (10 x 10) = 600 > zero --- OK

• all diagonals are positive (20 and 35) --- OK For the exponential structure:

40 30 30 55

• determinant = (40 x 55) - (30 x 30) = 1300 > zero --- OK

• all diagonals are positive (40 and 55) --- OK

Since for all structures the determinants are greater than zero and the diagonals are positive, then the linear model of coregionalization is honored. (AI-Geostats)

Cokriging

Often there is more than one type of available data, and often these data are correlated. Also, the

sample nearly the entire volume. When there is more than one type of available data of interest is called the primary data and all other data is called secondary data. In kriging the spatial correlation of a single data type is used to make the best estimate at locations where there is no data. Cokriging differs from kriging in that it uses the spatial correlation of the primary data and the spatial correlation of primary data to secondary data to fortify the estimate of the primary variable. For example, we know it is hot and dry in Texas and that there is a good chance of finding cactus whereas it is cold and dry in Alberta and there is poor chance of finding cactus. Cokriging is particularly useful when there is fewer primary data than secondary data. The cokriging estimate, like the kriging, estimate is the linear combination of data. Cokriging however is the linear estimate of the primary and secondary data as shown in equation 11.20. The cokriging system of equations is derived in the same way as the kriging system of equations and will not be covered here.

(11.2)

and the system of equations given two primary and two secondary data is:

(11.3)