lab-inf-4_tasks / 4N

Copyright 2001, Society of Petroleum Engineers Inc.

This paper was prepared for presentation at the SPE Reservoir Simulation Symposium held in

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Abstract

Computer models of oil reservoirs have become increasingly more complex in order to represent geological reality and its impact on fluid flow. Memory and CPU time limitations by finite difference/volume simulators force a coarser resolution of reservoirs models through upscaling.

Upscaling can lead to significant difficulties in reservoirs studies: (1) while the fine-scale geological model is build from petrophysical, log, and seismic data, its dynamic behaviour is never checked. As a result, a coarse-scale reservoir study can be linked to a fine-scale geological model but the two might be inconsistent in their dynamic behaviour. (2) Conversely, the upscaled model cannot be properly tested since the flow and production behaviour at the fine-scale level is not available. There is no reference solution for guiding important decisions for building a consistent upscaled model. (3) A large number of sector models are required in designing optimal well patterns.

Streamlines simulation is now an attractive alternative to overcome some of these drawbacks since it offers substantial computational efficiency while minimising numerical diffusion and grid orientation effects. It allows the integration of fine-scale geological models into the reservoir engineering workflow.

In this paper, we demonstrate the usefulness and efficiency of a streamline simulator (3DSL5) in the reservoir engineering workflow. We evaluate its speed, memory requirements and scalability using tracer and black oil test data sets on an SGI Origin 2000 (250 MHz MIPS). Our data are based on real fields and range from 200000 to 7 millions cells with cells as small as 30x30x0;5 meters. We examine problems with preand post-processing of large data sets and visualising such simulations. Streamlines allowed us to check the validity of a

large geological model and to optimise wells patterns with more than 30 producers and injectors. Application of streamlines to the 10th SPE comparative solution project is also discussed.

We demonstrate how streamline-based simulation has matured from a research tool to an industrial application providing real benefits to engineers as a complementary tool to existing conventional simulation technology based on finite volumes.

Introduction

Dynamic flow simulation is still a bottleneck in most integrated reservoir studies that attempt to reconcile the geological model with seismic data and well data. Threedimensional, high resolution (3DHR) seismic data as well as improved 3D static modeling tools produce models that are ever more detailed and allow significantly more faults than the previous generation of static models. Today’s fine-scale models are commonly in the range of 1 to 10 million cells. On the other hand, flow simulation technology based on finite volumes (FV) or finite differences (FD) is mature. Any improvements are expected mainly from parallel processing of key modules such as the simultaneous solution of the linearized flow equations or PVT calculations. As a result, only relatively small dynamic models ( 100000 active cells) can be considered in routine engineering studies. Dynamic flow simulation has also suffered from recent cost cutting by reserving large-scale computing power (machines with more than 1000 processors) to seismic processing, while shifting most other simulations to PC clusters with a limited number of processors (8 to 32).

Upscaling fine-scale geological models remains a reality for most studies causing significant deterioration in the geological model. In many cases, the fine-scale and coarsescale models do not superimpose, with coarse blocks being traversed by fine-scale faults. Under realistic reservoir conditions, rigorous upscaling becomes difficult forcing the engineer to make dubious approximation (fault location and transmissivity, layer resampling, etc). The fact that these approximations often cannot be quantified since a fine-scale reference solution is not available makes matters worst. A methodology that allows solutions on the original geological model is therefore desirable, allowing some quantification of errors due to upscaling. Streamline-based reservoir flow simulation is one alternative currently available1,2 .

Streamlines simulation versus Finite Volume simulation

Streamline-based flow simulation has made significant advances in the last ten years. Today’s simulators are fully 3D1,3, account for gravity1,4 as well as complex well controls. Most recent advances also allow for compressible flow5 and compositional displacements6,7. A number of recent

publications demonstrate how streamline-based simulation is now coming into the mainstream13,14,15,16.

