Parameterization in the History Matching Process

The reduction of the number of parameters from infinity (for a continuous system) to a finite number is called parameterization (e.g. Yeh 1986). The process presents the choice of the parameters and preference of modification representative for the spatial distribution of reservoir properties, such as net:gross, and permeability, in the model.

The success in history matching depends on suitable choice of parameterization and the range of parameter values. An inadequate set of parameters would result in inaccurate uncertainty estimations, and introduce bias errors. A large number of parameters with a wide range requires a large computational time, increases the variance errors and decrease the stability of the solution. Usually the reservoir parameterization stage is subjective. The type, number, and range of parameters are determined based on petro-physics, well test, and seismic and geological interpretations and experiences. Common parameterization approaches are demonstrated in the following paragraphs.

Individual Grid Cells: this technique considers the value of properties, such as permeability in every grid block to be independent parameters. The limitations of this approach are that it may not keep the knowledge of geological deposition. There are a large number of unknowns and results in a lack of spatial continuity in the reservoir model (Floris et al. 2001). A recent application of this method on a synthetic case was studied by Dadashpour et al. (2007) however this method is not suitable for real reservoir cases.

Zonation or Regions: this technique is a common practice in model construction and involves the assumption that reservoirs can be divided into several regions of uniform properties, distinguished by a Flow Zone Indicator (FZI) (see Figure 2.11). The benefits of using this method is to reduce the number of parameters (Gavalas et al. 1976, Makhlouf et al. 1993, Abacioglu et al. 2001, Huang et al. 2001), and it also incorporates in some degree the geological knowledge of the reservoir (Le Ravalec-Dupin et al.

2001, Aanonsen 2005). Appropriate zones account for layers, genetic or hydraulic units within layers, impermeable and permeable streaks, and drainage areas of wells. The drawback of the method is that it may not be sufficient for describing the actual heterogeneities of the properties, and may generate abrupt changes at the borders of regions, also some preconceived idea about the regions are not exact.

Figure 2.11: Zonation method principal.

Pilot Points Method: the pilot point method with kriging (de Marseily G. 1984) was developed to modify underground properties to enable continuous spatial variation in heterogeneous properties such as permeability between the prefixed locations or groups of locations called ‘pilot points’. The method consists of three stages. First, the initial value for the property at pilot points are obtained by prior geo-statistical realizations conditioned to a variogram and the observed fix point values. Then, the property at pilot points is perturbed by the inversion routine. The third stage is to propagate the perturbation induced by pilot points to the nearby grid cells in the reservoir model using a spatial interpolation scheme such as kriging.

In fact such approaches (Landa and Horne 1997, Bissell et al. 1997, Roggero 1997, Arenas et al. 2001, Wen et al. 2002) consist of calibrating an initial kriged reservoir

FZI₁ FZI₂ FZI₃

FZI₄

FZI₁ FZI₂ FZI₃

FZI₄

generated from the measurement of that property and create a set of synthetic conditional simulation realizations at selected unmeasured locations with pilot points at centre (see Figure 2.12), preserving statistical mean, standard deviation, and the spatial correlation of the properties. Generally, the pilot points are selected at locations with large uncertainties in properties, and the method solves most of the troubles encountered by zonation approach. The technique provides a practical tool to be incorporated in history matching (Ravalec-Dupin 2007).

Figure 2.12: Simulation of one a dimensional case when modifying the initial realization (thin curve) using two Pilot Points at locations 150 and 450. The perturbations induced by Pilot Points are local. The modified regions for different realizations (thin curves), cantered at Pilot Point locations, have radius equal to the correlation length (Ravalec-Dupin and Hu 2007).

Object Based Parameterization: in this approach the permeability or net:gross at each simulation grid is a function of a set of implied parameters related to geological features of the reservoir, referring to each as an object, such as channels, channel margins, facies and etc. The purpose in object modeling is to preserve the large scale geological entities to reduce the dimension of the problem. For example a channelized reservoir can be parameterized with only a few parameters per channel (such as d₀, d₁ and d₂ in Figure 2.13), which are smaller than the number of grid cells that define the channel size, spacing and shape. An example of this approach is the Boolean methods (Haldorsen and Damsleth 1988, Deutsch and Wang 1996, Lantuéjoul 1997). The advantage of this

method is that the shape of objects resembles actual channels. The drawback of the method is that objects are notoriously difficult to condition simply since there are many parameters and it requires a realistic geological and statistical interpretation of the characteristics of the size and shape distribution of the objects (Vargas-Guzmán and Al-Qassab 2006).

Figure 2.13: Principal of object base modeling. Channelized reservoir can be parameterized with few independent parameters, such as d0, d1 and d2 defining channel sizes, spacing and shape.

Structural Faults and Barriers or Baffles: the existence of barriers significantly influence the depletion performance of reservoirs by inducing flow and pressure discontinuities, particularly in compartmentalized and channelized reservoirs (Yielding et al. 1999a). The flow barriers can be horizontal, such as shales, impermeable streaks, and vertical, such as faults or sub-vertical such as shale drapes. The barrier’s distribution, location, thickness and transmissibility are hard to identify correctly from well test and log data. Although, barriers represent the borders of channels or faults may be identified in 3D/4D seismic maps. Properties of barriers are a source of uncertainty for reserve estimation (Lia et al. 1997) and are often picked as the parameters for calibrating reservoir models in history matching (kruijsdijk 2001, Stephen 2006, and Edris 2009).

Gradual Deformation: this method developed by Hu et al. (2000) and makes possible the gradual global transformation of the initial reservoir geo-statistical realization with a set of parameters which act as weights (Figure 2.14). For any value of this parameter the process provides a new property distribution. The simplest gradual deformation scheme consists of combining two independent Gaussian random functions Y₁ and Y₂ with mean y_o and identical covariance:

) t sin(

] y Y [ ) t cos(

] y Y [ ] y ) t ( Y

[ _{o 1 0 2 0} (2.9)

where t is a deformation parameter between [0,π/2], cos(t) and sin(t) are the combination coefficients making the method depend on only one parameter, t. Such combinations provide continuous chains of realizations ensuring that Y(t) is also a random Gaussian function with the same mean and the same covariance as Y1 and Y2. Instead of using two, it is possible combining several independent realizations which provides more flexibility for deforming realizations in history matching (Roggero and Hu 1998). Then the number of deformation parameters equals the number of complementary realizations added to the starting one.

Figure 2.14: An example of a continuous train of realizations for a 2-dimensional continuous Gaussian Random Function by gradually deformation, here two realizations at top right and bottom left are combined (after Roggero et al. 2005).

The advantages of gradual deformation are that the key statistic (i.e. variogram and variance) identify the target statistics and the scheme allows for reduction of the number of the parameters and thus it is computationally efficient. Its drawback is that it is mostly efficient in multi-Gaussian random fields (Le Ravalec-Dupin 2005).

The Probability Perturbation Method: this method shares some of the ideas behind gradual deformation (Strebelle 2000, Caers 2003 and 2004) and works with Gaussian fields and more geologically complex reservoirs where multiple-point statistics are required to describe the geology sufficiently. The method exploits the structure of sequential simulation algorithms which make use of local conditional probability functions. Where gradual deformation perturbs properties at the grid block scale directly, this technique perturbs probabilities at the grid block scale. Then a sequential simulation results in a perturbation of the desired field properties. The method does not rely on any assumptions and is suitable therefore, for any kind of geology that can be realized by sequential simulation. Examples of application of this method can be found on reservoir flow model calibration studies by Hoffman and Cares (2003), Berrera and Srinivasan (2009), and Li and Reynolds (2009).