Rotation and affinity invariance in multiple-point geostatistics

Transcription

1 Rotation and ainity invariance in multiple-point geostatistics Tuanfeng Zhang December, 2001 Abstract Multiple-point stochastic simulation of facies spatial distribution requires a training image depicting the geometric patterns deemed present in the study volume. That training image is the equivalent of the variogram model used in traditional 2-point statistics-based simulation. Therefore the training image should be reasonably stationary, that is invariant by any translation. However, even within a homogeneous formation, actual geological patterns may change gradually in space, e.g., in terms of direction and aspect ratio. Such location-dependent patterns can be simulated using location-dependent linear transforms (rotation + ainity) of the stationary patterns read from the training image. The linear transform of training patterns is discussed in this report and its application to simulation of non-stationary fan deposits is shown. 1 Introduction Training images as source of structured data are taking an important role in geostatistics. Under the useful decision of stationarity, multiple-point statistics can be lifted from a training image and used to model a target area. The training image should reflect the patterns of spatial variability deemed to prevail over the target area. These patterns must be repeated often enough over the training image so that they can be captured by multiple-point statistics. Statistics, no matter their order and the number of locations (multiple points) they involve, can only represent stationary features or patterns; they reflect average patterns not location-specific unique features. Geostatistics, as any other spatial statistics, can only capture stationary patterns from a training image and can only model and simulate these stationary patterns. The specific locations and local transforms 1

2 of these stationary patterns can only be obtained through local data: this is the task of conditioning and the essential dierence between non-conditional simulation which can only reproduce stationary patterns and conditional simulation which, in addition, also aims at locating them in space. Any unique (non repeated) pattern, such as a single fracture, is either detected by local data and can be deterministically reproduced, or it is not and no statistics could ever model it. A training image, that is a representation of repeated spatial patterns deemed present over the target area, need not and should not carry any location-specific characteristic. Training patterns are defined relative to each other and their statistics should be independent of the coordinate system used. More precisely, they should be independent of any linear transform of the coordinates. Training images can be adapted to the target area by a series of local rotations and ainity of the coordinates system, much like an isotropic variogram model can be made location-specific by a succession of rotations and ainity transforms of the coordinates, Xu (1996). Consider the training image (TI) of Figure 1a which features EW elongated s with little varying NS width. This TI can be used to model s of any direction, see Figures 1 b-c-d, and of any width, see Figures 2 and 3. The training patterns taken from the TI are made rotation and scale-invariant, more precisely independent of any linear transform of the coordinates system. The impact of stationarity of statistics on stochastic simulation is now investigated. 2 Multiple-point statistics Let fs(u); u 2 D)g be the stationary random function modeling the spatial distribution of an attribute s(u) over an reservoir D. IfS(u) has a finite number of outcomes: s 1 ;s 2 ; :::; s K and a conditional data set fs(u ); =1; 2; :::; ng is available, the probability of the variable S(u) at any unsampled location u can be obtained by considering the correlation between the single-point event S(u) and the known multiple-point (mp) data event B = fs(u ); = 1; 2; :::; ng. The outcome of the multiple-data event B conditions the probability of the single-point event S(u). The mp data is considered as a whole instead of being considered one datum location at a time. Such mp data allows capturing patterns within the data set. This is dierent from the traditional two-point geostatistics approach, in which only the correlation between any two points is considered to construct the probability of S(u). Traditional geostatistics only captures two-point correlation characteristics of the reservoir attributes. In contrast, multiple-point statistics can model curvilinear and large scale structures such as undulating s. Consider the binary random variable I B associated with occurrence of the data 2

3 event B: I B = ( 1 if B occurs 0 if not I B is called an event indicator variable. If the event B involves a single point u in space, I B is the ordinary indicator variable usually denoted by I(u). IfB is a multiple-point event, I B is then an indicator of the specific pattern constituted by all single data constituting B. Consider now the two events A k = fs(u) = s k g and B = fs(u ); = 1; 2; :::; ng. Simple kriging can be used to estimate the probability of the unknown event A k given that data event B has occurred: P (A k jb) =E(I Ak =1ji B )=E(I Ak )+ [1 E(B)] (1) A single normal equation provides the weight : V arfbg = CovfA k ;Bg (2) It is easy to check that equations (1) and (2) exactly identify Bayes relation, Strebelle (2000): P (A k jb) = P (A k;b) (3) P (B) The equivalent relations (1) and (3) show that inference of the conditional probability P (A k jb) requires knowing the probability of the joint occurrence of the two events A k and B as well as that of B alone. In the usual situation of sparse data, it is not possible to infer directly from actual data these two probabilities which are much beyond variograms and any other two-point statistics. An alternative is to identify these probabilities to proportions read from an exhaustively sampled training image, assuming that this training image has the same multi-point statistics as the reservoir under study. An easy and fast way to store and then retrieve the proportions of occurrence of any training pattern is provided by the concept of search tree Strebelle (2000). The training image is scanned only once with a specified template and the corresponding proportions of pattern occurrence are stored. Once the search tree is created, the probabilities attached to all patterns collected from the training image within the specified template can be easily retrieved. 3

