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Streamline simulation provides an alternative to cell-based grid techniques in reservoir simulation. Streamlines represent a snapshot of the instantaneous flow field and thereby produce data such as drainage/irrigation regions associated with producing/injecting wells and flow rate allocation between injector/producer pairs that are not easily determined by other simulation techniques.
Streamline-based flow simulation differentiates itself from cell-based simulation techniques such as finite-differences and finite-elements in that phase saturations and components are transported along a flow-based grid defined by streamlines (or streamtubes) rather than moved from cell-to-cell. This difference allows streamlines to be extremely efficient in solving large, heterogeneous models if key assumptions in the formulation are met by the physical system being simulated (see below). Specifically, large relates to the number of active grid cells.
The computational speed and novel solution data available have made streamlines an important, complementary approach to traditional simulation approaches to:
- Perform sensitivity runs
- Quantify the impact of upscaling algorithms used to move models from the geomodeling scale to the simulation scale
- Visualize the flow field
- Perform more reliable full-field simulations where sector models would normally be used
- Enable the ranking of predicted field behavior of given multiple production scenarios and input parameters
- Evaluate the efficiency of injectors and producers
- Reduce turnaround time in history matching
- Perform other established reservoir engineering tasks
A comprehensive overview on streamline-based flow simulation has recently been presented by Thiele.
Applicability of streamline simulation
The power of streamline simulation lies in its simplicity. The main objective is to capture how injected reservoir volumes (usually water and/or gas) displace resident reservoir volumes given the following:
- Well locations
- Well rates
- Reservoir geometry
- Geological description
One of the key underlying assumptions in streamline simulation is that the system be close to incompressibility. This decouples saturations from the underlying pressure field and allows each streamline to be treated as being independent from the streamlines next to it.
Many fields under waterflooding or other pressure maintenance schemes are excellent candidates for streamline modeling and have been successfully modeled in this way. Forecast simulations under the assumption of voidage replacement are another good example where streamlines can be very effective. Even miscible gas injection schemes have been successfully modeled. At high pressures, the displacement of resident oil by gas is primarily an issue of simulating local sweep efficiency and channeling, something streamlines are designed to model without incurring numerical difficulties associated with other formulations. An early application to streamdrive projects was presented by Emanuel. Crane and Blunt used streamlines to model solute transport. More recently, streamlines have been shown to be very effective in modeling fractured reservoirs using a dual porosity formulation.
Streamlines have been in the petroleum literature as early as Muskat and Wyckoff’s 1934 paper. In 1937, Muskat presented the governing analytical solutions for the stream function and the potential function for 2D domains using the assumption of incompressible flow. Since then, streamlines and streamtubes have received repeated attention as a way to numerically predict the movement of fluids, even after the advent of finite-difference methods in the early 1960s. Important early contributions were made by Fay and Pratts, Higgins and Leighton, Bommer and Schechter, Martin and Wegner, Lake et al., and Emanuel et al.
In the early 1990s, streamlines were revived because advances in geological modeling techniques were producing models that were too large for finite differences to simulate in an acceptable time frame. For streamlines to be applicable to real field cases, important advances were made that extended streamlines to 3D using a time-of-flight variable allowed for streamlines to be periodically updated and included gravity.
Mathematics of the streamline method
The streamline method and the underlying mathematics for incompressible multiphase flow are briefly outlined here. For a detailed discussion as well as additional references describing streamline methods, see Batycky et al., Batycky, and Blunt et al.
Governing IMPES equations
The streamline method is an IMPES-type formulation with the pressure field solved for implicitly and the oil/gas/water saturations solved for explicitly along streamlines. The governing equation for pressure, P, for multiphase incompressible flow without capillary or diffusion effects is given by
where D is the depth below datum, g is gravitational acceleration constant, k is the permeability tensor, krj is the relative permeability, μj is viscosity, and ρj is the phase density of phase j. The total velocity, , is derived from the 3D solution to the pressure equation and application of Darcy’s law. The explicit material balance equation for each incompressible phase j is then given by
Each phase fractional flow, fj, is given by
and the phase velocity resulting from gravity effects because of phase density differences is given by
The difference between finite-difference simulation and streamline simulation is the way the explicit material balance equations (Eqs. 2 through 4) are solved. In finite difference, the material balance equations are solved between gridblocks, whereas in streamline simulation the material balance equations are solved along streamlines. How this is done is explained next.
Solution to the transport equation
In a standard finite-difference method, Eq. 2 is discretized and solved on the underlying grid on which the pressure field is computed. The solution to Eq. 2 is governed by the grid Courant-Friedrichs-Lewy (CFL) condition, which can lead to prohibitively small timestep sizes, particularly for models with high permeability contrasts and/or high local flow velocities. With streamlines, this grid CFL limit is avoided completely by solving Eq. 2 along each streamline using a time-of-flight (TOF) coordinate transform.
