Directional permeability upscaling of a discrete fracture network
A method for estimating an upscaled directional permeability of a formation includes: receiving a discrete fracture network (DFN), aligning the DFN in a desired direction; cropping the DFN to a desired size having boundaries. The method further includes identifying a fracture plane or fracture planes that are connected with each other and establish a path between the boundaries of the cropped DFN and creating a pipe model of the identified fracture plane or planes that includes one or more conduits connecting one boundary to another boundary. The method further includes creating a system of equations representing flows through the conduits, applying boundary conditions to the system of equations; solving the system of equations for steady-state flow to estimate the upscaled directional permeability of the formation. The method steps can be iterated for another desired direction if the upscaled directional permeability is wanted for the another desired direction.
1. A method for estimating an upscaled directional permeability of an earth formation, the method comprising:
receiving a discrete fracture network (DFN) representing the earth formation, the DFN comprising a plurality of rock fractures and a location, size, orientation and aperture of each of the rock fractures;
aligning the DFN in a desired orientation or direction;
cropping the DFN to a desired size having boundaries;
identifying a fracture plane or fracture planes that are connected with each other and establish a path between the boundaries of the cropped DFN;
creating a pipe model of the identified fracture or fractures, the pipe model comprising a node at an intersection of fracture planes and a conduit connecting two nodes together such that one or more conduits alone or in combination establish the path between the boundaries of the cropped DFN, the pipe model further comprising a node at each conduit intersecting a boundary;
creating a system of equations representing flows through the conduits;
applying boundary conditions to the system of equations;
solving the system of equations for steady-state flow to estimate the upscaled directional permeability of the formation;
iterating the aligning, cropping, identifying, creating a pipe model, creating a system of equations, applying and solving for another desired orientation or direction in response to a signal indicating the upscaled directional permeability is to be estimated for the another desired orientation or direction;
performing a borehole and/or formation-related action using the estimated upscaled directional permeability of the formation and associated action-equipment;
wherein the aligning, cropping, identifying, creating a pipe model, creating a system of equations, applying, solving and iterating are performed using a processor.
2. The method according to claim 1 , wherein cropping comprises cropping the DFN to a selected area or volume having at least four intersecting boundary lines.
3. The method according to claim 1 , wherein the node is placed at a midpoint along a complete length of the intersection of the fracture planes.
4. The method according to claim 1 , wherein the system of equations models fluid flow through a slot formed between two parallel plates separated by an aperture distance a.
5. The method according to claim 4 , wherein the system of equations comprises the following equation:
Q
=
1
12
μ
(
p
1
-
p
2
)
l
a
3
where μ is the fluid viscosity, p 1 and p 2 are the pressures on either side or end of the slot defined by the parallel plates, l is the length of the plates in flow direction, and a is the distance between the two plates.
6. The method according to claim 1 , wherein applying boundary conditions comprises setting a pressure at a node.
7. The method according to claim 1 , wherein a node at a boundary comprises one or more nodes at one boundary and one or more nodes at another boundary.
8. The method according to claim 1 , wherein solving the system of equations for steady-state flow comprises providing a flow rate through each of the conduits in the pipe model and a pressure at each node.
9. The method according to claim 8 , wherein solving the system of equations for steady-state flow to estimate the upscaled directional permeability of the formation further comprises using the following equation for the cropped DFN:
κ
=
∑
Q
i
L
y
μ
Δ
P
/
L
x
where Q i is the volume flux of conduit i that intersects with the boundary along L y , μ is the fluid viscosity, and ΔP is the pressure difference across L x where L y and L x are boundary lengths.
10. The method according to claim 1 , wherein the upscaled directional permeability comprises a plurality of upscaled directional permeability values for a plurality of orientations or directions.
11. The method according to claim 10 , further comprising plotting a polar plot illustrating the plurality of upscaled directional permeability values for a plurality of orientations or directions.
12. The method according to claim 1 , wherein the borehole and/or formation-related action comprises controlling a flow rate of fluids extracted from a borehole using a controller.
13. The method according to claim 1 , wherein the borehole and/or formation-related action comprises stimulating the formation using a reservoir stimulation system.
14. The method according to claim 1 , wherein the borehole and/or formation-related action comprises drilling a borehole with a trajectory that leads to a location in the formation having a maximum directional permeability value using a drill rig.
15. A system for estimating an upscaled directional permeability of a formation, the system comprising:
a memory having computer-readable instructions;
a processor for executing the computer-readable instructions, the computer-readable instructions comprising:
receiving a discrete fracture network (DFN), the DFN a plurality of rock fractures and a location, size, orientation and aperture of each of the rock fractures;
aligning the DFN in a desired orientation or direction;
cropping the DFN to a desired size having boundaries;
identifying a fracture plane or fracture planes that are connected with each other and establish a path between the boundaries of the cropped DFN;
creating a pipe model of the identified fracture or fractures, the pipe model comprising a node at an intersection of fracture planes and a conduit connecting two nodes together such that one or more conduits alone or in combination establish the path between the boundaries of the cropped DFN, the pipe model further comprising a node at each conduit intersecting a boundary;
creating a system of equations representing flows through the conduits;
applying boundary conditions to the system of equations;
solving the system of equations for steady-state flow to estimate the upscaled directional permeability of the formation; and
iterating the aligning, cropping, identifying, creating a pipe model, creating a system of equations, applying and solving for another desired orientation or direction in response to a signal indicating the upscaled directional permeability is to be estimated for the another desired orientation or direction;
equipment configured to perform a borehole and/or formation-related action using the estimated upscaled directional permeability of the formation.
16. The system according to claim 15 , wherein the upscaled directional permeability comprises a plurality of upscaled directional permeability values for a plurality of orientations or directions.
17. The system according to claim 16 , wherein the computer-readable instructions further comprise plotting a polar plot illustrating the plurality of upscaled directional permeability values for a plurality of orientations or directions.
18. The system according to claim 15 , wherein the borehole and/or formation-related action comprises controlling a flow rate of fluids extracted from a borehole and the equipment comprises a controller.
19. The system according to claim 15 , wherein the borehole and/or formation-related action comprises stimulating the formation and the equipment comprises a reservoir stimulation system.
20. The system according to claim 15 , wherein the borehole and/or formation-related action comprises drilling a borehole with a trajectory that leads to a location in the formation having a maximum directional permeability value and the equipment comprises a drill rig.