4 3 Search Tree The search tree is a powerful tool to store the number of replicates of any mp data event (i.e. pattern) found over the training image within the template retained. Related mp probabilities are then identified to these training proportions and retrieved. The building of a search tree proceeds as follows: l Specify the size and geometry of the template and order the corresponding grid nodes according to their distance to the center of the template. An anisotropic variogram distance can be used. Consider a template fi with n ordered nodes: fi = fh ; =1; 2; :::; ng where h = u u is the vector linking the center u of the template to the -th node location u. l Scan the training image for all occurrences of any data event within the previously specified template and any of its n sub-templates defined as: fi k = fh i ; i =1; 2; :::; Kg; K =1; 2; :::; n The smallest sub-template fi 0 corresponds to a single datum location at the center u of template fi. The next sub-template fi 1 includes that central location u and the next closest node u 1. Finally: fi 0 ρ fi 1 ρ fi 2 ρ ::: ρ fi n = fi l Starting from the smallest fi 0, these sub-templates are centered at each node of the training image: the corresponding training data event is recorded and a proper count is incremented. A data event consists of the sub-template geometry and data values: d k = fs(u + h i )=s i ; i =1; 2; :::; kg Once scanning of the training image is completed, a search tree has been generated. That tree stores the numbers of replicates of any data event found within the specified data template fi The search tree is divided into n +1levels, each level i corresponding to the sub-template fi i ;i = 1; 2; :::; n. The root of the search tree (level i = 0) stores the number of single-point occurrences of each category in the training image, that is the global proportion of each category. This is a single-point statistics. Level 1 of the 4

5 search tree records the number of occurrences of each category at the second node of the sub-template fi 1. From these first two levels of the search tree, one can derive the probability of occurrence of any data event within the sub-template fi 1. This is a twopoint statistics. Similarly, one can extract three-point, four-point,..., up to n-point statistics. Note that not all potential n-point data configurations may exist in any specific training image, in such cases some branches of the search tree will stop before the (n +1)th level. This feature significantly reduces the size of the search tree and RAM demand. In a simulation using multi-point statistics, we need to infer the probability of a specific event conditioned by specific data values arranged along a specific geometric template. Denote this conditioning data event by: d = fs(u + h j )=s j jj =1; 2; :::; mg In most situations, this specific data template fi m does not identify any one of the sub-templates fi i used to generate the search tree. In such case, retain the largest training sub-template fi i included in the conditioning data template fi m, with fi i ρ fi m. The probability associated to the conditioning data event d is then approximated by that corresponding to that largest training sub-template fi i. 4 Rotation and ainity transforms The rotation and ainity (squeezing) transform of a training image is now discussed. These linear coordinates transforms are associated with the rotation and scale invariance of multiple-point statistics. At first, consider a transform with constant rotation angle and constant ainity factors. Next, a transform with location-dependent rotation angles and ainity factors is considered. Application is made to a fan deposit simulated from a training image with straight s. All examples are 2D, but the algorithm is easily extended to 3D. All simulations are non-conditional to focus strictly on reproduction of the mp statistics. 4.1 Rotation Let ft (u); u 2 D L g be the training image defined over the grid D L. Corresponding unconditional simulations should display a single main direction of continuity but possibly dierent from that of the training image. Let be the desired direction of continuity measured in degrees clockwise from the continuity direction (East) of the training image. The simulated image is denoted by fs (u)ju 2 Dg. The necessary rotation can be implemented as follow: 5