Finite volume methods are based on the fundamental concept that fluids are moved from cell-to-cell. The problem with this methodology is an exponential increase in CPU time with a linear increase in model size. The reason for this is that larger models dramatically reduce time step sizes (both in implicit and explicit mode) due to reduced cell volumes and often increased heterogeneity. This means that locally higher fluxes have to be pushed through blocks with smaller volumes. Routine solutions of million-cell models with FV or FD technology are therefore out of reach for most practical applications. Even with significant simulation power, a single solution can take weeks. Data debugging and sensitivity calculations under these circumstances can become difficult.

Streamline-base simulation is an attractive alternative because of the fundamentally different approach in moving fluids. Instead of moving fluids from cell-to-cell, streamline simulation breaks up the reservoir into one-dimensional systems, or tubes. The transport equations are then solved

along the one-dimensional space defined by the streamlines using the concept of time-of-flight9,10. By decoupling the

transport problem from the underlying 3D geological model, fluids can be transported much more efficiently. Large timesteps can be taken, numerical diffusion is minimized, and cpu time varies near linearly with model size.

Description of the streamline simulator

Modern streamline-based simulation rests on 5 key principals:

(1) tracing streamlines in a velocity field9; (2) writing the mass conservation equations in terms on time-of-flight10 (TOF); (3) numerical solution of conservation equations along streamlines17; (4) periodic updating of the streamlines18,2; and

(5) operator splitting to account for gravity4. Details of the methodology can be found elsewhere1. We only give a brief overview here.

The TOF along a streamline traced form a source to a sink is given by

sφ (δ )

τ= ò0 uvt (δ ) dδ ,

allowing to rewrite the conservation equation for incompressible, immiscible flow as

where Gj accounts for flow due to density differences only. This equation is solved using a two-step approach (operatorsplitting): (1) first saturations are transported along streamlines ignoring any gravity effects and then (2) saturations are allowed to segregate due to density differences. Recent key advances have pushed the technology to allow for 3phase, compressible and compositional flows5,7.

The elegance of streamline-based simulation is that once a streamline paths have been defined, any transport problem is reduced to a one-dimensional problem. Furthermore, by decoupling the actual transport problem from the underlying grid, CFL constraints imposed by small grid blocks and/or locally high velocities are completely circumvented. This allows for the solution of large problems efficiently while minimizing grid orientation effects and numerical dispersion. Freeing the choice of the 3D time step from grid stability considerations allows to significantly reduce the number of overall time steps required and makes streamline-based simulators so much more efficient. However, for nonlinear displacements there still exists an upper limit to the time step size in order to obtain physically meaningful solutions (converged solutions) by capturing changing streamline paths, gravity segregation, and for compressible problems the change in pressure of the overall field. An example of such convergence behavior is illustrated in Figure 18. Choosing uniform half-year time steps does not allow to capture the early non-linear behavior of the problem.

Tests cases

We present three test cases. Case A is a full field model taken from an industrial reservoir study; Case B looks at well pattern implementations; and in Case C we look at the 10th SPE comparative solution project.

Case A : Validation of the upscaling process

This case deals with a turbidic field with channel deposits. Initially, a coarse full field model was built to evaluate different development schemes. However, geological heterogeneity and field development schemes motivated a detailed construction of a 8,5 million cell model with 2,7 millions active cells. Figure 1 depicts the permeability map PERMX (XY view) at a given layer.

The fine-scale geological model was constructed from 3DHR seismic, well logs (2 appraisal wells) and geostatistical input. Consistent facies and porosity fields were generated. Permeability fields were generated as functions of porosity, resulting in values differing by several orders of magnitude throughout the field. The resulting 200x544x80–grid geological model was consistent with known geological horizons.

At 200x544x80 the model was simply too big to simulate even a single production scenario in an acceptable amount of time using a conventional black-oil simulator. Test runs on a

waterflood simulation showed that 27 hours were required to simulate only 40 days of production using a single processor on a 16 Cpu Origin 2000 (250MHz MIPS). The large CPU requirement is only in part due to the size of the model. The small size of the elementary grid cells (about 37m x 37m x 1.25m), a lot of barriers, and zero permeabilities all contribute to forcing time steps in the range from 11 hours to 2 days even for a fully implicit numerical scheme.

