SIMPAT SIMULATION OF FACIES THICKNESSES INTERPRETED THROUGH SEQUENCE STRATIGRAPHY: APPLICATION ON A CARBONATED OUTCROP. Authors: Laura Dovera 1 and Jef Caers 2 1 Eni Exploration & Production division, San Donato Milanese, MI, Italy. 2 Department of Petroleum Engineering, Stanford University, CA, USA. Introduction Outcrop models can provide important information about reservoir architecture and heterogeneity. However, geological information from outcrop, because of its great variety and complexity, cannot be integrated to its fullest extent using traditional geostatistics. The conventional variogram-based modeling techniques typically fail to capture complex geological structures. Multiple-point geostatistics encompasses a set of innovative modeling techniques that perform simulation starting from a training image, a 3D conceptual visual representation of how heterogeneities are believed to be distributed in the actual subsurface reservoir. The training image forms a gateway for geological expertise and interpretation to be quantitatively used in reservoir modelling. Training images are in fact an explicit but merely conceptual rendering of the subsurface heterogeneity and need not be constrained to any specific reservoir data. Training images can be constructed using object-based simulation or using outcrop data. The latter will be the topic of investigation in this report. This report presents an application of simpat, a multiple-point stochastic simulation method for generating reservoir facies models by capturing the complex facies succession of a carbonate outcrop and anchoring it to subsurface well data. The outcrop consists of a complex depositional sequence of mound and lobe bodies with complex spatial relationships. The idea is to generate reservoir models that reflect the observed geological complexity, yet at the same time are constrained to any available reservoir data. In order to reproduce this complex architecture in such models and to make explicit use of the sequence stratigraphic depositional information, the depositional bodies were not simulated directly but with a new approach relying on the simulation of facies thicknesses interpreted through sequence stratigraphy. The main idea of this new approach is to separate thickness and facies information into two different 3D training images rely on the capabilities of simpat to jointly simulate these two properties. This approach can easily integrate complex information regarding bodies’ geometry and sequence stratigraphy and it can be applied to any type of geological depositional system. The report also looks into the issue of the within realization variability and between realization of simpat simulations. Since simpat literally copies patterns from the training

image, the models may not render “enough” uncertainty. We show how uncertainty can be increased by introducing more explicitly variability through randomization of either the training image or of the training image properties. These various ideas are applied to a real-case outcrop: the Cassis outcrop (South of France), which is introduced first. Cassis upper Cretaceous outcrop, Beausset Basin – Provence, France The upper cretaceous carbonated outcrop of La Marcouline (Cenomanian interval) is located in the Beausset Basin in Provence, France (Fig 1). This complex is constituted by different depositional environments: carbonated platform (North-East), shelf, slope and basin area (South-West) (Fig 2). The geological data related to the external shelf and slope parts have been integrated in a 3D gOcad model. The geological data include previous knowledge of the outcrop (from five existing PhD thesis) and sedimentological and stratigraphical analysis. The Gocad model consists of a depositional sequence of mound and lobe bodies with complex interpreted spatial correlations. The data set includes a gOcad S-grid for the entire outcrop and 21 gOcad S-grids for 21 different depositional objects. Each object is identified by an index depositional sequence and a facies code. The model covers an area of about 1.8 Km x 1.4 Km and has an average thickness of about 342 m. The following facies classification was identified: facies 1 = breccia;

facies 2 = lobes;

facies 3 = grainflows;

facies 4 = debris flows. The choice of the Cassis outcrop as test case is for the following reasons:

• it is analogue to Middle East and North Adriatic reservoirs;

• it is a continuous outcrop in 3D;

• it presents a large range of carbonated sedimentary environments and gradual to sharp spatial changes;

• it shows spatial heterogeneity at reservoir scale;

• it includes various lithologies from limestone to sandstone and marls;

• it is well studied in previous work. The idea of this application is to generate reservoir models that reflect the Cassis outcrop geological complexity, yet at the same time are constrained to any available reservoir data. However, the complex architecture of this outcrop requires for its modeling a new geostatistical algorithm because the conventional variogram-based modeling techniques are not suitable to capture complex geological structures.

Multiple-point geostatistics Variogram-based geostatistics When modeling a hydrocarbon reservoir from a static point of view the goal is:

• to provide a model that correctly expresses the believed geological heterogeneity;

• to integrate all the available information (different types of data coming from different sources at different scales), such as well-log and seismic data;

• to quantify uncertainty through generating multiple reservoir models. Traditional geostatistics relies on obtaining a variogram to model spatial variability of the reservoir variable in question. The variogram essentially measures the degree of correlation between any two sample values at various distances, hence quantifies the spatial correlation in the reservoir. Once the variogram is determined, a technique termed sequential simulation generates reservoir models that reproduce the variogram modeled from wells, hence reproduces the desired spatial correlation. Property modeling through sequential simulation can be applied to either facies and/or petrophysical properties. Sequential simulation generates values of reservoir properties between wells by random drawing of conditional distributions of previously simulated nodes and hard data. These conditional distributions are derived from kriging equations that use the variogram as input. In the traditional sequential simulations algorithms the main tool for spatial correlation modeling is the variogram. Since variogram is a statistical tool describing the dissimilarity of a variable observed at any two spatial locations only, limiting the complexity of the reservoir heterogeneity being captured by this simple statistical tool. Hence, this type of correlation, that is usually called two-point statistics, is too limited to reproduce complex geological structures. Such structures, with curvilinear or geometrically complex patterns, require modeling multiple-point correlations. Multiple-Point geostatistics Multiple-Point geostatistics is an innovative technique introduced by Stanford University as an alternative to traditional variogram-based geostatistics. Utilizing multiple-point statistics explicitly allows for the reproduction of complex geological structures. This statistical approach does not rely on variogram models but it obtains multiple-point correlation models from a training image. The training image is a conceptual visual representation (2D or 3D) of the geological heterogeneity. The training image can be selected or built by a geologist, hence reflects a prior geological concept. The training image need not be conditioned to any location-specific subsurface data. It contains the relevant geological patterns from which multiple-point correlations can be extracted and then the multiple-point geostatistics simulation anchors those patterns to conditioning data.