6 l Keep the original search tree. Rotate any conditioning data event by - within the data template prior to looking for the number of replicates in the search tree. The coordinates rotation is written: where u new and u old u new = R u old u =(x; y; z) (4) are the old and new coordinates of each single datum component in a data event. R is the rotation matrix. In 2D with u =(x; y), this is written: ( x new y new = x old = x old Because the rotated location (x new cos yold sin sin + yold cos (5) ;ynew ) does not usually fall on a grid node of the training image, it is relocated to the nearest grid node. Note that this relocation entails changing somewhat the original data geometry, see Appendix. A training image (Figure 1a) and a square data template with 7 7 = 49 grid nodes are used. Only two facies are considered: in black with training proportion 0.3 and in white with proportion 0.7. Three rotation angles =0 i ; 45 i, and 90 i are considered. The corresponding unconditional realizations of size are displayed in Figures 1b to 1d. The realizations do reproduce the desired direction of maximum continuity. The lack of full continuity across the simulated area is not related to the rotations done. It is a consequence of using a template of limited size (7 7) 4.2 Ainity We can also squeeze the given training image (an ainity transform) to change the aspect ratio of its geometric patterns. In a fluvial sedimentary environment, this amounts to change the width and sinuosity of the training image s. Two ainity factors a x, a y along the two desired major and minor directions of continuity control the magnitude of the ainity. If the elongation in the training image is horizontal (x-direction), then a y > 1 leads to thinner s and a x > 1 leads to more sinuous s.the corresponding coordinates transform is written: u new = Au old u =(x; y; z) (6) where u new and u old are the old and new coordinates of each single datum location of the mp data event. A is the ainity matrix: 6

7 A = In 2D with u =(x; y), this is written: 0 a x a y a z x new y new = a x x old = a y y old Using the training image of Figure 1a, the following ainity factors were considered: a x =1.0, a y = 2.0, 3.0, 4.0, 5.0. The corresponding four realizations are displayed in Figure 2. The simulated s are increasingly thinner as a y increases. The four realizations corresponding to a x =1.0, a y = 0.9, 0.8, 0.7, 0.6, are displayed in Figure 3. The simulated s are increasingly wider as a y decreases. If a y is fixed, increasing a x leads to increase sinuosity, see Figure 4. 1 C A (7) 4.3 Full linear transform We can simultaneously rotate and squeeze the patterns present in the training image. The corresponding coordinate transform is written: u new = AR u old u =(x; y; z) (8) where u new and u old are the old and new coordinates of each single datum location of the mp data event. R and A are the rotation and ainity matrices. In 2D, this is written: ( x new y new = a x [x old = a y [x old cos yold sin ] (9) sin + yold cos ] Figure 5a shows the sequence of rotation and ainity. The simulated realization, corresponding to a 30 i rotation angle and ainity factors a x =1:0, a y =3:0, is displayed in Figure 5b. That realization is similar to that of Figure 2b except for the 30 i rotation. With a 45 i rotation and a x = a y =3:0, the realization is displayed in Figure 5c; it is seen to be similar to that of Figure 4b. 4.4 Location-dependent transform Stationarity is a statistics requirement rarely exactly met by natural phenomena. Instead, geological patterns typically evolve gradually from one area to another; this 7

8 gradual deformation can be modeled by a series of location-dependent linear transforms (rotations and ainities). If these location-dependent transforms can be evaluated, e.g. from seismic data, the stationary patterns of the training image can be gradually transformed to deliver the locally variable patterns desired. For example, a fan deposit may be seen as consisting of three parts: upper fan, middle fan and lower fan. A main feeder (or a few feed s) split and produce many smaller braided s as they move forward from the upper fan to the lower fan. Hence, the s tend to deviate from the feeder direction and be increasingly smaller. The location-dependent transform is written similarly to expression (8): u new = A(u 0 )R (u0 )u old u =(x; y; z) (10) where u new and u old are the old and new coordinates of each single datum location of the mp data event; u 0 is the center of that data event; R (u0 ) and A(u 0 ) are the rotation and ainity matrices which are location u 0 dependent. In 2D, this is can be developed as: ( x new y new (u) =a x(u 0 )[x old cos( (u 0)) y old (u) =a y (u 0 )[x old sin( (u 0))] sin( (u 0)) + y old cos( (u 0))] Consider the simulation of a fan deposit in which the feeder s are known to move from northwest to southeast. A large stationary training image, twice larger than the simulated grid size of is used, see Figure 6a. At each simulated grid node, a dierent set of rotation angle and ainity factors is used to impose local characteristics of that fan deposit. The maps of local rotation angles and ainity factors used are shown in Figures 6b and 6c. Figure 6d gives one resulting simulated realization. The format of the data file containing rotation angles and ainity factors as well as the corresponding comments are given in Appendix of this report. 8