Upscaling: Upscaling was performed using a uniform aggregation rate of 4x4x4 resulting into a 50x136x20 coarse grid. Several methods were used to upscale the permeabilities:

1.Confined boundary conditions (TotalFinaElf internal FEM upscaler based on a 27 point scheme),

2.Unconfined BC (TotalFinaElf internal FEM upscaler),

3.Diagonal tensor with sealed conditions (commercial upscaler based on 7 point scheme),

4.Diagonal tensor with open conditions (commercial upscaler based on 7 point scheme),

5.Square root of harmonic-arithmetic and arithmeticharmonic average (analytical method)

The first four methods are based on a pressure solution for incompressible flow. They differ by the numerical scheme (6 or 26 neighbours for a given grid-block) and by the boundary conditions used: “confined” stands for no-flow boundary conditions on the opposite faces of the averaging domain, while “unconfined” considers a linear pressure gradient between opposite faces.

The fine-scale model contained five different sets of relative permeability and capillary pressure curves. The rock type associated with each coarse grid-block was determined by considering the dominant rock type in the 4x4x4 subgrid through an arithmetic discrete mean weighted by the pore volume. The “averaged” rocktype number was then used to determine the shape of the KR/PC curve. The end-points were re-calculated for each block of the dynamic model. Saturation endpoints were obtained by pore volume weighted arithmetic means, while relative permeability endpoints were computed by a TotalFinaElf internal FEM upscaler. Upscaling of well productivity indexes was done by assigning to each perforated coarse grid-block a KH value equal to the sum of the finescale KH values.

The streamline simulator5 was then used to compare the dynamic behavior of the fine-scale and coarse-scale grids under different flow conditions.

Tracer simulation / incompressible waterflood simulation

Given the elongated shape of the reservoir in the Y-direction (channel deposit), 13 injectors and 15 producers were disseminated along the reservoir pattern (Fig. 2).

The reservoir was filled with oil and water was injected at the injector wells. All fluids were assumed incompressible.

Since the aim of these first runs was to check only the upscaling process of the absolute permeabilities, the relative

permeabilities effects are ignored by setting water and oil relative permeabilities to be straight lines. Each injector was assigned a fictitious rate of 2500 m3/day (reservoir conditions) with a maximum BHP pressure of 350 bars. All producers were set to a constant BHP pressure of 180 bars. We compared total field oil rates and water-cuts for the 5 different upscaling methods (Fig. 3).

Field oil rate versus time

Time (days)

Time (days)

The single phase flow-based method (FEM) with confined boundary conditions gave the best fit in terms of global production, although local well-by-well comparison were not necessarily as good. Well GA is such an example (Figure 4). As we want to use this model for optimizing the well pattern, the well positions have not been yet defined. The aim of the uspcaling process is to get a representative flow model on the entire grid.

Oil rate for well GA

Time (days)

Fig. 4 - Oil rate comparisons at well GA

We therefore also compared the pressure and oil saturations maps between the fine grid and the coarse grid model. To quantify the difference, we computed an error norm:

Where Paveragefine(i) is the pore volume weighted arithmetic average of the 4x4x4 fine grid pressures corresponding to the coarse cell i and Pupscaled(i) is the computed pressure value on the coarse grid-block i. A small error norm indicates a good match in terms of global pressure and saturations maps between the fine and coarse grid. The pressure norm and the oil saturation norm at 5 years using the 5 different upscaling methods are given in Table 1:

Table 1 - Error differences in pressure and oil saturations with the fine grid

Again, the confined method gave the smallest norms (pressure and oil saturation). The sealed boundary conditions for the pressure solver based methods are better then the open boundary conditions, while the analytical method is significantly higher than the others. The pressure differences seem to be a better indicator of the quality of the fit and separate more easily the various methods. Since the error norm comparison is an indication of global spatial results, we prefer this type of criteria in ranking the upscaling results.

