Calculating shut-in bottom-hole pressure in numerical reservoir simulations
Disclosed are methods, systems, and computer-readable medium to perform operations including: generating a grid representation of a reservoir model comprising a simulation well, the grid comprising a plurality of grid-blocks, wherein the simulation well is centered in grid-block (i) of the plurality of grid-blocks; simulating, using the grid representation, grid-block pressures for the plurality of grid-blocks over a plurality of time steps; calculating an equivalent radius (ro*) for a grid-block (i+1), wherein the grid-block (i+1) is adjacent to the grid-block (i); and calculating, based on the equivalent radius and a grid-block (i+1) pressure, a shut-in bottom-hole pressure (SBHP) for the simulation well.
1. A method comprising:
generating a grid representation of a reservoir model comprising a simulation well, the grid comprising a plurality of grid-blocks, wherein the simulation well is centered in a first grid-block of the plurality of grid-blocks;
simulating, using the grid representation, grid-block pressures for the plurality of grid-blocks over a plurality of time steps;
calculating an equivalent radius (r o *) for a second grid-block adjacent to the first grid-block; and
calculating, based on the equivalent radius and a pressure of the second grid-block, a shut-in bottom-hole pressure (SBHP) for the simulation well.
2. The method of claim 1 , wherein the equivalent radius for the second grid-block is a distance from the simulation well at which actual pressure is equal to an average pressure of the second grid-block, and wherein the equivalent radius is calculated according to the equation:
r o *=0.5Δ x+r o ,
wherein Δx is a width of each of the plurality of grid-blocks and r o is a conventional equivalent radius for the second grid-block.
3. The method of claim 2 , wherein the conventional equivalent radius is a radius at which the second grid-block pressure is equal to a pressure from an analytical solution for steady-state single-phase flow in the second grid-block, and wherein the conventional equivalent radius is calculated according to the equation:
r o =0.32Δ x.
4. The method of claim 1 , wherein calculating, based on the equivalent radius and the second grid-block pressure, the SBHP for the simulation well comprises:
calculating the SBHP using the equation:
P
(
r
w
,
Δ
t
)
=
P
(
r
o
*
,
Δ
t
)
-
[
θ
(
-
E
i
(
-
α
r
o
*
2
Δ
t
)
)
-
θ
(
-
E
i
(
-
α
r
o
*
2
(
t
p
+
Δ
t
)
)
)
-
θ
(
-
E
i
(
-
α
r
w
2
Δ
t
)
)
+
θ
(
-
E
i
(
-
α
r
w
2
(
t
p
+
Δ
t
)
)
)
]
,
wherein:
−Ei(x)=−ln(1.78*x),
r w is a wellbore radius,
t p is a shut-in time,
θ
=
7
0
.
6
q
β
μ
κ
h
,
α
=
(
-
948
ϕ
μ
c
t
κ
)
,
q is an oil flow rate,
β is a formation volume factor,
μ is a viscosity of oil at reservoir conditions,
k is reservoir permeability,
h is reservoir thickness,
ϕ is reservoir porosity fraction, and
c t is total compressibility.
5. The method of claim 1 , wherein the SBHP is validated by an analytically derived SBHP, the analytically derived SBHP calculated using the equation:
P
i
-
P
(
r
w
,
Δ
t
)
=
7
0
.
6
q
β
μ
κ
h
Ln
(
t
p
+
Δ
t
Δ
t
)
.
6. The method of claim 1 , further comprising:
comparing the SBHP to actual shut-in gauge pressures from a reservoir corresponding to the reservoir model; and
based on the comparison, calibrating the reservoir model.
7. A device comprising:
one or more processors; and
a non-transitory computer-readable storage medium coupled to the one or more processors and storing programming instructions for execution by the one or more processors, the programming instructions instructing the one or more processors to perform operations comprising:
generating a grid representation of a reservoir model comprising a simulation well, the grid comprising a plurality of grid-blocks, wherein the simulation well is centered in a first grid-block of the plurality of grid-blocks;
simulating, using the grid representation, grid-block pressures for the plurality of grid-blocks over a plurality of time steps;
calculating an equivalent radius (r o *) for a second grid-block adjacent to the first grid-block; and
calculating, based on the equivalent radius and a pressure of the second grid-block, a shut-in bottom-hole pressure (SBHP) for the simulation well.
8. The device of claim 7 , wherein the equivalent radius for the second grid-block is a distance from the simulation well at which actual pressure is equal to an average pressure of the second grid-block, and wherein the equivalent radius is calculated according to the equation:
r o *=0.5Δ x+r o ,
wherein Δx is a width of each of the plurality of grid-blocks and r o is a conventional equivalent radius for the second grid-block.
