Omar Al-Hinai, Jing Ping, Mary F. Wheeler

Reservoir production management and optimization requires the characterization of the uncertainty in reservoir description. For fractured reservoirs, the connectivity of fracture distributions is crucial for predicting production characteristics. In this case, since the rock property fields are highly non-Gaussian, a method that combines vector-based level-set parameterization technique and ensemble Kalman filter (EnKF) or estimating fracture distributions is developed. The mimetic finite differences approach is utilized as forward model.

Mimetic finite differences approach

Modeling fluid flow through fracture networks is challenging due to the geometric characteristics of fractures. Subsurface fractures are thin inclusions with large surface areas that can intersect in random configurations. Using traditional hexahedral or tetrahedral mesh generation is not a tenable option, as it is difficult to maintain mesh quality and reasonable number elements. In addition, the context of uncertainty quantification adds further challenges. Correctly sampling realizations requires the ability to select uniformly from the space of possible configurations. This means that a very large space of possible fracture configurations is to be modeled without user intervention.

For these reasons, a different approach is needed. Our approach has been to circumvent traditional mesh generation by using methods that allow for general polyhedral elements. First, a baseline rectangular grid is generated over the computational domain. Then, using an in-house utility, simple polygon division operations incorporate the fracture into the mesh. The utility also allows for mesh quality enhancements through simple vertex shifting operations.  This produces a very fast and robust system for meshing complex fracture networks.

The meshes are then passed into Mimpy for forward modeling of the two-phase fluid system. Mimpy is a open source three dimensional implementation for solving two-phase flow equations through porous media. Mimpy uses the Mimetic Finite Difference method, which is a conservative technique similar to the Mixed Finite Element method and the Finite Volume method.

Parameterization using vector-based level-set method

For reservoirs with complex geology, a good estimation of the geological structures is very important for predicting and optimizing reservoir production. As the property fields of complex reservoirs are usually of bimodal or multimodal distributions, the Gaussian limitation is a major challenge for the application of the EnKF in the estimation of complex reservoirs [1, 2]. In order to improve the performance of the EnKF for highly non-Gaussian problems, we developed vector-based level-set parameterization method. The work is motivated by the method proposed in [3], in the sense that the level set function is applied as an indicator by which the facies fields can be transformed into smoother parameter fields. However, in the vector-based level set parameterization method, not only is the level set function used to describe the facies fields, a group of proper parameters combined with the level set function, which is called the parameter vector, is applied to depict complex shaped facies domains [4]. Compared to the original multimodal distributed parameters, the transformed parameters are in better agreement with the EnKF Gaussianity limitation, which can be updated using the standard EnKF. This approach is flexible. For different types of complex geology, different parameter vectors can be used to describe the features of the reservoirs appropriately.

Applications

We apply this method on a synthetic two-dimensional two-phase fractured reservoir. Figure 1 is the reference field with three main fractures. Figure 2 shows fracture realizations and water saturation profiles before and after updating. Figure 3 shows the matches of production data. After updating, the features of fracture distribution in reference field could be captured, and the matches of production data are more reliable.

In conclusion, the combination of mimetic finite differences approach, level-set parameterization and EnKF provides an effective solution to address the challenges in the history matching problem of highly non-Gaussian fractured reservoirs.

Figure 1. Reference field. Black lines denote fracture. Red points are producers. Green point denotes injector.

Figure 2. Fracture realizations and water saturation profiles (A) before and (B) after updating.

Figure 3. (A) oil production rate and (B) water cut from well 1 before and after updating. The red curves denote the responses computed with the reference field. The blue and black curves are the responses from fracture realizations before and after updating, and the green curves denote the average.

References:

[1] Aanonsen, S. I., Nævdal, G., Oliver, D. S., Reynolds, A. C., and Vallès, B., 2009. The ensemble Kalman filter in reservoir engineering–a Review. SPE Journal, 14 (3): 393–412. doi:10.2118/117274-PA.
[2] Oliver, D. S., & Chen, Y., 2011. Recent progress on reservoir history matching: a review. Computational Geosciences, 15(1), 185-221.
[3] Chang, H., Zhang, D., and Lu, Z., 2010. History matching of facies distribution with the EnKF and level set parameterization. Journal of Computational Physics 229 (20), 8011–8030.
[4] Ping, J., & Zhang, D., 2013. History matching of fracture distributions by ensemble Kalman filter combined with vector based level set parameterization. Journal of Petroleum Science and Engineering, 108, 288-303.
This research has been presented at several conferences and is currently being prepared for publication.