Figure 5 depicts streamlines colored by the TOF for the coarse grid using two different permeability fields: one upscaled by the “confined FEM” method and the second using the “Commercial sealed BC” method. Figure 6 shows the drainage volumes of the producers between the fine grid (1 streamline represented over 200), the coarse grid upscaled by “confined FEM” method (bottom left) and the grid upscaled by an analytical method (top right). These types of graphical representations –which are specific to a streamline simulator– allow a quick qualitative comparison between coarse and fine grid by indicating zones where the upscaling is not satisfactory. Streamlines are best viewed in color and will loose much of their visual usefulness if viewed in black-and- white. Both Figure 5 and Figure 6 are good examples of how streamlines can be used to help an engineer crosscheck upscaling methodologies as well as sensitivity to other data manipulation.

Other streamline-specific data is shown in Figure 7, which quantifies the relative flow percentage of injectors contributing to producer well P3C as well the relative flow percentage of producers receiving water from injector well ID1.

Injector well ID1

Fig. 7- Allocations factors for injector well ID1 and producer well P3C

The relative influence of injectors and producers illustrated by streamlines is similar to the results obtained using a tracer option in a conventional finite volume simulator by injecting a tracer of different color in each injector well. But it is much easier to produce this type of data using streamlines rather than a FV/FD simulator. In addition, the runtimes for the 8 million fine cells grid are considerably lower as shown hereafter.

We conducted single-phase tracer simulations using a finite volume simulator (Eclipse 100). The fine grid required 20 hours for simulating 60 days of production, with 33 time steps ranging from 12h to 4 days. The simulation was conducted on a single processor of a 16 CPUs Origin 2000 (250MHz MIPS) with 13 GB of memory. We estimated the CPU time

necessary to run the full 5 year simulation with Eclipse to be approximately 240hrs (about 430 time steps).

The reason for the large amount of time steps in single-phase run is that Eclipse is a general purpose simulator that does not perform incompressible runs. Pressures are recomputed at each time step, compressible fluids and rock are mandatory, and the CFL condition will in general force small time steps in areas of high-flow/small pore volume. The same 5 years simulation was run in 5h42mn with 3DSL (1 main timestep (pressure solution) and 7 checking residual and output steps).

Even in the case of an industrial study, the CPU times required by a general purpose FV simulator in single phase mode remain outside the useful limits needed for checking upscaling studies of the type considered here. The usual procedure would be to either use a specific single phase tracer simulator and/or to split the fine grid in 5 sector models containing 1.4 millions of grid-blocks (about 400000 actives cells). Significant file manipulations would be necessary to split the fine grid in parts, and some sort of automation is usually necessary to avoid errors while translating the wells positions from the full field model to the sector models. In practice, the validation is generally performed only on one sector model due to time constraints.

As a result of the excessive CPU time for the fine grid, we could compare 3DSL and Eclipse only on the coarse grid model. Although the exact type of results are different (Eclipse outputs a tracer concentration inside a water flow while 3DSL outputs oil and water flow rates and water-cut), we can relate the normalized tracer concentration from Eclipse with the 3DSL water-cut.

To summarize, streamline-based simulation is a powerful tool to validate the upscaling process of absolute permeabilities and to determine rapidly the best upscaling method. Results in terms of production curves but also saturations and pressure maps can be compared between the geological grid (several millions of cells) and the simulation grid. Two major advantages emerged: (1) simulation on the fine-scale geological model can actually be performed in a reasonable CPU time allowing to generate a reference solution; and (2) quick and useful graphical comparisons using time-of-flight, drainage time or well pore volumes can be used to quality check the upscaling process. General purpose fine volume simulators are not suitable to run large geological models even in single phase flow since they are not optimized for this type of problem and lead to unrealistic of CPU times.

CPU time and visualization aspects

Table 2 indicates the CPU time and total memory consumption for the various 3DSL and Eclipse runs. The memory data were obtained using the Unix command “pmem, indicating the sum of the physical and the virtual memory used by the process.

