This example demonstrates the use of pore pressure cohesive
elements and fluid pipe elements to model the initiation and opening of
hydraulically induced cracks near an oil well borehole.
With the technique illustrated in this section you can assess
the quantitative impact of the hydraulic fracture process on wellbore
productivity.
The hydraulic fracture process is commonly used in the production of oil and
natural gas reservoirs as a means of increasing well productivity and extending
the production lifetime of the reservoir. The objectives of a hydraulic
fracture treatment are
to create more surface area that is exposed to the hydrocarbon bearing
rocks, and
to provide a highly conductive pathway that allows the hydrocarbons to
flow easily to the wellbore.
The productivity of a hydraulically fractured oil or gas well is directly
related to the extent of the fracture and to how well the wellbore is connected
to the fracture. Some rock formations contain natural fracture systems that can
further increase a well’s productivity, provided that the generated hydraulic
fracture can grow such that it intersects these natural fractures.
A hydraulic fracture job consists of pumping fluids into a well at very high
pressures so that the tractions created on the well-bore face reduce the
in-situ (compressive) stress in the rock so much that the rock fractures. Once
a fracture initiates in the rock formation it is possible, given enough
hydraulic fluid, to propagate the fracture for a considerable distance,
sometimes as far as a hundred meters or more.
Pumping stages
The initial stage often involves pumping a rather small amount of polymer
laden fluid, typically 1–20 barrels (.15 to 3.2 cubic meters) so that data can
be gathered on the pressure needed to fracture the formation and the rate at
which the fluid will “leakoff” from the fracture into the pore space of the
rock. The data gathered are used to plan subsequent stages of the job. The main
stage of the job might consist of anywhere from one hundred to several
thousands of barrels of hydraulic fluid. The size of this stage is determined
by the target fracture size, the leak-off rate, and the capacity (rate) of the
pumps.
During the next stage of the fracture job, solid material, known as
proppant, is added to the injected fluid and is carried into the fracture
volume. Chemicals, typically polymers, are added to the fluid in each stage of
a fracture job to produce the necessary properties in the fluid (viscosity,
leakoff, density). In the last stage of the job, chemicals are pumped into the
fracture that help breakdown the polymers used in the previous stages and make
it easier to flow fluid back through the fracture without disrupting the
proppant material.
Geometry and model
The domain of the problem considered in this example is a 50 m (1969 in)
thick slice of oil-bearing rock with the well borehole modeled. The domain has
a diameter of 400 m (15,748 in). Three sections of rock are considered: a
region where production oil recovery is targeted, and two surrounding regions
of shale, as shown in
Figure 1.
Due to symmetry only a half of the domain is modeled.
Figure 2
shows the finite element model. The rock is modeled with C3D8RP elements, and the borehole casing is modeled with M3D4 elements.
The unopened fracture is modeled along the entire height of the model
domain. Cohesive elements (COH3D8P) are used to model a vertical fracture surface.
The flowing fluid in the wellbore is modeled using fluid pipe and fluid pipe
connector elements. The fluid pipe elements are tied to the formation to induce
hydraulic fracture in a more realistic fashion. The valve behavior is activated
on the connector element with user subroutine
UFLUIDCONNECTORVALVE to turn off flow to the formation after the pumping stage
is completed.
Rock constitutive model
A linear Drucker-Prager model with hardening is chosen for the rock, while
the casing is linear elastic.
Fracture constitutive model
The fracture model consists of the mechanical behavior of the fracture
itself and the behavior of the fluid that enters and leaks through the fracture
surfaces.
Fracture mechanical behavior
The elastic properties of the bonded interface are defined using a
traction-separation description, with stiffness values of
===
8.5 × 104 MPa. The quadratic traction-interaction failure criterion
is chosen for damage initiation in the cohesive elements; and a mixed-mode,
energy-based damage evolution law is used for damage propagation. The relevant
material data are as follows:
=
= 0.32 KPa, =
==
28 N/mm, and =
2.284.
Fluid model
Tangential and normal flow are both modeled in the fracture zone cohesive
elements. The following parameters are specified:
Gap flow is specified as Newtonian with a viscosity of 1 ×
10−6 kPaS (1 centepoise), roughly the viscosity of water.
Fluid leakoff is specified as 5.879 × 10−10 m/(kPa s) for
the early stages. In the final stage, when the polymer is dissolved, the fluid
leak-off coefficient is increased to 1 × 10−3 m/(kPa s). This
step-dependent fluid leak-off coefficient is set in user subroutine
UFLUIDLEAKOFF.
Initial conditions
An initial geostatic stress field is defined using the user subroutines
SIGINI and
UPOREP. A depth varying initial void ratio is specified using
user subroutine
VOIDRI. Gravity loading is specified; and an orthotropic
overburden stress state is imposed, with the maximum principle stress in the
formation aligned orthogonal to the cohesive element fracture plane.
Loading and boundary conditions
The analysis consists of four steps:
A geostatic step is performed where equilibrium is achieved after
applying the initial pore pressure to the formation and the initial in-situ
stresses. The bottom hole shut-in pressure is applied as a traction to the
well-bore face.
The next step represents the hydraulic fracture stage where the main
volume of fluid is being injected into the well. Flow at a rate of 2.4
m3 (15 barrels) per minute is injected along a targeted 8 m extent
in the target formation in the model, and the cohesive elements adjacent to the
wellbore along this length are defined as initially open to permit entry of
fluid. The duration of this stage is 20 minutes.
Following the hydraulic fracture, another transient soils consolidation
analysis is conducted. The injection into the well is terminated, and the
built-up pore pressure in the fracture is allowed to bleed off into the
formation. An additional boundary condition is applied at this stage, fixing
the fracture surface open to simulate the behavior of the proppant material
that was injected into the fracture.
In the final step a drawdown pressure of 20 kPa is applied to the
well-bore nodes of the fracture cohesive elements. This step ends after 100
days of production or when steady-state conditions prevail, defined as pore
pressure transients below 0.05 kPa/sec in the model.
When the fluid pipe elements are used to simulate fluid flow in the
wellbore, the loading conditions in the analysis are modified to specify the
flow as an inlet condition at the top of the wellbore. The bottom of the
wellbore is tied to the formation to enable automatic injection of the fluid.
User subroutine
UFLUIDCONNECTORVALVE is used to shut off flow to the formation during and after
the third step of the analysis as described above.
Results and discussion
The flow injected during Step 2, the pumping stage, initiated and grew a
crack extending outward from the wellbore.
Figure 3
shows the resulting geometry of the fracture at the end of the 20 minute
pumping period. These results show that a fracture initiated within the target
formation zone tends to avoid the lower shale region, where the compressive
stresses are higher, but does infiltrate the upper shale region, where it can
decrease well yields.
Figure 4
shows the fracture opening profile at various times during the pumping stage;
the final profile, shown at 1200 s, is then fixed in the subsequent steps,
holding the fracture open with the proppant material.
Figure 5
shows a similar history of the pore pressure across the fracture face and
indicates that the pore flow has stabilized.
Figure 6
shows the resulting well-bore yield following the hydraulic fracture process.
This is compared to an equivalent model where the hydraulic fracturing did not
occur. In this simple example the hydraulically fractured wellbore shows a
marked improvement, with a flow rate more than 100 times the unfractured
configuration.
Figure 1. Position of the target formation and surrounding shale. Figure 2. Near well-bore mesh. Figure 3. Fracture geometry following the injection stage, with the deformation
magnified 50 times. Figure 4. History of the fracture opening profile. Figure 5. History of the fracture pore pressure profile. Figure 6. Well yield before and after the hydraulic fracture process.