There are many important advantages in performing a reservoir characterization using multiple-point geostatistics:

• with multiple-point geostatistics it is possibile to integrate significantly more information than the locally sampled data available at the well location;

• multiple-point geostatistics can integrate geological information with great complexity that it is not easily captured with two-point statistics;

• multiple-point geostatistics can reproduce different geological scenarios by considering multiple alternative training images and consequently it can more adequatly express and quantify geological uncertainties.

Two different algorithms for multiple-point geostatistics Currently several different algorithms for multiple-point geostatistics are available in 3D, we will focus on now routinely used snesim algorithm and the more recent simpat algorithm. The filtersim algorithm of Zhang (2006) is another important contribution that could have been considered, however our approach relies on some simpat features that are currently not available in filtersim. Snesim and simpat are both sequential simulation algorithms. They merely map random numbers into samples using each simulated value as hard datum value for the simulations in nodes visited later in the sequence. While snesim uses probability theory to define the simulation mapping and consequently it strongly relies on the principle of stationarity in inferring multiple-point statistics, simpat does not use probability theory and is less limited by this principle, but pays a price in limited variability. Both of these algorithms require a training image to extract multiple-point statistical information, but each does it in a different way. Snesim follows very closely the sequential simulation principle. At each node to be simulated one searches for nearby data from wells and also any previously simulated values. This set of values is termed “data event”. For this data event, snesim searches for replicates of that event in the 3D training image and then retrieves the corresponding histogram of the central value. However, instead of repeating this CPU costly scanning operation, snesim stores, prior to starting simulation, all possible data events in a dynamic data structure called “search tree”. During simulation, at each visited location, the correct probability can be immediately read from the search tree and a value can be simulated. In other words, snesim reads the training image patterns proportions as probabilities and simulates new values according to those probabilities. Involving probability theory, to extract multiple-point statistics, a training image for snesim is necessarily bounded by the principle of stationarity. The requirement of stationarity of the training image may be considered as an advantage and disadvantage at the same time. Snesim extracts only the stationary essence from the training image, it filters non-repetitive details from the training image. Hence, under this principle of stationarity the training image has to contain only patterns with stationary features, other will be ignored. The disadvantage is that such non-repetitive details may

be geologically relevant. Indeed, one can expect that most geological concepts contain some non-stationary features. There are two solutions to this problem in snesim. One is to repeat the non-stationary pattern, but this would require a very large training image and hence large memory demand (search tree is large). The other solution is to use trend maps for proportions (areal or vertical, see for example Harding et al., 2004), or locally varying anisotropy directions to simulate channels meandering in different directions. The latter solution only handles certain types of non-stationary information. Simpat, or SIMulation with PATterns, is a new multiple-point algorithm that is not developed with explicit reference to probability theory. Simpat considers the training image as a collection of patterns from which one selects the patterns that locally match the reservoir conditioning data. In simpat, one defines a 3D template that is used to extract all patterns by scanning the template over the training image. The patterns are stored in a pattern database and during the simulation the method looks for training image patterns that are “similar” to the data event and pastes the patterns into the simulation grid. Instead of exporting multiple-point statistics through conditional probabilities, the problem is redefined in simpat as an image processing problem and the traditional probability calculations of multiple-point statistics are replaced with similarity calculations. The main advantage in the direct and explicit reproduction of multiple-scale training image patterns instead of multiple-point statistics is the possibility to reproduce realistic highly complex training images with non-stationary features in a more intuitive way. In simpat the training image construction is not limited by the principle of stationarity. When building a simpat training image a geologist has simply to draw explicitly and exhaustively the patterns to be reproduced without caring about their stationarity. Furthermore, simpat can deal easily with multiple categorical values as well as continuous valued training images. Snesim and simpat assign a different meaning to the training image. Snesim captures the essence of the training image and, by choosing snesim, a geologist has to think of the training image as a representation of that stationary essence. Simpat is very faithful to training image patterns and a geologist should expect the resulting simulations to contain patterns exactly similar to the training image ones. The disadvantage of simpat is that it may be too faithful to the training image in the sense that it exactly reproduces the same patterns, no filtering or morphing takes place. The latter may also be a problem when conditioning to data. Because of this, both these two algorithms are useful and the choice between snesim and simpat depend on the different circumstances of the application. If, as for the deterministic model of the Cassis outcrop, we know exactly the patterns to be reproduced and if these patterns present complex geometries, simpat appears to be a better choice than snesim. Also simpat contains several useful features that pertain to simulations based on the Cassis outcrop as will be discussed next.