Table 2 - CPU time and memory for incompressible runs

The streamline simulator uses 3 to 4 time less memory than Eclipse, and is estimated to be 20 times faster than Eclipse for the full grid model (2,7 million active cells) but only 1,5 times for the coarse grid (80 000 active cells). The CPU time for the coarse grid (6 mn versus 4mn) is not significant because it comprises much more I/O time than calculation time.

It is important to underline that the numbers mentioned here are meant as a practical indication for the reservoir engineer to estimate the time necessary to perform validation simulations in an upscaling study, and are not meant as general comparison of the efficiency of the finite volume scheme versus streamline-based simulation. The streamline simulator we used has a specific option for incompressible runs, while the FV simulator is not suited for this type of simulations. Moreover, all the runs were not performed on a dedicated machine and some CPU times may have been overestimated while the simulations were running in an overloaded environment. The processors on our SGI Origin 2000 are 250 MHz old-fashioned MIPS processors. For example, the 5h40 streamline CPU time can be reduced to less than 1h using a more recent 866 MHz Pentium III dedicated PC.

Production curves were viewed using Eclipse’s postprocessor GRAF (load user option), while visualization of TOF, pressures, and saturations maps was done using Gocad user interface files. While post-processing of grid properties such as pressure and saturations maps is becoming easier for large grids, post-processing of streamlines results (time of flight TOF, drainage zones for producers PW or injectors IW) is not considered standard output and therefore lacks an efficient output standard. For now, 3DSL interfaces with Gocad’s p-line ASCII format. Storing the streamlines plus properties such as TOF and well pore volumes produces large ASCII files (1,4 GB per output step for the fine grid) causing difficulties with our Gocad version limited to 1,5 GB of memory. It is important to note that in many cases attempting to view all streamlines is actually beyond the resolution of the screen. The solution is to reduce strongly the number of outputted streamlines (for example 1 every 200). For now, this reduction needs to be performed inside the output section of the streamline simulator and not inside the 3D modeler tool.

3D modelers must account for post-processing streamlines simulations results coming from large models. Indeed, treating simulation results in 3 dimensions with a very high resolution grid can lead to the following problems encountered in most commercial 3D modelers and post-processing softwares:

-size of the disk files (ascii interface format is not adequate),

-long time to load the data,

-refreshing views can take a long time.

Compressible waterflood simulation

In the next set of runs we added compressibility to the oil and water phase. A water-oil contact was set, and the fine grid model used 5 different relative permeability curves (5 rocktypes) with no capillary pressure (a limitation of 3DSL). For the coarse model we used the dominant facies for each gridblock with the appropriate relative permeability curves.

Six producer wells were set in the central upper part of the reservoirs, and seven injectors were set on the flanks to maintain pressure and ensure a good sweep. Figure 8 compares the field oil and water rates between the coarse grid and the fine grid. Results are not satisfactory. Oil flow rates are significantly different and breakthrough times are much earlier on the coarse grid than on the fine grid.

Fig. 8 - Field oil and water rates

Other pseudoisation techniques as steady-state viscous dominant pseudos and pore volume weighted dynamic pseudo were also compared for the water-oil system. The dynamic pseudos could be computed using pressures and saturations maps computed on the fine grid by the streamline simulator.

Use of pseudos instead of dominant facies significantly

improved the fit. Details on all these pseudoisation techniques can be found elsewhere 19,20 and are beyond the scope of this

paper.

In the same way, comparisons as the error norms in pressure and saturations maps as indicated in the previous section were also carried out to select the best pseudoisation techniques.

The waterflood simulation was tried with Eclipse on the fine grid using on a single processor on the SGI. A 100 days simulation (52 timesteps) ran in 50h. Time steps ranged from 5 hours to 2,7 days. We extrapolated the CPU time required for a 5 year simulation to be roughly 25 days.

Table 3 - CPU time and memory for dead-oil simulations

As in the previous case, 3DSL uses significantly less memory (3 to 10 time less) than Eclipse (Table 3). Introducing a limited amount of compressibility in rocks and fluids (dead-oil problem) doubles the streamline CPU time and increases by a factor of 2.5 the estimated FV simulation time. By allowing for compressibility, exact voidage replacement is no longer necessary and the field average pressure is allowed to change over time. This means that 3DSL is now solving for pressure at every timestep.