Streamlines are traced from sources to sinks based on the underlying total velocity field. As each streamline is traced, compute the TOF along the streamline, which is defined as
and leads to the definition
Using Eq. 1, rewrite Eq. 2 as
Because the gravity term is not aligned along a streamline direction, Eq. 7 is split into two parts (operator splitting), giving two 1D equations. The convective portion of the material-balance equation along streamlines is given by
while the portion resulting from phase-density differences solved along gravity lines is given by
Both Eqs. 8 and 9 represent 1D equations that are solved using standard finite-difference numerical techniques. There are still CFL limits that restrict timestep sizes in these equations, but these are local to each streamline or gravity line, rather then at the 3D grid level.
In field-scale displacements, the streamline paths change with time because of the changing fluid distributions and the changing well conditions. As a result, the total velocity field is periodically updated, and new streamlines are recomputed to reflect the nonlinear nature of the displacement.
To move the 3D saturation distribution forward in time between successive streamline distributions from time Ti to Ti+1 = Ti+ dTi, the algorithm pictured in Fig. 1 is used.
The basic algorithm for streamline-based flow simulation is as follows: (1) Given initial conditions (i.e., pressures and saturations for each active cell in the system) and well conditions, the pressure is solved implicitly for each cell, as is done in conventional finite-difference methods (Eq. 1). (2) With the pressures known, the total velocity for each cell interface can be determined using Darcy’s Law. The total velocity is then used to trace streamlines using Pollock’s algorithm. (3) 1D mass conservation equations are then solved along each streamline, independently of each other (Eq. 7). The initial conditions for the streamlines are obtained by a mapping from the underlying 3D grid onto each streamline. The mass-transport problem is marched forward in time along each streamline for a pre-specified global timestep dTi, and then the solution is mapped back onto the 3D grid. Gravity is included by considering a vertical segregation step along gravity lines after movement along all streamlines (Eq. 8). While simple in its approach, important details must be considered. In particular:
- The algorithm is similar to an IMPES approach, in that the pressure is solved implicitly for a new time level n+1 assuming saturations at level n. The saturations at time n are given by mapping back solutions from each streamline onto the 3D grid at the previous timestep. Because of the implicit nature of the pressure solution, there is no limitation on the timestep to reach n+1. However, for compressible systems numerical convergence problems might limit the actual size of the timestep. This is no different than in finite difference (FD) simulation.
- The tracing of the streamlines using Pollock’s algorithm assumes Cartesian cells. Nonorthogonal corner-point cells require an isoparametric transformation for tracing streamlines.
- For incompressible systems, streamline will start at injection wells and end at production wells. For compressible systems, streamline can start/end anywhere in the system, because any gridblock in the system might act as a source (volume expansion) or a sink (volume contraction). Multiphase gravity effects can give rise to circulation cells for both incompressible and compressible systems.
- Initial launching of streamlines from wells can be proportional to the total flux at the wells, though this will in general leave many cells in the system without a streamline passing through them. For missed cells, tracing begins at the center of the missed cell and then traced backward until a source is encountered. If a cell does not have a streamline pass through it, then it is not possible to assign an updated saturation back to that cell.
- In practice, it is not possible to have all streamlines carry the same flux and ensure at least one streamline per cell. Thus, streamlines do not carry the same flux. Furthermore, for incompressible problems the flux along each streamline is a constant, while for compressible systems it is not.
- The tracing of streamlines using the TOF variable produces a highly irregular 1D grid along each streamline. To numerically solve the 1D problem efficiently, the 1D grid must be regularized, solved using an implicit approach, or regridded in some way to allow for a more efficient solution.
- The tracing of the streamlines relies on an accurate solution of the velocity field. Excessive distortions of the grid (nonorthogonal) or a pressure solution that has not been solved to a small enough tolerance can cause problems in tracing streamline paths.
Computational efficiency of streamlines
One advantage of streamline simulation over more traditional approaches is its inherent efficiency, both in terms of memory and computational speed. Specifically, streamline-based simulation can exhibit a near-linear scaling in run times as a function of active cells in the model. Memory efficiency is a result of two key aspects of the formula: streamline-based simulation is an IMPES-type method and therefore involves only the implicit solution of pressure, and tracing of streamlines and solution of the relevant transport problem along each streamline is done sequentially. Only one streamline needs to be kept in memory at any given time.
Computational speed, on the other hand, is achieved because the transport problem is decoupled from the 3D grid and instead solved along each streamline. Because transport along streamlines is 1D, they can be solved efficiently. Because the number of streamlines increases linearly with the number of active cells, and streamlines only need to be updated infrequently, the computational time exhibits a near-linear scaling with increasing number of gridblocks
The number of global timesteps is related to how often the flow field (streamlines) requires updating. Specifically, changing flow paths are a function of:
- Mobility changes
- Changing well conditions
For many practical problems, it is the changing well rates that introduced the greatest impact on a changing flow field and is therefore the limiting factor in deciding on global timestep sizes. Grouping well events into semiyearly or yearly intervals and assuming that the streamlines remain unchanged over each period is reasonable. This is why field simulations with 30- to 40-year histories are successfully and routinely simulated with 1-year timesteps.
