Cellular Automata Applied to Static Reservoir Modeling in Carbonate Reservoirs

Claude-Alain Hasler
Carbonate Research, Shell Global Solutions, Kessler Park 1, 2288 GS Rijswijk, The Netherlands

Over the past decades, reservoir models have constantly evolved. Their resolution and complexity have grown in parallel with computing power and the number of available modeling tools, but also, and not least, in response to the need for robust models as a result of the rising number of mature hydrocarbon fields. The reservoirs hosted in carbonate platform deposits are an excellent example of this evolution. The “layer cake” type models which were systematically used, with success, during the past decades are no longer appropriate today, when IOR/EOR techniques are becoming the norm in numerous fields in the Middle East.

Cellular automata (CA) present important advantages as static modeling tools for reservoirs, particularly carbonate ones. The CA method is based on a regular grid of cells (lattice) for which each cell is in one of a finite number of states at a specified time. The dimension of the grid cells is constant but can be any finite number. Most CA are scale independent, meaning that the underlying grid can be defined at the scale of the relevant heterogeneities within a subsurface reservoir. CA can be implemented into subsurface modeling packages and use directly their grids as CA’s lattice. During a discrete time step, every cell in the CA’s grid is updated according to a fixed set of postulated rules that directly depend on the states of a finite number of neighboring cells. Every process which can be discretized in space could theoretically be modeled by a cellular automaton. CA models can be built combining rules incorporating for instance both depositional and early diagenetic processes.

This abstract presents 3 applications of CA for 3D outcrop or subsurface modeling based on 3 main approaches: a forward model where sedimentary bodies are built through time taking into account both depositional and early diagentic processes, a stochastic cellular automaton based on Markov transition matrixes, and a geometric approach used to model sinkholes in karst setting.

Many carbonate fields in the Middle East comprise carbonate shoals. While it has been demonstrated that individual sand bodies are correlatable over large distances (more than 10 km) and can have uniform stratigraphic thicknesses, their external landscape geometries are diverse, and internal sedimentologic and diagenetic partitioning of texture, fabric and pore types intricate. The interwell facies modeling method has been developed based on CA, aiming to model processes associated with shoal deposition. The model is designed to reproduce facies distribution at decimeter-scale vertically and at decameter-scale horizontally. This approach is process-oriented in order to take into account the combined results of the sedimentology and early diagenesis. The present day shoal environments of the Bahamas have been used to understand and establish the CA rules corresponding to these depositional and early diagenesis processes. The final outcome of the CA is a 3D grid, reproducing the geological stacking recorded through time in such environments for a given relative sea level setting. Outcrop data from the Khuff Formation in the Jabal al Akhdar (Oman), where excellent outcrop continuity enables the observation and measurements of the vertical and lateral facies distribution, have been used to validate the model’s prediction.

Traditional variogram-based methods used to populate rock properties in static reservoir models are not able to capture very complex geometries. One way to improve these statistical approaches and produce models that are more geologically intuitive is to use a Markov chain approach. The intrinsic characteristics of a Markov chain (discrete state-space environment, memoryless probability which depends only on the current state) suit particularly well the CA’s structure. Markov chains have been used far and wide in sedimentology to characterize vertical variability. Coupled with Walther’s Law, Markov chain theory allows working in 2D or 3D using transition probability matrices. Transition probability matrices are here transformed into CA statistical rules. The outcome of the CA is a 3D distribution of facies conditioned to vertical and/or horizontal well data.

CA are used here to reproduce trends and geometries due to fault or surface related processes. Early and burial diagenesis are the main processes modifying carbonate rock matrix properties. Fluids that are involved in these diagenetic processes are often driven by faults or fracture networks and along lithological contacts. Petrographic studies on cores or field analogues allow geologists to assess the intensity of peculiar processes as one moves away from the fluid feeder surfaces. These lateral evolutions can be expressed as trends, which are then transformed into CA rules. In this case, rules are based on simple arithmetic averages and are therefore purely deterministic. Such rules have been used, for instance, to reproduce sinkholes geobodies in an unconventional reservoir. The occurrence of the sinkholes is related to a stratigraphic unconformity which corresponds to the surface feeder. The location of sinkholes is mapped as polygons on the unconformity based on the seismic data. The cells of the lattice included in the polygons are used to condition the model. The result of the CA is a 3D reconstruction of the sinkholes, starting from the surface unconformity and taking into account the lithologies found below it.

During the last decade, CA have been used in various scientific disciplines but their application in static reservoir modeling remains anecdotic. The main message of the present paper is to underline the great flexibility that CA may offer to modelers. Using deterministic rules, CA are able to help geologists create their conceptual models in 3D. Using stochastic rules, CA may be great alternatives to classic variogram-based methods. Combining rules is also an elegant way to model multi-process derived systems such as carbonates rocks. Finally, conditioning on existing hard data is also straightforward using CA, since the same rules are applied to every cell, which in turn is a function of its neighbors, and subsurface data can be set as fixed neighbors directing the distribution of modeled geobodies.

AAPG Search and Discovery Article #120034©2012 AAPG Hedberg Conference Fundamental Controls on Flow in Carbonates, Saint-Cyr Sur Mer, Provence, France, July 8-13, 2012