The CPU time obtained with 3DSL on the fine grid model (12h on a single processor overloaded machine) is high but still acceptable. We could have checked the two phase upscaling process with an incompressible model, which would have significantly reduce the CPU time. Running the fine volume simulator on the other hand would require a large parallel machine (more than 30 dedicated processors if we assume a linear speed-up)11. For the coarse grid, the CPU times (25mn) are similar for both simulators, with 46 timesteps required by Eclipse and 11 for 3DSL. The I/O time spent for writing results files in Gocad format may be an improvement issue for 3DSL. The fact that the streamline simulator needs 4 times less timesteps to perform the run although the time needed for an individual timestep is higher then for Eclipse partly explains the significant differences in CPU times expected for the full grid model.

From a practical point of view, a streamline simulator can now help the reservoir engineer check relative permeability pseudos by a direct comparison of the upscaled reservoir simulation model with the geological (or geostatistical) model.

Black-oil simulation

We also ran the fine and coarse grid using black-oil fluid properties. As in the deadoil case, the fine grid could only be simulated with 3DSL which took 72h for a 5 year simulation (Table 4). On the coarse grid 3DSL used 33mn. For Eclipse the CPU time doubled on the coarse grid with respect to the deadoil case to 46mn. Assuming the same doubling on the fine grid, we estimated the Eclipse run time there would be approximately 50 days.

The value of 72h is only indicative and could be reduced on dedicated, more recent PCs; generally, blackoil simulation CPU times are strongly depend on the fluid properties and on the frequency of the changes in the well constraints. Since

predictive runs are usually for periods much longer than 5 years (i.e 20 or 30 years), even streamline-based simulation would produce excessive run times in this case.

Table 4 - CPU time and memory for black-oil runs

For a general blackoil simulation, the streamline simulator is forced to re-compute the pressure field much more frequently in order to correctly model its variation through time. For the fine-scale model, each timestep now requires approximately 4h on the SGI.

This last figure is significant and shows that a streamline simulator faces similar computational problems as a finite volume simulator for large models when the fluids are highly compressible or when the number of wells events is important, although the streamline simulator retains the advantage of not being forced to take small timesteps due to CFL restrictions imposed by small grid blocks/high flow regions.

Case B: Implementing a well pattern

Streamlines are particularly useful in visualizing flow and give information that is unique for balancing well patterns. We used a coarse grid model (70000 active cells) to optimize the well pattern. The mobile oil in place map is shown in Figure 9, and initial well placement was guided by contours showing a central target area. As a first guess, six horizontal wells were set in the maximum zones of oil in place, and seven water injectors were set in the flanks above the oil-water contact in high permeability zones. We used the dead-oil model for a 5 years simulation. The same model was run with Eclipse and results were similar.

Optimization of the well patterns with a standard FV simulator is not simple. Once one obtains a set of wells with a large production plateau and a late individual water-breakthrough, sensitivity studies to quantify the influence on the global production curves in changing the well position or the number of wells require a large number of simulations, given that both the well position and the total number of wells must be considered.

Since the basic principle of a streamline simulator is to trace streamlines from injectors to producers, three types of results are generated useful for balancing patterns:

1.The time of flight as shown in Figure 10. The TOF indicates the time necessary for an injected water particle to reach a given point in the reservoir, and is an instantaneous image of the swept area. Red zones are zones that will be swept

while the white zones are not. For the considered test case, the injectors seem to maintain the pressure on the flanks but will not sweep much oil to the central area towards the producer. An injector pattern with some injectors in the central area is likely to be more satisfactory to increase oil recovery.

Superimposing the time of flight (Fig. 10) with the mobile oil zone (Fig. 9) and the positions of producers and injectors wells also allows to detect potential injectors that would cause early water breakthrough.