9. The device of claim 8 , wherein the conventional equivalent radius is a radius at which the second grid-block pressure is equal to a pressure from an analytical solution for steady-state single-phase flow in the second grid-block, and wherein the conventional equivalent radius is calculated according to the equation:
r o =0.32Δ x.
10. The device of claim 7 , wherein calculating, based on the equivalent radius and the second grid-block pressure, the SBHP for the simulation well comprises:
calculating the SBHP using the equation:
P
(
r
w
,
Δ
t
)
=
P
(
r
o
*
,
Δ
t
)
-
[
θ
(
-
E
i
(
-
α
r
o
*
2
Δ
t
)
)
-
θ
(
-
E
i
(
-
α
r
o
*
2
(
t
p
+
Δ
t
)
)
)
-
θ
(
-
E
i
(
-
α
r
w
2
Δ
t
)
)
+
θ
(
-
E
i
(
-
α
r
w
2
(
t
p
+
Δ
t
)
)
)
]
,
wherein:
−Ei(x)=−ln(1.78*x),
r w is a wellbore radius,
t p is a shut-in time,
θ
=
7
0
.
6
q
β
μ
κ
h
,
α
=
(
-
948
ϕ
μ
c
t
κ
)
,
q is an oil flow rate,
β is a formation volume factor,
μ is a viscosity of oil at reservoir conditions,
k is reservoir permeability,
h is reservoir thickness,
ϕ is reservoir porosity fraction, and
c t is total compressibility.
11. The device of claim 7 , wherein the SBHP is validated by an analytically derived SBHP, the analytically derived SBHP calculated using the equation:
P
i
-
P
(
r
w
,
Δ
t
)
=
7
0
.
6
q
β
μ
κ
h
Ln
(
t
p
+
Δ
t
Δ
t
)
.
12. The device of claim 7 , wherein the operations further comprise:
comparing the SBHP to actual shut-in gauge pressures from a reservoir corresponding to the reservoir model; and
based on the comparison, calibrating the reservoir model.
13. A non-transitory computer-readable medium storing instructions executable by a computer system to perform operations comprising:
generating a grid representation of a reservoir model comprising a simulation well, the grid comprising a plurality of grid-blocks, wherein the simulation well is centered in a first grid-block of the plurality of grid-blocks;
simulating, using the grid representation, grid-block pressures for the plurality of grid-blocks over a plurality of time steps;
calculating an equivalent radius (r o *) for a second grid-block adjacent to the first grid-block; and
calculating, based on the equivalent radius and a pressure of the second grid-block, a shut-in bottom-hole pressure (SBHP) for the simulation well.
14. The non-transitory computer-readable medium of claim 13 , wherein the equivalent radius for the second grid-block is a distance from the simulation well at which actual pressure is equal to an average pressure of the second grid-block, and wherein the equivalent radius is calculated according to the equation:
r o *=0.5Δ x+r o ,
wherein Δx is a width of each of the plurality of grid-blocks and r o is a conventional equivalent radius for the second grid-block.
15. The non-transitory computer-readable medium of claim 14 , wherein the conventional equivalent radius is a radius at which the second grid-block pressure is equal to a pressure from an analytical solution for steady-state single-phase flow in the second grid-block, and wherein the conventional equivalent radius is calculated according to the equation:
r o =0.32Δ x.
16. The non-transitory computer-readable medium of claim 13 , calculating, based on the equivalent radius and the second grid-block pressure, the SBHP for the simulation well comprises:
calculating the SBHP using the equation:
P
(
r
w
,
Δ
t
)
=
P
(
r
o
*
,
Δ
t
)
-
[
θ
(
-
E
i
(
-
α
r
o
*
2
Δ
t
)
)
-
θ
(
-
E
i
(
-
α
r
o
*
2
(
t
p
+
Δ
t
)
)
)
-
θ
(
-
E
i
(
-
α
r
w
2
Δ
t
)
)
+
θ
(
-
E
i
(
-
α
r
w
2
(
t
p
+
Δ
t
)
)
)
]
,
wherein:
−Ei(x)=−ln(1.78*x),
r w is a wellbore radius,
t p is a shut-in time,
θ
=
7
0
.
6
q
β
μ
κ
h
,
α
=
(
-
948
ϕ
μ
c
t
κ
)
,
q is an oil flow rate,
β is a formation volume factor,
μ is a viscosity of oil at reservoir conditions,
k is reservoir permeability,
h is reservoir thickness,
ϕ is reservoir porosity fraction, and
c t is total compressibility.
17. The non-transitory computer-readable medium of claim 13 , wherein the SBHP is validated by an analytically derived SBHP, the analytically derived SBHP calculated using the equation:
P
i
-
P
(
r
w
,
Δ
t
)
=
7
0
.
6
q
β
μ
κ
h
Ln
(
t
p
+
Δ
t
Δ
t
)
.