Accurate numerical modeling of evolving saturated/unsaturated interfaces is very difficult with traditional techniques due to numerical dispersion that translates into artificial smoothing of the saturation fronts.
Traditional Finite Element and Finite Difference approaches resort to
a finer mesh/grid in a neighborhood of the saturation interface to control
numerical diffusion (artificial spreading) of interfaces. This approach
is effective in limited cases, because very often the interfaces move across
the simulation domain, and dynamic mesh/grid adaptation is difficult. In
many cases the size of the problem or the characteristic size of the mesh/grid
is such that too many levels of local grid refinement or mesh adaptation
would be required to reduce artificial diffusion effects to an acceptable
level. For all of these reasons, the Generalized FEM is a better approach
to the solution of saturated/unsaturated interface problems.
A rock core 100 ft high is subject to 100 psig of fluid pressure at the bottom. Initially only the lower 10% of the rock core is saturated with fluid. Specific gravity of the fluid 0.5 psi/ft. See full description of the problem and analytic solution.
The numerical technique used to simulate near-wellbore flows is a special form of the GFEM with special basis functions. No productivity index or any other well model is used; the near-wellbore pressure is modeled by the use of special basis functions. This technique is very accurate yet inexpensive, because higher accuracy is obtained without resorting to fine grids around wells. For this reason, this technique is very well suited to simulate wells that are turned on/off frequently. When wells come online or are shut off, the wells are automatically added/removed from the model. This translates in net saving at no cost, because the GFEM does not incur in the extra cost of having a fine mesh where is not needed.
Conventional reservoir simulators based on finite difference methods
(FD) have difficulties to simulate the performance of wells that are not
aligned with the directions of the underlying computational grid. Even
for wells aligned with the grid, many reservoir simulators produce bad
approximations to the wells' bottom hole pressure (BHP) when reservoir
properties are not uniform, when well blocks have aspect ratios outside
the range of applicability of the well model, or when other underlying
assumptions of the well model are not satisfied. On the other hand, simulators
that circumvent the use of analytical or numerical well models by using
local grid refinements aligned with the well trajectories (Finite Volume
and Finite Element methods) have serious difficulties to simulate reservoirs
with a large number of wells. This is due to grid complexity and the increased
number of grid blocks.
Many techniques have been developed to relate the flowing BHP to the
well block pressure for a number of FD approximations on orthogonal grids.
The technique developed by Peaceman is one of the best applications of
numerical postprocessing to extract BHPs from well blocks pressures obtained
with specific FD stencils on orthogonal grids of fixed aspect ratio. An
important factor in the use of post processing techniques based on numerical
or analytical well models is their range of applicability. They can be
safely applied (and in fact produce very good results) when all the necessary
conditions are satisfied, namely, specific FD stencil on a
specific type of grid, well orientation with respect to the gird, uniform
reservoir properties (anisotropy is allowed in some
cases), quasi steady state assumptions, etc. The GFEM is capable of
finding the best approximation to problems characterized by singularities
of known type, regardless of the underlying grid (orthogonal, fully unstructured,
etc.), anisotropy and non uniformity of
reservoir properties, well trajectory, adjacency of wells to reservoir
boundaries, time dependent conditions, etc.
Steam injection is used in the Kern River field (California) to increase
the temperature of the reservoir rock and thus lower the oil
viscosity and facilitate pumping/production. A strong aquifer acts
on the southwest corner of the field and is the source of approximately
300,000 BWPD (barrels of water per day) moving updip towards the production
sector of the reservoir, with the consequent cooling effect. One of the
goals of this simulation is to predict the efficiency of interdiction wells
in reducing the influx of
aquifer water into the producing sector of the field.
In 1996, the average combined production of 4900 wells was approximately
700,000 BPD, and the average steam injection with 1250 wells was approx.
150,000 BWPD. Production and injection performance for each well in the
field is available on a monthly basis. A water interdiction strategy consisting
of 17 wells with a combined total production of 410,000 BPD was designed
to reduce the influx of water into the producing area.
The Kern River field model contains 6196 wells.
Aquifer interdiction wells are shown with higher well markers. A
global view of the underlying mesh. Even though this mesh consist of
122145 tetrahedral elements, this is a very coarse mesh for the number
of wells. Observe the large number of wells
contained areally in a single base mesh element. Steam injector are
displayed blue, producer are displayed red.
Using the reservoir data and production history of 1996, the simulation study was meant to forecast the efficiency of the interdiction wells and also provide insights for future interdiction strategies. Reservoir properties are available in GridStat format with a resolution of 44x56x341 cells. Data for each cell included: temperature every 3 months, oil saturation, and absolute permeability.
Pressure field on a horizontal plane
after the 17 water interdiction wells start to produce a total of 410,000
BWPD. This produces a very large pressure drawdown in the vicinity of the
interdiction wells, but still a considerable amount of aquifer water reaches
the producing area through the gap left in the central area of interdiction.
Water table and pressure isosurfaces
in
a neighborhood of the interdiction wells. Isosurfaces
of pressure around the interdiction wells and pressure on a cutting plane.
Well defined water table cones around the interdiction wells show qualitatively
the advantage of using the GFEM for saturated/unsaturated interfaces.
The CPU time required for the simulations described above is approximately 1 hour (PII 400MHz) per year of simulated time. In other words, to predict the interdiction efficiency during the first year takes 1 hour of CPU time.
Acknowledgment: We thank the management and technical personnel of Texaco
California Business Unit and Texaco Upstream Technology for making available
the reservoir description and wells production history to demonstrate the
efficiency of the GFEM for the simulation of water tables.
The GFEM presented is capable of resolving saturation interfaces without artificial diffusion, even when the interface is arbitrarily located on the underlying grid/mesh. To achieve simulation results of comparable quality with the classical FEM/FVM would require simulation grids with extremely small elements. Such meshes would be very difficult to generate and would include a very large number of degrees of freedom. If adaptive refinement is used, many levels of local refinement would be required to capture the saturation interface, and this would also translate into a very large number of degrees of freedom and discretization errors associated with mesh gradation.
An accurate solution of the pressure equation is the foundation of almost
every oil reservoir simulator. The GFEM technique produces accurate pressure
distributions without mesh clustering around the wells, and without resorting
to simplifying assumptions such as those necessary to safely apply productivity
indexes. This well model is in fact a new discretisation technique that
takes into account the relevant parameters that characterize the near-wellbore
pressure distribution.
The aquifer interdiction simulation of an oil field model (Kern River
field, Texaco) with more than 6000 wells is an example of the use of this
new technique. The GFEM well model is very easy to use because the computational
mesh does not have to represent the wells. Wells of any type (vertical,
horizontal) can be added to the oil reservoir model just by defining the
well's trajectory and pertinent information such as diameter, completion,
production, etc. Comparisons with analytical results validated the accuracy
of this technique.
In short, for the simulation of fluid saturation interface applications
the GFEM is superior to the FEM and FVM, both in terms of solution accuracy
and CPU time. For the same accuracy, the GFEM requires between one and
two orders of magnitude less CPU time than classical FEM/FVM techniques.