Simpat The main reference for this section is the PhD thesis “Sequential simulation with patterns” by G. Arpat in 2005. Simpat unconditional simulation Simpat is a sequential simulation algorithm that scans the training image for patterns that are most similar to the data event. Define )(uti as a value of the training image ti where

tiGu ∈ and tiG is the regular Cartesian grid discretizing the training image. )(utiT indicates a specific multiple-point vector of )(uti values within a template T centered at node u ; i.e., )(utiT is the vector:

{ })(,),(,),(),()(Tn21T huhuhuhuuti ++++= titititi �� α

where the αh vectors are the vectors defining the geometry of the Tn nodes of template T and Tn,,1 �=α . When the training image ti is scanned using the template T , the corresponding multiple-point vectors )(utiT are stored in a data base. Each such )(utiT vector is called a “pattern” k

Tpat of the training image and the database is called the

“pattern database” and is denoted by Tpatdb . Such patterns are referred to as kTpat where k denotes the particular k-th configuration of the previous vector of values

)(utiT of the training image ti , with each value now denoted by )( αhpatTk , where

Tpatnk ,,1 �= and Tpatn is the number of total available patterns in the pattern database

Tpatdb . Once the pattern database Tpatdb is constructed, the algorithm proceeds with the simulation of these patterns on a realization re . In sequential simulation, a data event

)(udevT is defined as the set of hard data and previously simulated values found in the template T centred on a visited location u where T is the same template used to scan the training image. Then:

{ })(,),(,),()(TnTT1TT huhuhuudev +++= devdevdev �� α

where )()( αα huhuT +=+ redev and re is the realization. During simulation, the nodes

reGu ∈ of the realization re are randomly visited and at every node u , the data event

)(udevT is extracted. This data event is compared to all available patterns kTpat in the

pattern database Tpatdb using a predefined similarity criterion ⋅⋅,s . The aim is to find

the pattern *Tpat ‘most similar’ to the data event )(udevT , i.e. is the pattern that

maximizes ks TT patudev ),( for Tpatnk ,,1 �= or, equivalently, that minimizes the

dissimilarity kk sd TTTT patudevpatudev ),(),( −= for Tpatnk ,,1 �= . Once this most

similar pattern kTpat is found, the data event )(udevT is replaced by this pattern, i.e. the

values of the most similar pattern *Tpat are pasted on to the realization re at the current

node u , replacing all values of the data event )(udevT . In the current simpat version the (dis)similarity criterion is based on a single-point dissimilarity function (measuring similarity between any two pattern by point-to-point comparison), namely the Manhattan distance. The Manhattan distance between a data event and a pattern is defined as:

�=

−+=T

hhupatudev TTTT

nkk patdevd

1

)()(),(α

αα .

While convenient, this measure of distance does not always measure adequately the distance between patterns. For example, this measure is very dependent on the marginal distribution of pixels within the patterns being compared. As a result, use the single-point distance functions in simpat often results in simulated realizations that do no reflect well the training image multiple-point properties. In particular, the Manhattan distance does not reproduce well training images where the global proportion of a particular category or value is significantly less than that of any other category or value. In the binary case, this problem can be avoided using distance transforms option in simpat. The limitation of using this option on binary training images was overcome with the introduction of “bands” which made it possible to distance transform and simulate multiple-category training images. We will discuss this in more detail. Multiple-category training images The main reference for this paragraph is the paper Re-addressing the missing scale using edges by L. Stright and J. Caers in 2005. A single band describes the occurrence of a single category in a multiple-category training image, ti , where the number of bands is equal to the number of categories (or bands) bn and each category is identified by an index, ib , i.e. ibti =)(u where bni ,,1�= .

The multiple-bands of the training image can be represented as a vectorial variable )(uti such that for a specific category index ib , the vectorial variable has entities 0),( =bti u if

ibb ≠ and 1),( =ibti u otherwise. A “training image band” is then defined as the vector

)( ibti of all vectorial values ),( ibti u , tiGu ∈∀ where tiG is the training image grid.

A data event )(udevT is now comprised of several single data events each belonging to a particular band:

{ }),(,),,(,),,(),,()( 21 bni bbbb udevudevudevudevudev TTTTT ��= with

{ }⋅++++= ),(,),,(,),,(),,(),( iiiii bdevbdevbdevbdevb

TnTT2T1TT huhuhuhuudev �� α

A pattern,

{ }),(,),,(,),,(),,()( 21 bni bbbb hpathpathpathpathpat TTTTT ��=

is scanned from the pattern database, and the Manhattan distance to measure dissimilarity between patterns and data events is defined as the sum of all absolute differences of all bands:

�=

+=T

hhupatudev TTTT

n kk patdevdd

1

)(),(),(α

ααα

and

�=

−+=bn

ik

i bpatbdevyxd1

),(),(,α

ααααα hhu TT .

Where Tn,,1 �=α is the number of nodes in the template and the least dissimilar pattern is the most similar. The simulation is now performed based on the multiple-category training image )( ibti . For an unconditional simulation the process is as follows:

1. a random path is defined on the simulation grid;

2. at the first node of the random path a vectorial pattern k

Tpat is randomly selected from the pattern database and pasted into the grid;

3. step 2 is repeated until some of the previously simulated nodes start to appear within the bounds of the template at which point the pattern database is searched

to match to most similar )(udevTk

based on the Manhattan distance ααα yxd , ;

4. the most similar vectorial pattern k

Tpat is pasted into the simulation grid and the simulation proceeds to the next node.

The final result is vectorial since the patterns k

Tpat are vectorial. The original implementation of banded simulation was such that distance transforms could be calculated on a training image that contained more than one category since this transform was applicable to binary images. In this application we use this option in a different way.

