Three (3) dimensional one-phase fluid (oil) reservoir simulator using a partially implicit method with lagging coefficients. Skin damage after drilling the production well was expected; the impact of the skin damage and the necessity of well stimulation was assessed

1. PTE 7397 Midterm Project Report
Jeffrey Daniels
Tuesday 30th
March, 2021
1 Introduction
In this study, an operating company is working on a three (3) dimensional single phase
oil reservoir. Skin damage after drilling the well is expected and the company wants to
verify the impact of the skin damage and the necessity of well stimulation. Therefore,
this simulation study seeks to forecast oil production in three cases:
a. Standard case (skin = 0)
b. Damaged well case (skin = 3)
c. Stimulated well case (skin = -1)
2 Reservoir Description
2.1 Reservoir Properties
The oil reservoir under study is approximated by a rectangular prism The geometric
measurements of this prism is length Lx = 450 ft, width Ly = 450 ft and height
Lz = 150 ft. The reservoir is assumed to have a constant porosity of φ = 0.25, a hori-
zontal permeability in both the x and y-directions of kx = ky = 70 mD and a vertical
permeability of kz = 7 mD. The formation compressibility factor of cr = 3x10−6
psia−1
is assumed to be constant.
2.2 Production Well
One well is drilled in the reservoir first, and its location on the grid is at xw = 435 ft,
yw = 15 ft, zw = 6900 ft. The well serves as a producer in this reservoir simulation
study and partially penetrates the reservoir. The penetration thickness is zt = 30 ft.
It has a wellbore radius of rw = 0.35 ft. The well is produced at a constant pressure of
Pwf = 2700 psia
2.3 Reservoir Fluid Properties
The fluid flow is one-phase which is oil. At a reference pressure of 3000 psia, the fluid
properties of oil are viscosity µo
= 0.99 cp, oil compressibility co = 10−5
psia−1
, forma-
tion volume factor Bo
= 1 rcf/scf and density ρo
= 45 lbm
ft3 .
1

2. The oil properties are defined as:
Density (lbm/ft3
) : ρ(P) = 45 exp[Co(P − 3000)]
Formation Volume Factor (rcf/scf) : B(P) = 1.0 exp[−Co(P − 3000)]
Viscosity (cP) : µ(P) = 6.00961538 ∗ 10−9
P2
− 9.13324176 ∗ 10−5
P + 1.21
3 Creation of Simulation Grid Blocks
In this study, the reservoir is three dimensional and was discretized into 1125 simula-
tion cells (i.e characterized by 15x15x5 grid blocks). Each grid block has a length of
∆X = 30 ft, a width of ∆Y = 30 ft and a height of ∆Z = 30 ft. The volume of a grid
block is 27000 ft3
.
4 Partial Differential Equations (PDEs) With Initial and Bound-
ary Conditions
We start with the oil material balance equation which is as follows:
Rate of mass accumulation = Net rate of mass flow in − Sink
In mathematical terms this is expressed with a partial differential equation (PDE) as
follows:
∂(φρ)
∂t
= ∇[
ρ
µ
k(∇P + ρg∇Z)] − Q
The density and formation volume factor at reference pressure (Po
) are defined as ρo
and Bo
respectively. Hence density can be computed at any pressure as follows:
ρ =
ρo
Bo
B
Substituting this into the second equation listed, we obtain
∂
∂t
(
φ
B
) = ∇[
k
Bµ
(∇P + ρg∇Z)] −
Q
ρoBo
where
Mobility in x,y and z directions : λx =
kx
Bµ
λy =
ky
Bµ
λz =
kz
Bµ
Flow Rate : e
q =
Q
ρoBo
Pressure gradient : γ = ρg however for this project γ(psi/ft) = ρ/144
2

3. Therefore in terms of mobility and pressure gradient our partial differential equation
becomes
∂
∂t
(
φ
B
) = ∇[λ(∇P + γ∇Z)] − e
q
Since we are using a three (3) dimensional simulation grid, the PDE to be discretized is
as follows:
∂
∂t
(
φ
B
) =
∂
∂x
[λx(
∂P
∂x
+ γx
∂z
∂x
)] +
∂
∂y
[λy(
∂P
∂y
+ γy
∂z
∂y
)] +
∂
∂z
[λz(
∂P
∂z
+ γz
∂z
∂z
)] − e
q
We have no flow boundary conditions which are mathematically represented as follows:
∂P
∂x
|x=0ft = 0, ∂P
∂x
|x=450ft = 0
∂P
∂y
|y=0ft = 0, ∂P
∂y
|y=450ft = 0
∂P
∂y
|z=6825ft = 0, ∂P
∂z
|z=6975ft = 0
Initial condition: P(z = 6900ft)|t=0 = 3000 psia
5 Implicit Block-Centered Grid System Discretization with
Lagging Coefficients
We will derive the finite difference equation for the lefthand side (LHS) and right hand
side (RHS) of the PDE separately, starting with the RHS which deals with spatial dis-
cretization. The Finite Difference Equation (FDE) for the PDE derived is as follows:
∂
∂t
(
φ
B
)|i,j,k =
λn
i+1,j,k
∆x
(
Pn+1
i+1,j,k − Pn+1
i,j,k
∆xi+1,j,k
+ γn
i+1,j,k
zi+1,j,k − zi,j,k
∆xi+1,j,k
)
+
λn
i−1,j,k
∆x
(
Pn+1
i−1,j,k − Pn+1
i,j,k
∆xi−1,j,k
+ γn
i−1,j,k
zi−1,j,k − zi,j,k
∆xi−1,j,k
)
+
λn
i,j+1,k
∆y
(
Pn+1
i,j+1,k − Pn+1
i,j,k
∆yi,j+1,k
+ γn
i,j+1,k
zi,j+1,k − zi,j,k
∆yi,j+1,k
)
+
λn
i,j−1,k
∆y
(
Pn+1
i,j−1,k − Pn+1
i,j,k
∆yi,j−1,k
+ γn
i,j−1,k
zi,j−1,k − zi,j,k
∆yi,j−1,k
)
+
λn
i,j,k+1
∆z
(
Pn+1
i,j,k+1 − Pn+1
i,j,k
∆zi,j,k+1
+ γn
i,j,k+1
zi,j,k+1 − zi,j,k
∆zi,j,k+1
)
+
λn
i,j,k−1
∆z
(
Pn+1
i,j,k−1 − