In recent years, continuous **improvements in computing and software technologies** have considerably enhanced several aspects of reservoir simulation techniques. Simulation runtimes have sharply decreased due to advancements in hardware and processing power. In spite of this, the level of detail built into a geological model continues to exceed the current computational capabilities.

Typical **reservoir simulators** can operate in the order of **10 ^{5}-10^{6 }**simulation cells. The exact number will vary depending on the type of simulation and the available computer hardware. On the other hand, geological models may contain cells in the order of

**10**[2]. Therefore, reservoir simulators are currently not capable of handling such geological details and this limitation makes it necessary to forego some of these details and recreate a

^{7}-10^{8}**coarser version**of the model at a larger scale.

As a way to circumvent this limitation, **upscaling techniques** were developed to allow for flow simulations by upscaling the detailed description of a reservoir model into a reservoir grid. Most simulators use coarse-grid cells with lack of spatial resolution to compute reservoir heterogeneities. The equations used to solve flow throughout the grid are expressed in terms of partial differential equations, written in a finite-difference form, and derived from fundamental principles such as Thermodynamics and Darcy’s Law. This was briefly discussed in the first post.

According to Wang [6], upscaling consists of **integrating all properties** defined in a fine-grid system to equivalent properties defined in a coarse-grid, such that the two systems will act similarly. As exact upscaling of a heterogeneous reservoir is not possible, the technique aims to convert the properties with minimal loss of information by using suitable procedures. Among the main goals of this challenging process, the following are worth mentioning:

: it is crucial to maintain important fine-scale flow characteristics from the full model, which include flow rates at given pressures and breakthrough times for fluids injection.*Accuracy*

: it refers to the coarsening degree achieved by an algorithm. Also, upscaling techniques yield intrinsic costs; thus, efficiency must be considered while ensuring that accuracy is not compromised.*Efficiency*

: primarily refers to the level of processing independence. The main goal is to apply the same coarse properties at different flow conditions; for example when considering the sensitivity degrees of an upscaled model.*Robustness*

Durlofsky [2] argues that different upscaling procedures are used for different situations. An ideal approach depends upon: **I)** the simulation question being addressed, **II)** the production mechanism, and **III)** the level of details to be upscaled into a coarser-grid.

Additionally, several factors in reservoir characterization present varying degrees of uncertainty, which raises many concerns regarding **risk assessment**. These uncertainties can be assessed by simulating various geological realizations and scenarios. On the other hand, it is rarely feasible to run thousands of simulations using a fine-scale grid. Thus, upscaled models are essential to fully address risk and uncertainty evaluation.

Most of the upscaling techniques proposed in the literature are used for single-phase flow. This is a **fundamental step** towards solving multiphase flow problems. An equivalent flow in different scales is usually based on the following assumptions: steady-state, uniform, and single-phase flow. Hence, it may overlook several important cases such as radial and transient flow. Durlofsky *et. al.* [3] discuss that in the near-well region, many of the upscaling methods do not provide satisfactory results. As the standard upscaling methods generally assume steady-state flow, that is clearly not true around the wellbore.

In terms of how upscaled parameters are computed, the **upscaling procedures** are classified as *local* or *global*. In a local procedure, coarse scale parameters are computed by only considering a predetermined fine scale region. On the other hand, a global approach considers the entire fine scale model to calculate the coarse scale parameters. [2]

The **change of scales** presents the challenge of how to compute permeabilities for a coarser-scale based on the geological model. Differently from other geological parameters, permeability is not an additive variable [4], *i.e*, it is not an intrinsic property of the heterogeneous medium. Instead, it depends on the boundary conditions and the distribution of heterogeneities, which in turn depend on the volume being considered [1].

The ** equivalent permeability** corresponds to a constant tensor which represents a heterogeneous medium. There are several approaches available in the literature to determine the equivalent permeability, which include heuristic, deterministic and stochastic methods. The major methods in upscaling include power-law averaging methods, arithmetic-, geometric- and harmonic mean techniques; renormalization techniques; pressure-solver method.

The application of** mean-value** is the most obvious choice in upscaling. Although these techniques suffer from some applicability limitations, they are very fast and efficient to perform and are shown mathematically as follows

**Arithmetic Mean**:

**Geometric Mean**:

**Harmonic Mean**:

**Power Law**:

The Power Law allows us to **determine the effective permeability for different values** of ω ranging between -1 and 1. For values of ω = 1, 0, −1, the equation yields arithmetic, geometric and harmonic averages, respectively. In general, an arithmetic average provides an upper bound of “k*”, and the harmonic average provides a lower bound. [5]

It may be clear by now that upscaling techniques are sophisticated methods which require tools capable of representing the necessary finer details of the original model to maintain a proper level of accuracy and efficiency. One such tool is Kraken, which allows the user to select the property to be upscaled and the desired method, by choosing a number of grid blocks in each direction (x, y, z-axes). The figure below exemplifies an upscaled model generated by Kraken:

The grid shown to the right of the figure is the upscaled model of the original reservoir grid (left). It is apparent that this upscaled model has considerable fewer cells. The **reduction** in the number of cells considers an upscaling factor and the number of active cells. The next example is a comparison of the same region in both original and upscaled grids. The upscaled property used is porosity and the method used is the geometric mean. The featured black square is the upscaled cell from the original grid and comprises other four smaller cells. The value of porosity in those cells are 0.2847, 0.2847, 0.2878 and 0.2877, respectively.

The following picture highlights the same upscaled cell with the equivalent value for porosity. It is noticeable that the number “0.2862” equals to the geometric mean of the porosity values mentioned above.

Considerable and special attention has been given to **upscaling research** because of the need for coarsening highly detailed models while maintaining the fine-scale effects, computational efficiency and optimizing the power of the current hardware. Additionally, it is imperative to have a good understanding of the errors introduced during upscaling. Literature has shown that there is not a single optimal upscaling technique suitable for all fine-scale models. Therefore, for each model, a sensitivity study with different methods should be conducted to determine the least error introduced into the process and consequently, the most appropriate method for that case.

**REFERENCES**

- Begg, S. H., Carter, R. R. and Dranfield, P. “Assigning effective values to simulator gridblock parameters for heterogeneous reservoir”.
*SPE Reservoir Engineering*. November, 1989. - Durlofsky, L. J., “Upscaling and Gridding of Fine Scale Geological Models for Flow Simulation”. Department of Petroleum Engineering, Stanford University. 8th International Forum of Reservoir Simulation Iles Borromees, Stresa, Italy, 2004.
- Durlofsky, L.J., Milliken, W.J. and Bernath A., “Scale up in the Near-Well Region”, SPE 51940, Proceedings of the 15th SPE Symposium on Reservoir Simulation, Houston, February 14-17, 1999.
- Renard, P., & De Marsily, G. (1997). Calculating equivalent permeability: a review.
*Advances in Water Resources, 20*(5-6), 253-278. - Villa, J.R., Salazar, M.O. “Permeability Upscaling Techniques for Reservoir Simulation”. SPE 106679, SPE Latin America and Caribbean Petroleum Engineering Conference. Buenos Aires, April 2007.
- Wang, K., “Upscaling and Parallel Reservoir Simulation”. The University of Texas at Austin, 2011.