Here, as suggested by Stright and Caers, we simulate two different properties as part of a vectorial training image using the dual band option to generate a coupled geostatistical realizations of the two attributes. Hard data conditioning In order to generate conditional realizations, the simpat algorithm first copies the hard data onto the realizationre , i.e. the realization re is not longer fully uninformed at the beginning of a simulation but contains the hard data instead. Since the hard data are copied onto the realizationre , a data event )(udevT automatically contains the hard data event )(uhdevT . The unconditional algorithm previously described is then modified such that it accepts the hard data as hard constrains, i.e. during the simulation, for each data event )(udevT with u in the vicinity of hard data point (defined by template T ), the conditional algorithm attempt to find a most similar pattern *

Tpat such that the pattern *Tpat reproduces exactly the hard data event )(uhdevT on the realization re . The

problem being that sometimes no such pattern is found, hence one cannot exactly reproduce the hard data event, resulting in discontinuity near some hard data. In other words, conditioning to hard data does not change how the most similar pattern is found but only changes the pattern database on which the similarity search is performed. Performing multiple-banded simulations, the current simpat implementation allows the hard data conditioning only for the first band of the training image. This means that if we simulate different properties using vectorial training images, only the first property can be constrained to well data. Future implementations of simpat will fix this problem. Simulation Fitness maps The simulation fitness maps are a way to quantify the overall pattern reproduction of a realization and a method to check data conditioning. Since the simpat conditional algorithm calculates the similarity ⋅⋅,s for every node of a realization, one way to quantify and visualize the quality of pattern reproduction in a single realization is to utilize the similarity error (i.e. the distance ⋅⋅,d ) as a measure of mismatch. Every time a lookup of the pattern database provides a most similar pattern that does not exactly match the current data event, a “pattern mismatch” occurs, i.e. 0, ≠⋅⋅d . The so-called simulation fitness maps are calculated through cumulative summation of individual node errors ⋅⋅,d . Note that snesim reports an equivalent of this error: snesim drops values from the data event if no replicates of that data event occur in the training image. Snesim reports, for each simulated node, how many nodes were dropped. In this scheme, the good realization has a very low overall error, i.e. high fitness, whereas the “bad” realization has conditioning problems i.e. low fitness. Furthermore, since the error is comparative and always positive (or zero), one can simply take the mean of the simulation fitness map as a single, scalar indication of overall fitness of a realization, allowing quantitative comparisons between different realizations, or better between different alternative training images.

Application of multi-point geostatistics to Cassis This report presents an application of the simpat multiple-point algorithm to the Cassis carbonated outcrop, in particular to the Cenomanian interval “La Marcouline”, using the outcrop 3D deterministic model as reference. Outcrop analogues provide important knowledge and information in terms of size, internal heterogeneity and connectivity of reservoir flow units. Utilizing the deterministic outcrop model to verify and set-up new methods of reservoir model population allows:

• to optimize a workflow for multiple-point geostatistics in the carbonated reservoir characterization and modeling;

• to test and better understand new multiple-point geostatistics algorithms, or, the advantages and limitations of existing techniques.

The Cenomanian carbonated complex of La Marcouline consists of a depositional sequence of mound and lobe bodies with complex spatial correlations (Fig 3). In particular, the deterministic model shows a sequence of 21 different bodies stacked on top of each other. Each body is identified by an index depositional sequence and a facies code. Complex geometries and facies associations are evident looking at 3D sections of the model (Fig 4). These complicated features of the outcrop model require a multiple-point geostatistics approach for their modeling. In testing the methodology, we will work only with this deterministic outcrop data. No specific reservoir was used. The goal is to reconstruct the outcrop patterns, with reasonable variation in the various realizations generated. Data conditioning to synthetic well data will be tested. The outcrop model cannot be used directly as input to a multiple-point algorithm, be it snesim or simpat. The standard approach applying multiple-point geostatistics on this outcrop would be to simulate directly the depositional bodies building a 3D training image consisting of depositional facies with their correct shapes and areal associations. In case of snesim, the outcrop model would need to be transformed in a stationary or set of stationary training image models and a set of areal proportion maps and vertical proportion curves. Harding et al. (2004) suggest a method where geologist draw by hand based on wells and geological interpretation, areally the various zones of deposition, for example, one zone contains facies x and y but not z, another zone contains only facies x and z, but not y. These areal interpretations are combined with vertical proportion curves to generate a 3D probability cube that is input into snesim and serves as a 3D constrained to the simulated realizations. The workflow proposed in Harding et al. actually uses a training image that contains the various areal and vertical trends. This workflow works well if the interpretations of such zones are indeed available and reliable. Indeed, the resulting simulation will be strongly constrained to such areal maps and if they are uncertain, some randomization of them may be required. The workflow of Harding et al.