A good example to demonstrate the efficiency of streamline (SL) simulation is Model 2 of the 10th SPE Comparative solution project. The total run time, T, of any streamline simulation is approximately proportional to
A near-linear scaling arises because:
- The number of timesteps (streamline updates) is independent of the model size, heterogeneity, and any other geometrical description of the 3D model. It is mainly a function of the number of well events and the actual displacement physics. For the SPE10 problem in Fig. 2, all cases were run with the exact same number of streamline updates—24.
- An efficient pressure solver is expected to have a near-linear behavior as well.
- The number of streamlines tends to increase linearly with the number of gridblocks, all else being equal. Fig. 2 illustrates this behavior.
- The time to solve the transport problem along each streamline can be made efficient by regularizing the underlying TOF grid and choosing the number of nodes to use along each streamline regardless of the size of the underlying 3D grid.
The linear behavior with model size is the main reason why streamline simulation is so useful in modeling large systems. In FDs, finer models not only cause smaller timesteps because of smaller gridblocks but usually face problems because of increased heterogeneity as finer models tend to have wider permeability and porosity distributions. The usual workaround is to use an implicit or adaptive-implicit formulation, but for large problems these solutions can become prohibitively expensive, both in terms of CPU time and memory.
Fig. 3 – Streamlines automatically allow the determination of the allocation of flow between wells by summing the flux of all streamlines associated with a particular well, well pair, or group of wells. Using this information and the visual display of streamlines allows patterns to be balanced correctly and efficiently. From left to right: rates are progressively changed to yield a balanced pattern.
Novel data produced by streamlines
Streamlines produce new data not available with conventional simulators. Because streamlines start at a source and end in a sink, it is possible to determine which injectors are (or which part of an aquifer is) supporting a particular producer, and exactly by how much. A high water cut in a producing well can therefore be traced back to specific injection wells or boundaries with water influx. Conversely, it is possible to determine just how much volume from a particular injection well is contributing to the producers it is supporting—particularly valuable information when trying to balance patterns ( Fig. 3) or optimize water injection over a field.
Streamlines can also identify the reservoir volume associated with any well in the system, because a block traversed by a streamline attached to a particular well will belong to that well’s drainage volume. It is therefore possible to divide the reservoir into dynamically defined drainage zones attached to wells (Fig. 3). Properties normally associated with reservoir volumes can now be expressed on a per-well basis, such as oil in place, water in place, and average pressure, just to mention a few.
Applications of streamlines
Streamlines are a powerful complementary tool to more traditional simulation techniques, and they are expected to play an important part in optimizing field production and management in the future. Specifically, streamlines can be used to:
- Validate upscaling techniques by allowing to generate reference solutions of fine-scale models.
- Efficiently perform parametric studies
- Visualize flow
- Balance patterns
- Determine efficiency of injectors and producers using data provided by streamlines
- Aid in history matching
- Enable ranking of production scenarios/geological models
- Optimize and manage field injection/production
- Conduct reservoir surveillance
It is important to underline that the theory on which streamline simulation rests is firmly rooted in the incompressible formulation of exact voidage replacement. Thus, streamline simulation is particularly powerful for modeling systems that are not a strong function of absolute pressure, but are instead governed by a pressure gradient. In addition, the strong assumptions of independence between streamlines favors modeling displacements that are not a strong function of diffusive phenomena, such as capillary pressure, transverse diffusion, or compressibility. For example, streamline simulation offers little or no advantage over conventional simulation for modeling primary production. This is because the main feature of modeling primary production is to accurately capture the pressure decline over time, not the movement of a saturation front.
Future of streamline simulation
The next few years are expected to bring a further maturing and extended application of streamline-based flow-simulation technology. It is reasonable to expect that most companies using conventional simulation technology today will in one form or another use SL simulation in their future work. What remains uncertain is whether new user groups, such as geologists and geophysicists, will adopt the technology to bring a dynamic flow component to their analysis. Developments in the following areas are currently under way: the use of streamlines in conditioning static reservoir models to production data, extension of streamline simulation to compositional models, tracing of streamlines in structurally complex reservoirs, modeling of dual-porosity/dual-permeability models, and parallelization of streamline numerics for the solution of large models.
|D||=||future time, time units|
|g||=||acceleration constant (length/length/time) or gas phase, depending on use|
|kr||=||relative permeability, fraction|
|ρ||=||density, mass/volume or molar density, mols/volume unless noted otherwise|
|I||=||component number, index counter, or initial condition|
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