2.Another typical results from a streamline simulator are the pore volumes associated with each well as indicated in Figure 11, which shows reservoir volumes associated with the

different injection wells. The regions are much larger than the swept area shown in Figure 10, because the display of the pore volumes associated with each well is a purely geometrical construction. It says nothing about how long it would take to sweep the entire area. This type of figure helps to find the geometrical influence zone of the injectors.

For the considered test case with 6 injectors, we can notice that the injectors have little influence on the central area and that increasing the injected rate or enabling a longer period of injection will have little effect. For incompressible system, such detection might more difficult since taken together all injectors must exactly sweep the entire pore volume, although it might take a long time. Figure 11 also indicates that one injector (dark grey color - upper left) only has a very small pore volume associated with it, suggesting that this injector is not efficient and should be placed elsewhere. As the previous one, this second type of results is difficult to obtain using a conventional FV simulator.

3.A third type of typical result is the geometrical producer influence volume as indicated in Fig. 12. Each color indicates the volume of the reservoir tied to a given producer. Figure 12 depicts the reservoir volume associated with each producer for 6 vertical wells while Figure 13 for 6 horizontal wells having one end at the same XY position. In this case where all the faults are communicating, the horizontal well scheme has the potentiality to drain almost all the oil field but it doesn't indicate the time necessary to reach this objective. The theoretical drainage zone for horizontal well P4 (white color – Fig. 13) is enormous but since the streamlines are very elongated, the time indicated by the simulation for the oil in the northern part of the reservoir to reach the producer is too long and other economical constraints do intervene. The volumes associated with the 2 northern wells are smaller using horizontal wells. This observation is in line with the results of the simulation which demonstrates that implementing 2 vertical wells instead of 2 horizontal wells in the northern part of the reservoir was more efficient.

Superimposing Figure 13 with the oil in place map helps to detect undrained areas, and locate positions for infill wells. Misplacement of producers is easily detected when the

drainage areas are not clearly separated or when the drainage areas is very small.

Up to this point, we have used considered streamlines only for analyzing fixed well pattern. An interesting extension is to examine how the producer influence volumes evolve in time as new wells (injectors or producers) are opened.

depicts such an evolution in time.

Reservoir drainage volume percentage for producers

Time (in days)

Fig 14 - Reservoir drainage volume (%) for producers vs time

During the first year, well P4 is tied to 23% of the pore volume of the reservoir. In the second year, following the start-up of new wells P1 and P3, the pore volume tied to well P4 decreases to 10%, followed by 7% for the next year. It decreases again to 10% after start-up of well P2. Representation of the reservoir pore volume areas tied to wells can also be visualized as Fig 12. It is convenient for depicting the geographical evolutions of drainage areas. Unfortunately, such representation must be done in color. This type of information is extremely useful for preparing a first estimate of the well pattern of a development scheme and optimizing well management and production start-up over time.

Figure 15 shows the same graph for injectors and clearly indicates that injector I4 (dark area) is inefficient.

Fig 15 - Reservoir volume (%) tied to injectors versus time

Reservoir volume percentage for injectors

Obtaining this type of result using a conventional simulator would be much more complex, requiring the addition of different colored tracers while producing a significantly less powerful graphical representation of flow.

Case C: 10th SPE Comparative Solution Project

The main purpose for the 10th SPE comparative solution project8 (SPE10) is to offer a common data set on which to compare upscaling algorithms. Two problems were presented, but we only discuss the second one here as it more relevant to streamline-based simulation. SPE10 is 1.1 million cell model (60x220x85) describing a part of the Brent sequence originally generated for use in the PUNQ Project. The top part of the model (35 layers) is a Tarbert formation, the bottom part (50 layers) represents Upper Ness.

SPE10 offers a good example for demonstrating the nearlinear scalability of CPU time for flow simulations using streamlines. Fig. 16 shows simulation results for various grid sizes, with the fine-scale solution taking 280min on an 800MHz PC with 768MB of RAM.

Fig. 16 - CPU run time (in minutes) demonstrating the near-linear scaling with the number of active cells.

Linear vs. nonlinear flow. SPE10 is nonlinear example in that it is a non-unit mobility ratio displacement of water into oil. Figure 17 shows how the field water cut is a strong function of the mobility ratio but is not impacted by gravity. Thus, streamlines must be updated to capture the nonlinearity of the problem. Gravity, on the other, may be ignored if desired.