could be applied to this case as well, but would require more work in terms of generating randomized areal proportion maps and vertical proportion curved. Also, in complex carbonate environment, one may not have access to these interpretations. Moreover, we would like to make more explicit us of the stratigraphic and genetic information on carbonate mound growth provided by the Cassis outcrop interpretation to better constrain the geometry of the bedding, hence we opt for a more direct approach based on simulating bedding thickness, rather than directly simulating facies presence. The approach we take will allow the geologist to export the essential features of the Cassis outcrop to the subsurface modeling. If we consider this outcrop only as an example of complex geological architecture created from the depositional process of a certain number of bodies, we can pose a more general problem of how to build in a simple way a training image that reflects an observed geometry and that includes the sequence stratigraphic depositional information. This report presents a possible solution to this problem introducing a new approach relying on the simulation of facies thicknesses interpreted through sequence stratigraphy. The main idea of such an approach is that a body thickness contains all the information regarding its geometry and is also the simplest way to represent this geometry. Furthermore, in case the stacking of bodies can be interpreted as a sequence, using a sequence of bodies’ thickness maps, it will be possible to explicitly integrate the stratigraphic sequence information. Consider a geological architecture created from the deposition of n bodies. Each body has its own shape, index in the depositional sequence and facies code. Considering the following workflow:

• draw, for each object, a thickness map and a map with the corresponding facies code (Fig 6a – Fig 6b);

• build a 3D “thickness training image” by stacking the n bodies’ thickness maps according to a depositional sequence; similarly, stack the n facies code maps to form a 3D “facies code training image” (Fig 7);

• use these two training images as input for simpat and simulate jointly thickness values and facies codes using multiple-banded simpat simulations (Fig 8).

• transform the simulated thickness values facies codes to form a realization of actual shapes (Fig 8).

Many motivations and advantages justify this proposed workflow:

• a 3D training image consisting of thickness maps contains all the information regarding the geometry of the geological scenario considered;

• using two training images with thickness values and facies codes we can explicitly use the sequence stratigraphic information;

• the 3D simulations allows the integration of spatial correlations between depositional bodies as present in the outcrop;

• drawing thickness and facies codes maps is easy, intuitive and natural for a geologist, no need to deduce areal proportion maps

• the workflow can be applied to any type of geology that can be represented as combination of depositional bodies whose thickness maps can be drawn. For example, the stacking of major channel bodies.

This new workflow has been applied to the Cassis outcrop in three step. In the first step, the training image with facies codes was ignored and single-band simpat simulations were performed of the bedding thickness only. In the second step also the facies information was included. In the third step simulations conditioned to hard thickness data were performed. Detailed activity Data set processing The structural deterministic 3D gOcad model was imported into S-GeMS as GSLib file using a gOcad plugin (S-GeMS grid dimensions: 180 x 142 x 342) (Fig 4). The thickness maps (Fig 6a – Fig 6b) were calculated using the 21 Gocad Sgrid. In order to speed up the work all the tests have been performed using an upscaled grid. The dimensions of the reference upscaled model are: 45 x 36 x 342 (Fig 9). Also the thickness maps have been upscaled. First step: the Manhattan distance problem Consider a 3D “thickness training image” by stacking the 21 bodies’ thickness maps according to the original depositional sequence (Fig 11). We can perform a non-conditional simpat simulation on this training image. We can use a template size of 5 x 5 x 3 and automatic number of multiple-grids: in this step the aim is to reproduce the training image as much as possible. If we look at the simulation result (Fig 13) and if we compare its histogram (Fig 14) with the target histogram of the training image (Fig 12) we can see that the patterns with high thicknesses values are not well reproduced. This problem is related to the Manhattan distance. In fact the Manhattan distance fails to capture correctly the training image patterns when the global proportion of one value is significantly higher than any other values in the training image. In this case the target histogram has a spike at zero thickness values (no facies body occurring at the location during that time of deposition). During simulation, when a data event matches poorly to the patterns available in the pattern database, the Manhattan distance settles for a most similar pattern that has the highest number of matching nodes to the data event. The problem is that these patterns likely do not contain the low proportion category on the mismatching nodes, resulting in a bias toward the category with highest proportion or frequency of occurrence.

One possible solution to this problem is to transform the thickness values before the simpat simulation in a way such that the new thickness values will be correctly weighted in the similarity calculation and than to back transform the simulation result. In particular we have to choose a transformation that

1) increases the distance between the zero values and the other values;

2) reduces the range of the thickness values major than 0;

3) produces a uniform distribution for all the thickness values major than 0. According to this, we can transform the values as following:

1) assign thickness = -50 if thickness = 0;

2) rank transform in (0,1) all the thickness values larger than 0. Once the thickness values were transformed, we can create a new stacking of transformed thickness maps (Fig 15) whose histogram (Fig 16) satisfies the previous requirements sufficiently. We can perform the simpat simulation on this new training image, the result is shown in Fig 17. Comparing the histograms of Fig 18 with the histograms of Fig 16, we can see that the Manhattan distance works better on the transformed values than on the original ones. Back transforming the simulated values we can go back to the original thicknesses and get the final result (Fig 19) whose histogram (Fig 21) matches well the target of Fig 12. Using the original depositional sequence we can also generate facies bodies correspondent to the simulated thickness values (Fig 20) and build the final outcrop model (Fig 22). In the following, all the simulations will be performed using this transformation even if not explicitly said. This means that the general workflow includes a pre-simulation transformation of the training image and a post-simulation back transformation of the simpat result. We will not show these two intermediate steps in order not to confuse the treatment. Second Step: workflow application without conditioning data In this step we want to test the workflow previously explained without using conditioning data. Consider 2 training images: the 3D stacking of the 21 bodies’ thickness maps and the 3D stacking of the 21 facies code maps (Fig 23). A multiple-banded simpat simulation using these 2 training images was performed. Also in this case we used a template size of 5 x 5 x 3 and automatic number of multiple-grids: without conditioning data we give maximum weight to the prior geological information trying to reproduce the training image as much as possible. The simulation result is shown in Fig 25. We can observe that the shape of each object is well reproduced. Simpat keeps the TI concept but depositional bodies change positions vertically as well as areally when compared to the training image. The result is confirmed by a second simulation performed changing the random seed (Fig 27). The variations in the bodies’ positions are evident also looking at 3D sections (Fig 29) and 2D maps and slices (Fig 31) of the final model. Comparing the

