BPOPE Model
Basic Description of the Model [1], [2]
Planar fractures of arbitrary shape are modeled using a three-dimensional boundary element method. Fracture growth is governed by linear elastic fracture mechanics. Pore pressure and thermal stress changes are coupled in this model using a three-dimensional finite difference reservoir simulation of fluid and heat flow in the region around the well. At each time step during injection, the pressure, saturation and temperature are calculated in the gridblocks of the reservoir model, using the fracture as the fluid source term. At intervals, the stress state in the plane of the fracture is calculated and the fracture size is updated so that it is in equilibrium with the new stress field. The effect of face-plugging due to suspended solids is modeled as a static filtration process. Model constants can be determined from core flooding tests. This model for formation damage due to suspended solids was found reasonable for low solid content injection.
Fracture Model
A three-dimensional boundary element method is used to relate the fracture opening and the pressure for planar fractures of arbitrary shape.1 The fracture growth criterion is based on the computed stress intensity factor and the input fracture toughness. This fracture model is a true three-dimensional hydraulic fracturing model. The fracture model is coupled with a reservoir model to calculate temperature change (and thus thermal stress) and pore pressure change (and thus poro-elastic effects) on fracture growth.
Reservoir Model
The fracture model is interfaced with a reservoir model. The reservoir model is based on a three-dimensional finite difference method for solving temperature change and pore pressure change. Saturation changes and temperature effects on water relative permeability are considered in this model. Once the temperature and pore pressure changes are obtained from the reservoir model, the stress changes due to these changes can be obtained numerically by integrating a three-dimensional integral.1 This 3-D integral may be reduced to a 2-D integral through integration by parts. This improves the numerical performance of this coupled model. Having computed the stress changes due to thermal and poroelastic effects from the reservoir model, new stresses are applied to the fracture model to update the fracture geometry.
Recent additions allow representation of multiple fractures in a deviated or horizontal well, and couple the model with a thermal wellbore simulator for calculation of the injection temperature along the well. These features are important for optimization of the injection along the wellbore.
Representation of Injected Solids [3]
Injection fines are represented by the gradual build-up of a thin layer of low permeability skin along the fracture face, either on the surface or internal. If this layer has a thickness dsk and a permeability of ksk on some part of the fracture face, the pressure drop across it in that region will be:
where:
Q/A ................................... flow rate through unit area of the fracture face and,
m ........................................ water viscosity.
The buildup of this skin (on any region of the fracture surface) is assumed to depend on the cumulative flux of injected water through that region of the fracture face. For relatively low concentrations of fine solids, it is assumed that the face plugging can be described by the following equation:
where:
C .......................................................................... dimensionless constant and,
L ...................................................................... cumulative
flux in units of length
(m3 of injected water volume per m2 of fracture area).
The constant C is determined from core-flood experiments, and depends on the water quality and the formation properties, most notably the permeability. A typical test involves injection of several thousand pore volumes of representative water into a core plug of one inch length. If the effective plug permeability is found to be reduced 50% for 500 pore volumes of injection, then one can determine the corresponding C. For seawater, C usually varies from 0.01 to 0.1.
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