A novel flow-based geometrical upscaling method to represent three-dimensional complex sub-seismic fault zone structures into a dynamic reservoir model

Most hydrocarbon reservoirs contain faults, which are highly complex heterogeneous and anisotropic three-dimensional (3D) volumes of deformed rock. A major technical challenge in full-field flow simulation is to represent the effects of 3D fault zone structure within the two-dimensional (2D) fault planar surfaces using the industry standard commercial reservoir flow simulator due to its limited functionality. Therefore, a new flow-based geometrical upscaling (FBGU) method has been developed for capturing the effects of 3D fault zone structures in conventional low-resolution upscaled flow simulation models. Geometrical upscaling (GU) is the process of calculating across-fault and up-fault transmissibility arising from 3D flow paths through fault zones, and expressing these transmissibilities as implicit connections in a low-resolution upscaled flow simulation model. The high accuracy of the method is demonstrated by comparing the flow responses of high-resolution (referred as truth model in this paper) simulation models in which the 3D fault zone structure is represented explicitly in the grid geometry, with that of conventional resolution models in which it is upscaled using FBGU method. The flow results show that the newly developed FBGU method is extremely accurate and geometrically flexible.


Supplementary Note 1 Background of GU to modelling fault zone structures
Discrete sub-seismic fault zone components (e.g. breached and unbreached relay zones) may be present at different locations on a fault, but are difficult to incorporate into flow simulation models explicitly due to their size (small compared to upscaled model's grid block size), resolution (below the seismic data), and very complex geometry. GU solves this issue by representing complex sub-seismic fault zone structure implicitly into the flow simulation model as a function of connection transmissibilities that cannot be visualised at the resolution of the flow simulation model. Therefore, GU is referred as a connectivity-based algorithm that aims to upscale structurally complex 3D sub-seismic fault zone components to the resolution of a upscaled flow simulation model 18,28, 44 . The objective of the GU method is to estimate transmissibilities of different up-fault and across-fault flow paths present through a fault zone, and to include them as neighbour and non-neighbour connection tramsmissibilities in the lowresolution full-field flow simulation model in which the fault is represented as a 2D planar surface. In the GU approach, the connection transmissibilities are calculated from the centre of one cell to the centre of another cell associated with complex 3D fault zone structure (Supplementary Fig. S1a-b) and representing them into the upscaled flow model ( Supplementary   Fig. S1c). The GU method first devised by Manzocchi et al. 18 is a TBGU approach and implemented in TransGen software 30 , has rarely been applied in real reservoir studies documented in the public domain (an exception being 45 ). Geometrically, the method has constraints in terms of length and complexity of the fault zone structures that can be considered, 1 and the petrophysical properties within the zones. Specially, this TBGU method is applicable only for fault zone structure contained within two-cell stacks ( Supplementary Fig. S1), where the fault zone is smaller than a single grid cell and can't deal with anisotropic petrophysical properties and two-phase fault rock properties. Therefore, the principal objective of this study is to devise a novel FBGU approach to expand the flexibility of the existing TBGU method in terms of geometries, scales and properties that geometrical upscaling is capable of dealing with.
Moreover, the upscaling algorithm devised by Manzocchi et al. 18 contains numerous assumptions and simplifications, and has never been tested rigorously. In the existing TBGU approach, the connection transmissibilities are estimated geometrically 18,28,30 , but in the newly developed ignored. This means paths that pass vertically between layers in the ramp are ignored. This decision was, presumably, made to simplify the method, but has not been validated or justified.
All these classes of limitations are addressed using the newly developed FBGU method that is Overall, however, the existing TBGU method shows that the whole technique relies on defining a template for each type of fault zone component; it cannot be flexible for larger and more complex fault zones. For example, the template devised for two cell stacks is inappropriate for a relay ranging over four or more cell stacks, which would need a completely new template.
Therefore, an entirely different approach that does not rely on templates is desirable. A new FBGU method is devised to make the GU more flexible and is presented in this study.

Supplementary Note 3 Summary of the models for testing the method
Three different test models called reservoir A, B, and C are used in this study to test the FBGU algorithm ( Supplementary Fig. S3). The dimension, resolution and geometry of the models are shown in Table 1. Reservoir model A, with seven breached relay ramps ( Supplementary Fig.   S3a), and model C with eight breached relay ramps ( Supplementary Fig. S3c) both have total throws of 110m which is larger than the total 90m model thickness, therefore, there would be no communications between the injector and producer wells without these fault zone components.
Model B, by contrast, contains six breached relay ramps (Supplementary Fig. S3b) and has a total throw of 80m, which is smaller than the total 90m model thickness and therefore acrossfault fluid flow would be possible even in the absence of the relay zones. Supplementary Fig. S4 shows the upscaled models for which the FBGU has been performed (described in Method Section) to include the effects of the relay zones. The two injection wells and the two producing wells are all perforated throughout the formations and therefore the production is co-mingled from all layers.
Water and oil relative permeability curves used in this study are calculated using the equations of Christie and Blunt 46 , and are shown in Supplementary Fig. S5.
The dead oil PVT properties are reported in Supplementary Table S3.
The top of three test models is 1000m deep. Therefore, the reference depth is 1100m for reservoir A and C, 1085m for reservoir B and the initial pressure at reference depth is considered of 1000 bar. Surface oil, and water densities are 876 kg/m 3 , and 1000 kg/m 3 respectively and the rock compressibility is 10 -6 bar -1 . The well bore diameter is considered as 0.25m in this study.
The injection wells are constrained by a water injection rate of 1000 rm 3 /day, while the producing wells are controlled by fixed well bottom hole pressure (BHP) of 300 bar for each reservoir test cases. All the simulations are run for 50 years of production.

Supplementary Note 4 Inclusion of connection transmissibilities into a upscaled model
In this study, the Eclipse flow simulator 32 is used in the upscaling process. Therefore, the Eclipse simulator requires different keywords for representing neighbour or non-neighbour connection transmissibility, and the procedure for including them into flow simulator is discussed here. In a corner point grid (CPG) model, a connection between two cells is termed a neighbour connection if a single digit in all co-ordinates separates the two cells. Hence, the cell

Supplementary Tables
Supplementary Table S1. Distribution of petrophysical properties of three geological models.