histograms of Fig 30, we can see that we do not perfectly reproduce the target histogram of the reference model. Third Step: workflow application using thickness conditioning data In this step, we apply the workflow using hard thickness data. Consider the two training images of thickness maps and facies codes maps and a hard data set consisting of 1 well randomly sampled from the 3D “thickness training image” and then randomly positioned in the simulation grid (Fig 32). Ideally, the well should come from another model or from real data, but this is not available to us at this time. A multiple-banded simpat simulation using these two training images and the sampled well as conditioning hard data was performed. We used a template size of 5 x 5 x 3 and three multiple-grids: these parameters were found by testing several different parameter files and by comparing the results. The simulation result is shown in Fig 34. We can see that using 1 well the two training images are reproduced with small variations: the histogram of Fig 35 matches very well the target histogram of Fig 33. The simulation is also well conditioned to the well data (Fig 36 and Fig 37). Consequently the final model (Fig 38) looks very similar to the reference model (Fig 9) and perfectly reproduces its first order statistics (Fig 39 and Fig 10). We repeated this step also with a set of 5 wells (Fig 40) and using the same simpat parameters. Also in this case the wells have been randomly sampled from the training image and then randomly positioned in the simulation grid. The simulation result is shown in Fig 41 and the final model in Fig 45. Using a conditioning of 5 wells, the first order statistics are slightly different (Fig 42 and Fig 24, Fig 46 and Fig 10) and the simulation presents more variations even if it is still conditioned to the well data (Fig 43 and Fig 44) . These variations increase again when we use an hard data set conditioning of 50 wells (Fig 47 and Fig 48). Even if the simulation is conditioned (Fig 50 and Fig 51) and the statistics are quite well reproduced (Fig 49 and Fig 24, Fig 53 and Fig 10), the geometry of the reference outcrop (Fig 52) is completely altered in the simulation result. The reason is that increasing the number of hard data randomly sampled, we increase the inconsistency between data conditioning patterns and training image patterns. The deterioration of the patterns reproduction increasing the number of hard data is evident looking at the simulation fitness maps. In fact the absolute value of the range and the mean of the fitness maps increase consistently when the number of wells increase (Fig 54). The fitness error is evident also looking at the 3D images of the fitness maps (Fig 55) in fact the cell values with a fitness minor than -1 become more and more spread increasing the number of wells. Uncertainty assessment Simpat is designed to reproduce the training image patterns as much as possible, no filtering or morphing takes place in simpat. Consequently the disadvantage of this algorithm is that it may copy too much without producing enough variability. As a consequence, the algorithm is sensitive to inconsistency between the conditioning data and the training image.

However, this disadvantage may be turned into an advantage. First, one should question, where variability between simulated realizations comes from. It has too long been thought that the variability between simulated realizations from a fixed algorithm with a fixed parameter input realistically represents uncertainty. On the contrary, the major source of uncertainty lies first in the geological scenario adapted, then in the (geometrical) parameters describing that scenario. Since that variability is limited in simpat, in fact, it forces simpat users to explicitly state the various sources of variability that can then be integrated into simpat realizations. Simpat makes it clear that one cannot longer hide behind ergodic fluctuations (geostatistical “noise”) of simulated realizations alone. In most cases, these kind of randomizations are of lesser importance in reservoir modeling. Going from a geological scenario to the simulation results, we distinguish different levels of uncertainty/variability.

• Level 1: uncertainty related to the choice of the geological scenario itself (choice of the training image or outcrop model considered relevant for the reservoir under investigation).

• Level 2: uncertainty related to the “parameters” of that geological scenario (“parameters” of the training image as total facies proportions, dimensions, directions…).

• Level 3: even for given scenario and its parameters, the incompleteness of the given reservoir data allows one to generate multiple realizations that match the data and reflect the geological scenario (spatial uncertainty).

According to these types of uncertainty we can explicitly model variability at different levels. Changing the training image we can introduce variability at level 1, varying the total facies proportions of the training image we can model variability at level 2 and adding more or less constraining data we can control variability al level 3. A simple and general method to achieve variability at level 2 is to randomize explicitly certain aspects/properties of a given training image. Given a training image, called “mother” training image, constituted of certain parameters/properties, we can perturb one of the parameters and generate different “children” training images. Using these new training images as input for simpat, this explicit variability can be reproduced. Such training image randomization has been applied to the Cassis outcrop scenario. Detailed activity A 3D training image of thickness maps is defined by many parameters such as total thickness, thickness of individual beds, proportions of facies or bodies shapes. In this application we decided to introduce variability changing the thickness values for each