9 Assume the following notation: l A is the binary (indicator) random variable associated to the occurrence of state s k at location u. l B is the binary random variable associated to the occurrence of the data event constituted by the n hard conditioning data S(u + h )=s k, =1; 2; :::; n, considered jointly. l C is the binary random variable associated to the occurrence of the data event constituted by the n Y soft conditioning data Y (u+h 0 fi )=y fi, fi =1; 2; :::; n Y, considered jointly. If seismic information can be calibrated to provide the partial conditional probability P (AjC), it can be combined with the other partial probability P (AjB) obtained from hard data to give the full conditional probability P (AjB;C), Journel(2000). Two realizations corresponding to dierent combinations of P (AjB) and P (AjC) are shown in Figure 7; the second (Figure 7b) gives more influence to the seismic component P (AjC). A more complex example is shown in Figure 8. In this case, a composite of fan deposits is considered in which the continuity direction and width of the s are dierent in dierent lobes of the composite fan. The spatial distributions of local rotation angles and ainity factors are given in Figure 8b and 8c. The resulting simulated realization is displayed in Figure 8d. 4.5 Non-stationary examples A non-stationary representation of a delta fan cannot be used as a training image because it is too location-specific and its patterns are not repeated over the training area. Figure 9 shows one such non-stationary training image of a delta fan and one resulting simulated realization. The characteristics of the original training delta fan are not reproduced. Indeed, if a non-stationary training image is scanned, the location-dependent patterns of this training image are averaged out by the multiplepoint statistics and results in the average stationary patterns seen on the simulated realization. This fundamental remark is true for all statistics including the variogram models of 2-point geostatistics. 9

10 5 Preliminary conclusions From the above discussion and preliminary test runs, the following conclusions can be made: l Training patterns should be stationary with rotation and ainity invariance. l Location-specific patterns associated to a specific reservoir can be reproduced through coordinates linear transform (rotation and ainity). The multiple-point statistics can be read from a modular stationary training image. l Very dierent facies distributions, although all related to the same geological sedimentary style, can be generated from one single modular training image. Reference [1] S. Strebelle., Sequential simulation drawing structures from training images. SCRF report, 2000, and PhD thesis. [2] W. Xu., Conditional curvilinear stochastic simulation using pixel-based algorithms. Mathematical Geology, 28(7). [3] C.V.Deutsch and A.G.Journel., GSLIB - Geostatistical Software Library and User s Guide. Oxford University Press,

11 2500 a- Training image East b- Realization with angle = c- Realization with angle = 45 East 1500 East d- Realization with angle = 90 East 1500 Figure 1: Training image (a) and simulated realizations with dierent rotation angles (b), (c), and (d) 11

12 1500 a- Realization with ax=1.0, ay= b- Realization with ax=1.0, ay=3.0 East 1500 East c- Realization with ax=1.0, ay= d- Realization with ax=1.0, ay=5.0 East 1500 East 1500 Figure 2: Realizations with decreasing width (a y > 1) 12

13 1500 a- Realization with ax=1.0, ay= b- Realization with ax=1.0, ay=0.8 East 1500 East c- Realization with ax=1.0, ay= d- Realization with ax=1.0, ay=0.6 East 1500 East 1500 Figure 3: Realizations with increasing width (a y < 1) 13

14 1500 a- Realization with ax=2.0, ay= b- Realization with ax=3.0, ay=3.0 East 1500 East c- Realization with ax=4.0, ay= d- Realization with ax=5.0, ay=3.0 East 1500 East 1500 Figure 4: Realizations with increasing sinuosity (a x > 1) 14

15 a - Sequence of rotation and ainity Original data event Ainity u 4 u 3 w 3 u 1 θ u 2 w 4 w 1 w 2 v 3 v 4 v 2 v 1 Rotation b- Realization with angle=30, ax=1.0, ay= c- Realization with angle=45, ax=ay=3.0 East 1500 East 1500 Figure 5: Realizations with dierent rotation angles and ainity factors 15

16 5000 a- Large training image East b- Rotation angles c- Ainity factors East 2500 East d- One realization East 2500 Figure 6: Training image for a delta fan (a), maps of rotation angles (b) and ainity factors (c), one simulated realization (d). 16

17 5000 a- Large training image East b- P(A C) c- One realization (omega = 0.2) East 2500 East d- One realization (omega = 1.0) East 2500 Figure 7: Training image (a), seismic data given as P(AjC) (b), and two realizations with dierent emphasis given to the C-seismic data (c) and (d). 17

18 5000 a- Large training image East b- Rotation angles c- Ainity factors East 2500 East d- one realization East 2500 Figure 8: Training image (a), rotation angles (b), ainity factors (c), and one realization (d). 18

19 2500 Training image East One realization East 2500 Figure 9: Non-stationary delta fan used as a training image and one resulting simulated realization 19