Fig. 17 - Field watercuts assuming unit mobility ratio (TRACER), no gravity and gravity. SPE10 is sensitive to the mobility ratio but not to gravity.

Optimal Time Stepping: One key advantage of streamlinebased simulation is that large timestep can be taken. What the optimal number is cannot be know apriori, but simple numerical experiments can quickly give an indication. Figure 18 shows simulations using constant time step sizes starting with 6 months and decreasing to 10days. The dots show nonuniform timesteps: 10x25 days followed by 5x50days, 5x100days and finally 5x200days for a total of 25 timesteps.

We use this timestepping for all subsequent runs.

Fig. 18 - Sensitivity runs for determining optimal time step size (30x110x17 grid).

Results: Detailed results on SPE10 can be found elsewhere8. We present the field water cut for different grid sizes in Fig. 19. Upscaling was done using a simple combination of arithmetic and geometric upscaling. Arithmetic upscaling was used on Kx and Ky on the original fine grid to go from 60x220x85 to 60x220x17. Kz was upscaled using a geometric average. All smaller grids were progressively derived using geometric averaging on all permeabilities starting from the 60x220x17 grid. The 30x110x85 was also obtained by using geometric averaging on all permeabilities. Flow-based upscaling was not used.

On a field scale, Fig. 19 shows that results are very good even for coarse grids such as 12x44x17, which represents an upscaling by a factor of 122 (1094418/8976). Results on an individual well basis show a little bit more difference but with the same overall trend: only a weak dependence on upscaling.

Fig. 19 - Field water cut response for different gridsizes.

Conclusions

The usage of a streamline simulator for two types of typical reservoir simulation tasks has been presented: (1) validation of the upscaling process for absolute permeabilities and for pseudo relative permeabilities in water-oil system; and (2) aid in optimizing well pattern design.

In both cases, the streamline simulator’s key advantage is to allow runs on the original geological model even with modest computational hardware. The difference in CPU times between streamline simulation and finite difference simulation is due to the ability of 3DSL to take fewer timesteps while being more memory efficient.

Streamline simulation also brings new graphical features such as time of flight visualization and producers and injectors influence zones. This new three types of results can be helpful:

(1) for a quick check of the upscaling process; (2) for a quick check of the consistency of the geological model with the dynamic flow assumptions and observations. In particular, for helping the geo-scientists to visualize flows and to check the assumptions of communicating or non communicating faults;

(3) for a check of the ability of the geo-statistical model to permit flow between wells across permeability barriers; (4) for quickly visualizing well pattern influence of a development scheme and optimizing well management and production startup over time.

Streamlines simulators are chiefly useful for producing fasttrack reservoirs studies as a complement to the traditional simulator. It is also a means for integrating dynamic data into the geology and becomes a complementary tool for the reservoir geologist. Integration of such a tool inside 3D modelers arises.

Some limitations do exist: (1) they are less accurate in materials balance than finite volume simulators; (2) they ignore capillary pressure and consequently cannot be used in the rare cases in which this is dominant; (3) computing times can become similar to conventional simulators whenever highly compressible fluids or well events impose time steps on the same order as those used by a conventional simulator; (4) they do not offer all the possibilities and robustness of traditional simulators for processing wells and managing overall production constraints; (5) additional progress is required to improve the visualization process for large grids.

Other instances where streamlines can complement traditional reservoir simulators are: (1) aid in construction and optimization of gridding of the conventional finite volume model; (2) aid in analysis of the flow process; (3) quantification of the biases associated with the resolution mode of conventional simulators such as grid orientations effects; (4) history matching; (5) ranking models in the process for identifying and quantifying uncertainties among several versions of the geological model. The streamline simulator helps make a rapid selection of versions on dynamic criteria without requiring prior sampling of the distribution of any static parameter.

Acknowledgment

The authors thank TotalFinaElf and StreamSim for permission to publish this work.

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