body. In particular, we generated 21 thickness multipliers (one for sequence) using a 1D sgsim simulation. We used the target histogram shows in Fig 56 obtained generating 1000 number normal distributed with mean and variance equal to 1: in this way we impose only a small variation around the original thickness values. We chose a spherical variogram model with a range equal to 10. Each layer of the 3D “mother” thickness training image was multiplied by this simulated multiplier value. We normalized the transformed values to conserve the total thickness of the outcrop, but this constrained need not be enforced if the total reservoir thickness is uncertain. Repeating this procedure 5 times, we can generate 5 sequences of multipliers (Fig 57) and 5 new “children” training images (Fig 58) that differ from the mother one in terms of the thickness variation of the facies bodies. We applied the workflow using these 5 different training images. We performed 2 – banded simpat simulations associating the 5 training images with the 3D facies codes training image and we conditioned the simulation to 5 wells sampled from the original 3D thickness training image (Fig 58). The Fig 59 shows the 5 different simulation results while Fig 60 compares the histograms of each children training image with the histogram of its simulation. The first-order statistics are generally reproduced but in the third and fourth case the histograms match very well. Looking at the fitness maps histograms (Fig 61 and Fig 62), we can observe that with the fourth training image we get the best pattern reproduction: the mean of the histogram is the lowest and the cells with fitness values minor than -1 are less spread. Fig 63 compares the geometry of the final models.

Conclusions and future work In this report a new workflow to simulate complex geology represented as combination of depositional bodies was proposed and tested on a well known carbonated outcrop. This new approach does not simulate directly depositional bodies instead is based on simulating bedding thickness using the simpat multiple-point algorithm. Simpat allows to separate thickness and facies information in two different training images and to jointly simulate these two properties. Testing the workflow on the Cassis carbonated outcrop allowed us to verify advantages and properties of this new approach and to investigate the multiple-point algorithm in detail. The results show that by simulating thicknesses, we can easily reproduce complex geological geometries and using a 3D simulation we can explicitly integrate sequence stratigraphic information and spatial correlation between depositional bodies. Simpat allows us to work with continuous variables and the two-banded simulation of thickness and facies properties was successful. Furthermore the approach is simple, intuitive and suitable for many future improvements. However there is a great deal of work to be done to better understand and evaluate the results. The main problem in this application has been to quantify the value of the final models obtained. We tried to reproduce first order statistics and we used the so-called “simulation fitness maps” to quantify the overall pattern reproduction of a realization and to check hard conditioning but the final evaluation of the result mostly relies on visual

inspection. This difficulty is related to the approach itself but it also involves open and well known problems of the algorithm. In this sense the future work has to be a better understanding of the thickness simulations workflow following the multiple-point geostatistics research developments. More specifically the future work should deal with the following problems.

• In order to get the “enough” variability among the simulations it is necessary to find the tune well the simpat input parameters. Currently the only way to find this set of parameters is by testing several different parameters files and comparing results. The research should investigate for methods for evaluating ahead of time this “best set” of parameters, particularly the template size.

• In this application the training image thickness values were transformed to avoid the problem of the Manhattan distance in reproducing the correct target histogram when the proportion of one values is significantly lower than the other values. It is necessary to test and improve different similarity measures, similar to the proximity transform.

• The speed of simpat will need to be improved to make it practical for large

applications. The filtersim idea whereby the search for an exact patterns match is avoided by means of classification of the various patterns opens up an avenue to make simpat more efficient.

• The generation of the simulation fitness map is a way to quantify the pattern reproduction of a simpat simulation but this concept has to be better understand and developed.

• The disadvantage of simpat in copying to much the training image patterns has been avoided introducing explicitly variability through a randomization of training image properties. The problem of the uncertainty assessment in simpat and, more in general, in multiple-point geostatistics is an important question that has to be investigate by future research.

• New and different methods to randomize training images have to be tested and improved.

• Using two-banded simulations we could integrate correlation between thickness and facies property. It is necessary to better understand and quantify this kind of correlation.

• Performing multiple-banded simulations it is not possible to constrain to hard data all the properties. Future implementation of simpat should include this option.

• One natural constraint in outcrop modeling could be the external thickness of the outcrop. A possible next step of the application is to constrain the simulation also to this datum.

• When the seismic data will be available, the next work regarding the specific application on the Cassis outcrop will be the soft data integration.

References

1. G. Arpat, Sequential simulation with patterns, PhD thesis, Stanford University, Stanford, CA, 2005.

2. G. Arpat and J. Caers, A multi-scale, pattern-based approach to sequential simulation, SCRF Annual Meeting Report 16, Stanford Center for Reservoir Forecasting, Stanford, CA, 2003.

3. J. Caers, Multiple-point geostatistics: choosing for stationarity or similarity, SCRF Annual Meeting Report 17, Stanford Center for Reservoir Forecasting, Stanford, CA, 2004.

4. J. Caers, S. Srinivasan and A. Journel, Geostatistical quantification of geological information for a fluvial-type north sea reservoir, In SPE ATCE Proceedings, number SPE 56655, Society of Petroleum Engineers, October 1999.

5. J. Caers and T. Zhang, Multiple-point geostatistics: a quantitative vehicle for integrating geologic analogs into multiple reservoir models, SCRF Annual Meeting Report 15, Stanford Center for Reservoir Forecasting, Stanford, CA, 2002.

6. A.Harding, S.Strebelle, M.Levy, J. Thorne, D.Xie, S. Leigh,R.Preece, Reservoir facies modeling new advances in MPS, in GEOSTAT 2004 proceedings, Banff, Canada, October 2004, 7th International Geostatistics Congress.

7. S. Strebelle, Sequential simulation drawing structures from training images, PhD thesis, Stanford University, CA, 2000.

8. S. Strebelle, Conditional simulation of complex geological structures using multiple-point statistics, Mathematical Geology, January 2002.

