Résumé
Ce chapitre est consacré à une application des méthodes de simulation avec le modèle Gaussien discret présenté dans cette thèse appliquée au cas d’étude réel – champ X – un gisement de gaz offshore avec un maillage tartan. La méthode de simulation avec DGM est comparée à la méthode de simulation classique, qui ignore l’effet du volume dans la simulation. Les résultats de comparaison des méthodes développées dans cette thèse par rapport aux méthodes classiques sont analysés.
Field X is an offshore gas reservoir. The reservoir model covers a large subsurface region with dimensions 140𝑘𝑚× 200𝑘𝑚× 350𝑚. Due to the enormous area covered by the model, it uses irregular meshing with grid refinement in the most important areas (see Figure 7-1). The refinement step in the horizontal direction varies from 250𝑚 to 4𝑘𝑚.The type of meshing used for field X is referred to as tartan meshing. The resulting model is not unstructured, since one can introduce an IJK coordinate system on it and will be able to use the IJK coordinates of the blocks in order to retrieve their neighbors. The dimensions of the grid in the IJK coordinate system are 132 × 220 × 31. The grid remains irregular since it is composed of blocks of varying size.
112 Figure 7-1. Field X reservoir model.
The goal of this case study is demonstrating the difference between the classical approach to geostatistical simulations and the proposed DGM-based approach which considers the change of support effect. For both approaches quasi-point support distribution and covariance function are used as initial inputs, the simulations are unconditional. The modeled petrophysical parameter is porosity with beta distribution on [0.01,0.28] with parameters 𝛼= 2.15 𝛽= 2.1 (see Figure 7-2) and a spherical covariance function 𝜌 in the Gaussian scale with ranges 3𝑘𝑚× 3𝑘𝑚× 30𝑚.
Figure 7-2. Average block porosity distribution.
The classical approach for the tartan grids consists in performing the simulation in the IJK space and transferring the results to the original space. The IJK formalism is useful since it simplifies the algorithmic part of simulation (neighborhood queries), enables applying simulation methods for regular meshes (i.e. spectral simulation) and the most important – ensures the continuity of the simulated parameter along the geological layers of the model. In order to make the classical simulation the point support inputs are upscaled to the “average support” of the model and the upscaled inputs are used. For the field X the block of the average size has dimensions 1𝑘𝑚× 900𝑚× 12𝑚. The input beta distribution was upscaled with DGM 2 in order to obtain the distribution of the average block (Figure 7-2). Figure 7-2 illustrates also the average porosity distribution of the smallest and the biggest blocks in the model for given inputs. The drawback of the classical approach is that it ignores the volumetric differences between the blocks which can lead to incorrect filling of the reservoir model.
113 In order to challenge the classical approach, we apply a DGM 2 simulation algorithm backed with Gibbs propagation algorithm for simulating the multivariate Gaussian random vector. The simulation algorithm cannot be applied directly to the original grid from Figure 7-1 since in that case the continuity of the simulated property along the geological layers would not be respected. In order to respect the continuity along the geological layers, the simulation should be performed in the depositional space which corresponds to the input geometry. The depositional space was derived as proposed in Mallet (2014). The reservoir model in the depositional space for field X is depicted on Figure 7-3. For this test we consider that the input point-support covariance function 𝜌 is the covariance function of porosity in the depositional space.
Figure 7-3. Depositional space reservoir model for field X.
The reservoir model in the depositional space preserves the mesh structure of the original model, which enables treating the volume support effect in the simulation.
In addition to providing the continuity of the simulated property along the geological layers of the model, an argument for the idea of simulating in depositional space is that it is the space in which the hypothesis of the stationarity of the simulated physical parameter seems to be less invalid. Indeed, some similarity in the spatial structure of the studied physical parameter in two different regions of the reservoir is more likely to be expected if the depositional process took place in the same moment of time in the past.
114 The simulation results in the IJK space with classical approach (SGS with 50 neighbors) and in the depositional space with DGM 2 are demonstrated on Figure 7-4.
a) b)
Figure 7-4. Simulation result (top view) a) in IJK space simulated with SGS b) in depositional space simulated with DGM 2.
The mesh on Figure 7-4 is not appropriate since it obstructs the image analysis. As expected, for the simulation in the IJK space the volume support information was lost and the same character of variations is observed everywhere in the model. For the simulation in the depositional space the refinement area demonstrates high variations of simulated porosity with an abundance of small porosity values (black) and large values (red). When approaching the corners of the model with the coarse blocks, the amount of extreme variations of porosity diminishes and less variation is observed. After simulation in the IJK and depositional spaces, the results are transferred to the original grid (Figure 7-5).
115 a)
b)
116 Figure 7-5. Simulation results. a) Classical approach b) DGM 2 approach. Vertical
zoom factor 30.
Figure 7-6. Vertical cross-section through the center of the model for the DGM 2 simulation. Vertical zoom factor 50.
It is visible on Figure 7-5a that the result provided by the classical approach is strongly impacted by the mesh structure and the desired continuity of the porosity field is not reproduced. Transferring the results from the IJK to the original space in this case can be considered as “stretching” the cells of the IJK model so that they get the desired shape. The effect of this “stretching” is visible on Figure 7-5a in the refinement zones. Although a stationary covariance function 𝜌 was used for simulation, the resulting image does not demonstrate the patterns typical for stationary covariance models. On the other hand the DGM 2 simulation provides the desired model behavior. Transferring the simulation result from the depositional space to the original grid does not cause any artifacts. The refinement zone of reservoir demonstrates high variations of porosity whereas the coarse block regions look smooth. Figure 7-6 demonstrates a cross-section of the DGM 2 simulated model from Figure 7-5b. It is visible that the simulated porosity demonstrates continuity along the geological layers of the model. The example of case study X demonstrate that ignoring the volume support effect can lead to incorrect result for a geostatistical simulation on a reservoir model.
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