INFERENCE OF THE BOOLEAN MODEL ON A NON STATIONARY CASE

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1 8 th annual conference of the International Association for Mathematical Geology, September 2002, Berlin INFERENCE OF THE BOOLEAN MODEL ON A NON STATIONARY CASE M. BENITO GARCÍA-MORALES and H. BEUCHER Ecole des Mines de Paris, Centre de Géostatistique 35 rue Saint-Honoré, Fontainebleau, France benito@cg.ensmp.fr Abstract Object-based approaches are used to simulate oil reservoirs, in particular in a fluvial environment, or to simulate non reservoir lenses in a reservoir facies. One problem with this kind of simulations is the non stationary distribution of the objects in the reservoir. Classically in truncated gaussian methods (Beucher, 1993) non stationarity is given by the 3D distribution of the lithofacies proportions. In the case of object based methods, this matrix of proportions represents the non stationarity of the objects both in terms of distribution and size parameters. In this work we focus on the distribution variability considering that the objects belong to the same family over the whole space. Then the problem is to fit the Poisson intensity of the boolean model knowing the 3D proportion matrix. The result is a matrix of intensity, given the distribution law of the objects. The method described in this paper was first proposed by Schmitt (1996) and tested by Benito (2001) for a one dimensional non stationarity. Introduction The non stationarity in an object-based method can be reproduced by a regionalized intensity of the objects. The non stationary vertical intensity has already been computed from the vertical proportions curves (see Schmitt,1996). In this paper, we propose to generalize the method to 2D non stationarity. To do this, the computation of the intensity is based on a deconvolution operation which is solved by means of a typical image analysis method, the Wiener filter. The characteristics of this technique are presented as well as some particular features which had to be adapted for our particular case. Then we illustrate this method in the case study of a reservoir presenting heterogeneities to be modelled by a boolean set. Inference of the Poisson intensity by deconvolution The boolean model is defined as (see Lantuéjoul, 1998 and Benito, 2001): X = A x x x where is a Poisson point process of intensity a(x), and A(x) are the objects or primary grains introduced at the Poisson point x. The A(x) are compact non empty sets in 3 that do not depend on their position. Under these conditions, the distribution of the primary grains is characterised by its Choquet s capacity, that is the probability for a compact set K to intersect a set S: 3 U ( ) ; R (1) T ( K) = P( S K ) S 1) when S = A(x) and K is reduced to a point of the space, we obtain p 0 (y), the probability associated to a primary grain : T ({ y}) = P( A( x) { y} ) = P( y A( x)) = p ( y) Ax ( ) 0 167

2 2) when S = X and K is reduced to a point of the space, we obtain p(y), the probability associated to the boolean set : T ({ y}) = P( X { y} ) = p( y) X In our particular case the available information, provided by wells and seismic surveys, is the proportion over a certain volume of the studied facies which will be modelled by the boolean model X. This proportion represents the probability of the indicator facies on this support, p(y), and can be expressed as (2), where the intensity function depends on the location : p( y) = T ({ y}) = 1 exp( a( u) T ({ y}) du) = 1 exp( a( u) P( A( u) { y} ) du) = X 3 A( u) 3 R R = 1 exp( aup ( ) (( u y) Adu ) ) = 1 exp[ ( a p 3 0)( y)] R (2) Under the conditions previously defined, the proportion of the boolean model, p(y), the probability associated to the primary grain, p 0 (y), and the regionalized intensity function, a(y), are related by a convolution operation. From here it is then possible to infer the regionalized intensity. The regionalized intensity can be obtained directly by taking the Fourier transform (FT) of equation (2) and inverting the resulting expression : FT sx ( ) = ( a p)( x) = ln(1 px ( )) sˆ( ν ) = aˆ( ν) pˆ ( ν) (3) 0 0 The difficulty is that the FT of p 0 (y) may attain very small values at some frequencies, making the intensity diverge. A very simple way to stabilise this operation is by means of the Wiener filter, a classical method used in digital image processing (see González, 1992 and Pratt, 1978). Supposing the convolution affected by a white noise non correlated with the intensity, this filter gives the best estimator of the intensity function in the sense of minimum mean square error. In this approach, the intensity and the noise are considered to be random functions of known means and spectral densities. The filter and the FT of the intensity estimator can be written as : ˆ ˆ () f () v = aˆ %() v = fˆ()() v sˆ v (4) * p0 v 2 pˆ 0( v) + Snn( v)/ Saa( v) where S nn (ν), S aa (ν) are the spectral densities of the noise and the intensity respectively. The intensity is obtained by performing the inverse Fourier transform (IFT). As a first approximation we consider the ratio K(ν) = S nn (ν) / S aa (ν) to be constant and taken as small as possible compared to the other term in the denominator. This approximation is widely used in digital image processing when no information is available on the statistical characteristics of the function sought after. In our case it will be possible to study the impact of this factor by choosing a model of the FT of the auto-correlation of the intensity, that is its spectral intensity S aa (ν). Interpretation of the function a(y) in terms of intensity The Poisson intensity represents the number of Poisson points to be simulated in a given volume, therefore its values are strictly positive or null. However, as the linear convolution process does not put any constraints on the resulting function, some the values obtained can 168