9. S. Strebelle, K. Payrazyan and J. Caers, Modeling of a deepwater turbidite reservoir conditional to seismic data using multiple-point geostatistics, In SPE ATCE Proceedings, number SPE 77425, Society of Petroleum Engineers, October 2002.

10. L. Stright and J. Caers, Re-addressing the missing scale using edges, SCRF Annual Meeting Report 18, Stanford Center for Reservoir Forecasting, Stanford, CA, 2005.

11. Zhang, T., 2006. Multiple-point and pattern simulation using filter scores: the filtersim algorithm, PhD Thesis, Stanford University, California USA

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Fig 1: Geographical location of the study.

Fig 2: Cenomanian carbonated complex of “La Marcouline”.

Fig 3: Mound and lobes bodies of “La Marcouline”.

Fig 4: SGeMS reference model of “La Marcouline” - 3D section and facies classification.

Fig 5: Reference model facies histogram.

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Fig 6a: Sequence of thickness and facies codes maps for each depositional object - the top left figure is the facies at the bottom of the outcrop, the bottom right in Fig 6b represents the top most facies.

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Fig 7: The workflow – thickness maps and facies codes stacking.

Fig 8: The workflow – simpat simulation and back transformation.

Fig 9: SGeMS reference upscaled model – 3D section.

Fig 10: Reference upscaled facies histogram.

Fig 11: “3D thickness training image” obtained by stacking the 21 bodies’ thickness maps.

Fig 12: Training image target histogram – histogram of all the thickness values (left) and histogram of thickness values major than 0 (right).

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Fig 13: Simulation result using a template size of 5 x 5 x 3 and automatic number of multiple - grids.

Fig 14: Simulation result histogram – histogram of all the simulated thickness values (left) and histogram of the simulated values major than 0 (right).

Fig 15: Transformed training image (values equal to -50 are not shown) .

Fig 16: Transformed training image histogram – histogram of all the transformed values (left) and histogramof the transformed values major than -50 (right).

Fig 17: Simulation results using the transformed training image.

Fig 18: Simulation result histogram using the new transformed training image – histogram of all the simulated values (left) and histogram of the simulated values major than -50 (right).

Fig 20: Facies bodies generated using the original depositional sequence.

Fig 21: Back transformation result histogram – histogram of all the back transformed values (left) and histogram of the back transformed values major than 0 (right).

Fig 19: Back transformation of the simulation result of Fig 17.

Fig 22: Final model: 3D cross sections.

Fig 23: 3D “thickness training image” and 3D “facies code training image”.

Fig 24: Training images histograms – histogram of the thickness values (left) and histogram of the facies codes (right).

Fig 25: Simpat 2-banded simulation results – first simulation.

Fig 26: Simulation results histograms – histogram of the simulated thickness values (left) and histogram of the simulated facies codes (right).

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Fig 27: Simpat 2-banded simulation results – second simulation.

Fig 28: Simulation results histograms – histogram of the simulated thickness values (left) and histogram of the simulated facies codes (right).

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Fig 29: Reference model versus simulated models.

Fig 30: Reference model and simulated models histograms.

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Fig 32: 3D “thickness training image” randomly sampled in 1 well and 3D “facies codes training image”.

Fig 33: Training images target histograms – histogram of the thickness values (left) and histogram of the facies codes (right).

Fig 34: Simpat simulation results.

Fig 35: Simulation results histograms – histogram of the simulated thickness values (left) and histogram of the simulated facies codes (right).

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Fig 36: Well location on the realization.

Fig 37: Well location plotted onto slices of the realization.

Fig 38: Final model – 3D cross sections.

Fig 39: Final model histogram.

Fig 40: 3D “thickness training image” randomly sampled in 5 wells and 3D “facies codes training image”.

Fig 41: Simpat simulation results.

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Fig 42: Simulation results histograms – histogram of the simulated thickness values (left) and histogram of the simulated facies codes (right).

Fig 43: 5 wells locations on the realization.

Fig 44: Wells locations plotted onto slices of the realization.

Fig 45: Final model – 3D cross sections.

Fig 46: Final model histogram.

Fig 47: 3D “thickness training image” randomly sampled in 50 wells and 3D “facies codes training image”.

Fig 48: Simpat simulation results.

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Fig 49: Simulation results histograms – histogram of the simulated thickness values (left) and histogram of the simulated facies codes (right).

Fig 50: 50 wells locations on the realization.

Fig 51: Wells locations plotted onto slices of the realization.

Fig 52: Final model – 3D cross sections.

Fig 53: Final model histogram.

Fig 54: Fitness simulation maps histograms – 1 well, 5 wells and 50 wells conditioned simulations.

Fig 55: Fitness simulation maps – 1 well, 5 wells and 50 wells conditioned simulations.

Fig 56: Sgsim target histogram.

Fig 57: Sgsim multipliers – 5 realizations.

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Fig 58: Training image randomization – generation of 5 training images perturbing thickness values, 5 wellssampled from the original 3D “thickness training image”.

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Fig 60: Simulation results histograms – histogramm of the simulated thickness values (left column) and histogram of the simulated facies codes (right column).

Fig 61: Fitness simulation maps histograms – 5 different perturbed training images as input.

Fig 62: Fitness simulation maps – 5 different perturbed training images as input.

Fig 63: Final models – 3D cross sections, 5 different perturbed training images as input.

Fig 64: Reference model versus simulated model.

Fig 65: Reference model and simulated model histograms.

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