Published by the DTI Oil & Gas Directorate for the reservoir engineering and IOR community in the UK.
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Reservoir Simulation of IOR Techniques Using SURE

Leonhard J. Ganzer (lganzer@hoteng.com) of HOT Engineering/Veritas DGC, Leoben, Austria presents some of the IOR orientated features of the SURE simulator, particularly its ability to use different PVT formulations in different parts of the model and at different times during the course of a simulation (co author Andras Kiraly). (http://www.veritasdgc.com/what/reservoir/index2.htm)

The term multi- or general purpose to describe the numerical formulation of the flow equations appeared in the literature in the eighties. Many authors contributed to this idea. Attempts to develop these simulators based on an extended black-oil formulation were very soon replaced by models using a general compositional formulation.

Today, general-purpose reservoir simulators offer various numerical models adapted to simulate different flow processes. All of them are based on a compositional formulation. The main differences between a black-oil run and a true compositional run are the PVT treatment, the limited solubility and restricted number of components allowed in the black-oil formulation.

The compositional formulation is used in the simulator described in this article and provides the underlying basis for a variety of simulation models. Several models were introduced until the early nineties to extend the capabilities of the simulator. Due to developments in the gridding techniques (PEBI grid, windowing technique) during this time period, the initial concept of the simulator was refined and generalised.

The simulator is based on a general compositional formulation. In theory all isothermal and non-isothermal processes can be derived from this general formulation limiting the number of components and phases, defining equations of state and a set of property functions. As a consequence, all specific simulation models are simplifications of the general model and have an inherent compatibility.

This compatibility can be exploited to combine the different numerical formulations and allow their simultaneous existence in adjacent grid blocks. The resulting simulation model is called a mixed model. The engineer can use the numerical formulation of choice as a function of time and space to model the selected IOR method in the simulation study.

In this PEBI-gridding implementation, the grid is composed of several objects, namely the aquifer grid, the productive area grid and the window grids (Figure 1). The simulator is capable of modelling each grid object with the assigned numerical formulation. In addition, it is possible to convert those individual grid objects to specific numerical formulations during the simulation run.

Figure 1: PEBI-grid simulation model (click image for larger view)

All models within the simulator have a common basic formulation. It is obviously no problem to combine a single-phase, single component water model (e.g.: applied to the grid blocks in the aquifer) with a three-phase, three-component black-oil model. Problems arise, when the same mobile fluid phase is described using different components in communicating grid blocks - see Figure 2 below. For example, the oil phase is described with a different number of components in a black-oil model compared to a compositional model. (Even, if the number of the component were the same, their properties would be different!)

Figure 2: Idealised mixed model situation (click image for larger view)

The mixed model option was successfully applied in a number of field cases, where the history match period was performed in a black-oil formulation. For the predictive mode of the simulation model we chose a compositional formulation for designated parts of the model or for the entire field. The compositional PVT data that must be entered should be as close as possible to the true reservoir fluid behaviour at the time of the compositional conversion. The simulator iterates on the entered composition to match the black-oil saturation pressure in each grid block.

Application and Results
A test example was setup in order to evaluate the performance of different model formulations and mixed models. The grid is constructed following the PEBI method and uses features typically found in field-scale simulation. For example, grid refinement is used, vertical faults are modelled and windows are constructed. The simulation model is divided into several parts: the aquifer grid, the productive area grid comprising the area of hydrocarbon accumulations and a window area located within the refined zone of the productive area.

The simulation run was set up to simulate seven years of production from primary energy, followed by CO₂ injection within the window region at 15000 m3/day. This should repressurise a region that is partially enclosed by vertical faults. The total simulation time is ten years.

The solution method applied for all runs performed is an iterative orthomin accelerated linear solver using ILU decomposition as a preconditioner. In addition, the adaptive implicitness option is applied in all runs. Newton iterations are used to polish up the root found by the linear equation solver. To allow a simpler performance comparison, the timestep length is limited to 5 days. This is assumed to be a reasonable value for the compositional model, even though the timestep length could be considerable larger for the black-oil model runs. Several runs were performed using different model setup:

• Run1: Black-oil model. Instead of the CO₂ injection, gas will be injected. This run serves as a base run.
• Run2: Black-oil model combined with hydro model in aquifer region. The aquifer blocks are converted to single-phase water blocks saving some storage and calculation time
• Run3: Compositional model. In order to simulate the locally restricted CO₂ injection, the whole model was set-up in compositional using 11 components and the history match period was recalculated.
• Run4: Compositional model combined with hydro model in aquifer region
• Run5: Mixed model run. Combination of single-phase water aquifer model, black-oil productive area and only the window are simulated in compositional formulation. The window is converted at the beginning.
• Run6: Mixed model run. This setup is identical to Run5, but the window is converted at the time, when the CO₂ injection is started.

Figure 3 displays the number of gridblocks simulated in different numerical models for all runs. The block formulation distribution of Run6 is calculated on a timestep-weighted basis of blocks calculated in compositional or black-oil formulation. These six models were run on an IBM^â workstation (IBM RS6000) under AIX^â operating system.

Figure 3: Comparison of model formulation for different simulation runs (click image for larger view)

The CPU times and solver performance of the simulation models are summarised in Table 1. In the first column, the CPU times measured for completion of the run (742 timesteps) are reported. When normalising the pure black-oil model (Run1) to a speed factor of unity, the compositional runs take about 17 times the CPU time of Run1. The mixed model run is about five times slower than Run1, if the model conversion is done initially. If the compositional conversion is at the time, when CO₂ injection starts, the CPU-time necessary to complete the simulation run approximately doubles compared to Run1, but treating 23% of the gridblocks in compositional mode.

The second column shows the number of outer solver iterations (Newton-Raphson iterations) that were necessary to polish the solution to reach iteration limits. The average implicity is a measure of the number of implicit unknowns per block on average over the whole simulation run. The linear solver strongly depends on the number of Newton iterations, therefore columns 5 and 6, the linear solver iterations per timestep and per solution are more descriptive for the difficulty of the coefficient matrix. The number of solutions is the sum of total timesteps plus the number of Newton iterations performed in the run.

Table 1: Performance comparison for the different runs

The most impressive result from above is the speed factor obtained for the mixed model runs (Run5, Run6). Whereas the fully compositional runs take about 17 times the CPU time of the homogenous black-oil run, the mixed model runs could be completed in 1/8 of the time with the same qualitative results.

Conclusions
SURE allows the dynamic conversion of designated parts of the simulation grid to different model areas. These model areas - no matter in which formulation - are fully integrated and compatible with all other options available within the simulator.

The fluxes across the model boundaries are inspected and the simulator reports errors, when underlying principles are violated.

The program overhead introduced is negligible for conventional simulation studies in terms of computing time, but CPU savings can be enormous for mixed model runs.

Together with the gridding technology available today, the mixed models will yield a most valuable tool to the engineer. However, even with this degree of sophistication, it cannot replace good engineering judgement during the course of a simulation study.

References

1. ‘A Novel Approach for Multi-Purpose Reservoir Simulators Using Mixed Models,’ L.J. Ganzer. Paper presented at the 6th European Conference on the Mathematics of Oil Recovery, Peebles, Scotland, September 8-11, 1998.
2. ‘Reservoir Simulation Using Mixed Models,’ L.J. Ganzer and Z.E. Heinemann, SPE 37981, Paper presented at the 1997 SPE Reservoir Simulation Symposium held in Dallas, Texas, 8-11 June 1997.

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