3 be negative. This is all the more so as the probability associated to the primary grain, p 0 (y), is not adapted to the studied proportion data, p(y). From a practical point of view, due to scarce sampling, the primary grain parameters and then its associated probability, are chosen from analogues outcrops. The presence of negative values in the computed intensity may be a test for checking out the relevance of this choice. However, even if the primary grain is well fitted to the proportions, there is no guarantee that there are no negative values after deconvolution. Consequently, the resulting values have to be corrected before being considered as a Poisson intensity. For that, a simple smoothing is performed over a window, the size of which is related to the size of the primary grains. It is a simple method that provides a satisfactory result for further simulations as we shall see in the case study. Definition of the support of calculation To perform the deconvolution process, the values of p 0 (y) and p(y) must be on the same support either discrete or continuous. The support of p 0 (y), in particular its discretization, determines the size and precision of the primary grain. The support of p(y) depends on the studied domain and it corresponds to the matrix of proportions, computed, by definition, on wide blocks. Thus, the convolution will be calculated on a discrete support. In order to have an accurate definition of the objects, we must work on a fine grid and, in consequence, the proportion has been subdivided into smaller blocks. The intensity will be then calculated on this finer grid. Number of objects to be simulated The average number of objects to be simulated in a volume V is directly given by the product of this volume and the intensity. In a non stationary case we have : E[ N( V )] = a( x, y, z) dxdydz V In a first approach we assume that only the vertical section presents non stationarity, known from a 2D matrix of proportions on the XoZ plane. The deconvolution results in a 2D intensity, a 2D (x, z). To achieve the 3D simulations we must know the 3D intensity and the number of objects to be introduced in the corresponding volume. As the boolean model is stationary along the direction perpendicular to this plane, a 2D (x, z) and a(x, y, z) are related by the average length of the objects in this direction, L ob y (see Lantuéjoul, 1994): a2 (, ) (,, ) ob D x z = a x y z Ly Let L V y be the average length of the field in the Y direction, the average number of objects to be simulated is: 1 L E[ N( V )] = a ( x, z) dxdydz = a ( x, z) dxdz L V y ob 2D ob 2D V y Ly X Z (5) Case study In the studied case, the unit is composed of two well differentiated regions separated by a non reservoir zone as shown on the reservoir-facies proportion map (fig. 1). Moreover, the reservoir-facies proportion decreases from West to East. The heterogeneity of the unit can be obtained by simulating the reservoir-facies as objects in a non reservoir facies matrix. The distribution of these objects varies from top to bottom and from West to East. 169

4 Figure 1. Proportions map of the reservoir facies showing a non stationarity along the X and Z axes. Figure 2. Resulting intensity map obtained after the deconvolution process. For this example the objects are parallelepiped shaped with constant size. The result of the deconvolution process is presented in figure 2. In order to analyse the impact of the positiveness constraint for the intensity function, the direct convolution has been calculated with the intensity before and after smoothing. As expected, the raw intensity restores the initial proportions almost exactly (fig. 3) with a deviation smaller than The smooth intensity also returns a satisfactory proportion (fig. 4) with a mean deviation of However, the maxima and minima being the values the most affected by the smoothing process of the correction of the intensity, they are not so well recovered. To improve the restitution of extreme proportions, other methods for correcting the intensity can be tested. Nevertheless, we use this smooth intensity for further simulations. The theoretical proportion to compare with should be that obtained by direct convolution (fig. 4) which is smoother than the initial one. Figure 3. Proportion map resulting of the direct convolution using raw intensity 170

5 Figure 4. Proportion maps resulting of the direct convolution using corrected intensity. Simulations are then carried out with corrected intensity and the objects previously defined. An example of simulation is presented in figure 5.a. We can appreciate the non stationary distribution of objects in the XoZ plan. The matrix of proportion computed on the simulation is displayed in figure 5.b. This proportion should be compared to that obtained in figure 4. However, the reference proportion is the initial one, that is our input data (fig. 1). Proportions obtained from simulations are more erratic, as expected, but they respect the initial global value. This is an important result because in fact, this global value is often given by complementary studies on the reservoir and must be well restored. The results obtained over 50 simulations show that the deviation between the global proportion of simulations and that of the data is smaller than a. Example of simulation 5.b Proportions computed on the simulation Figure 5. Example of a simulation using smooth intensity and its corresponding proportion map. 171

6 Conclusions The method presented in this paper makes it possible to estimate the intensity from a 2D matrix of proportions and a given family of primary grain. This method provides good results when the objects are relatively small compared to the whole volume. From a theoretical point of view, computing a 3D intensity from a 3D proportion matrix does not pose any problem. However, several practical problems will have to be analysed, for instance the size of objects compared to the size of the studied domain, the choice of the probability associated to the primary grain References Benito García-Morales, M., 2001: Non stationnarité dans les simulations de type objet. Technical report N-26/01/G, Centre de Géostatistique, ENSMP. Beucher H., Galli A., Le Loc h G., Ravenne Ch. & Heresim Group, 1993 : Including a regional trend in reservoir modelling using the truncated Gaussian method. In : A. Soares (ed.) Geostatistics Tróia 92, Dordrecht : Kluwer, vol. 1. p González, R.C., Richard, E., 1992: Digital Image Processing, Addison-Wesley Publishing Inc. Lantuéjoul, Ch., 1998 : Conditional simulation of random sets. In Proceedings of the Section on Statistics and the Environment of the American Statistical Association, pp Lantuéjoul, Ch., Simaku, A., 1994: Simulation conditionnelle d un schéma booléen. Technical report N-24/94/G, Centre de Géostatistique, ENSMP. Pratt, W.K., 1978 : Digital Image Processing. John Wiley & Sons Inc. Schmitt, M., Beucher, H., 1997: On the inference of the Boolean model. In E.Y. Baafi and N.A. Schofield (eds.), Geostatistics Wollongong 96, Dordrecht : Kluwer,, vol. 1. p