By definition, a streamline is characterized as a line that is tangent to the local velocity field at all points at a given time. Applying it to the oil and gas context, a streamline is the path of fluid particles flowing from the injection well to the production well, representing a one-dimensional flow channel which changes progressively through time [3]. Streamline distributions with different flow velocities are governed by reservoir heterogeneities and displacement characteristics. In reservoir simulation, streamlines come up as an alternative approach to the broadly used finite-difference method.

The use of streamlines has been present in the petroleum industry since the 1930s and its key applications include:

- Reservoir-flow surveillance
- Flow simulation
- History matching
- Flood management

Modern streamline simulation techniques emerged in the 90s to overcome computational efforts faced with traditional finite-difference simulation when dealing with geological models with high-resolution, which present heterogeneous spatial distributions of static properties such as permeability, porosity and relative permeability [1].

The method is based on the solution of fluid flow equations in porous media and take into account reservoir heterogeneity, fluid mobility ratio and changes in injection or production. In a streamline-based flow simulation, the fluids move over a timestep along the streamline itself rather than from grids – as in traditional reservoir simulations. Therefore, streamlines account for an image of the instantaneous velocity field, constantly updated along time. In turn, the spatial distribution of the static properties as well as the injected and produced volumes will shape streamlines geometry.

By applying the streamline method, 3D-domain heterogeneity problems can be reduced to a series of 1D homogeneous and independent objects along which the transport equations are solved. Finally, all the results obtained are gathered and combined to reassemble a solution for the entire reservoir. This breakup and reassembly for each timestep enables the generation of new distributions of compositions and pressures and provides current insights of the displacement.

According to (Batycky and Fenwick), the concept of sweep efficiency can be very useful to model streamline simulations. It can be described as the product of the volumetric sweep (Ev) times the displacement efficiency (Ed). The volumetric sweep is the reservoir volume contacted by the streamlines associated with the injection well, while the displacement efficiency is determined by the transport equations solved along the streamlines.

In reservoir simulation, there is a substantial difference between traditional numerical simulation and streamlines techniques. In the first one, flow is solved along the reservoir grid cells; in contrast, when applying streamlines simulation, fluid transportation occurs in a dual-grid approach: an Eulerian (static) time-invariant grid is setup to calculate the total velocity field, while a Lagrangian (dynamic) time-variant is used to transport the components along streamlines. If the pressure field is a known parameter, the velocity field can be designed through Darcy’s Law, enabling the construction of a transport network where fluid is transported along each streamline.

From the described simulation method, streamlines are assumed to be fixed during a period Δt, and the fluid components are transported along the dynamic grid from t to t+Δt. At this point (t+Δt), the new system’s configuration is set and the process is repeated until it reaches the desired solution, which is defined by the user and might depend on the applicability of certain numerical constraints in order to ensure accuracy.

In traditional numerical reservoir simulation, there is a pressure-solve segment and a saturation segment. By finite difference method, pressure is solved and then flow is computed based on the pressure distribution. On the other hand, in the streamline method, saturation moves along each flow unit of the streamline, and pressure is solved for the entire reservoir. Therefore, it may reduce the effect of grid division and arrangement, improving the results accuracy and reducing time consumption, hence increasing computational speeds [3].

In addition, the fluid is transported in the pressure gradient direction and not between the grid blocks as occurs in conventional finite difference approaches. It makes it possible to use larger time-steps with less sensitivity to block size and orientation [1].

Other advantages can also be outlined by the useful information that they provide. First, streamlines can delineate the drainage and irrigation areas associated with producers and injectors, respectively. This makes it possible to infer which gridblocks are associated with each well at a given timestep. In terms of assisted history-matching, these regions can be used to analyze grid properties and investigate possible strategies to improve the match between simulated and historical volumes.

Streamlines also offer an efficient way to solve the problem of associating produced and injected volumes. From computing all the volumetric flow rates associated with all the injection/production pairs, it is possible to determine well-rate allocation factors, which are defined by the percentage of flow from one well to another with which it communicates. According to [2], these well-allocation data are essential to workflows that are based on pattern analysis and play a key role in flood management.

However, the streamlines simulation method presents some disadvantages. As it is best suited for problems dominated by convection, the ones dominated by diffusion are more challenging, as they do not have a defined flow direction. Also, the use of a dual grid requires repeated mapping of pressure and compositions, which results in a method that is not mass conservative. In addition, the assumption that streamlines are independent among themselves makes difficult to capture phenomena transverse to the flow direction such as gravity, transverse-capillary pressure, diffusion, compressibility and transverse-thermal effects [2].

Powerful reservoir data manipulation tools such as Kraken can routinely perform advanced analyses such as streamlines visualizations. By adding a well source and selecting a producer or injector well, it is possible to obtain a snapshot of the instantaneous flow which indicates well-associated drainage/irrigation areas.

These types of visualizations play a key role in understanding well-influence in a determined development plan. The following sequence of images illustrates the fluid flow in streamlines for subsequent timesteps.

Analysis capabilities as those mentioned above can be applied to several tasks when dealing with reservoir data management such as flood management, due to streamlines ability to relate produced-oil volumes to injected volumes on a well-pair basis. It enables determining the injection efficiency for that connected pair: the ratio of oil volume produced to the injected volume, which is usually water.

The success of streamlines application highly depends on the questions being addressed from the model, the assumptions taken by the engineers as well as the available time for the study. However, without a doubt, extensive research and literature have proven that this method provides important contributions and useful insights regarding reservoir’s behaviour and can be used within diverse reservoir engineering workflows.

REFERENCES

- Batycky R.P., Blunt, M.J., and Thiele, M.R., A 3D Field Scale Streamline-Based Reservoir Simulator; SPE Reservoir Engineering, pp. 246-254, November 1997
- Batycky, R. P., Fenwick, D. H., Thiele, M. R. “Streamline Simulation for Modern Reservoir-Engineering Workflows”. SPE 118608, 2010
- Jun, Y. “Streamline Numerical Well Test Interpretation: Theory and Method”. Gulf Professional Publishing, 2011.
- Satter, A., Iqbal, G, M. “Reservoir Engineering: the fundamentals, simulation and management of conventional and unconventional reservoirs”. Gulf Professional Publishing, 2016.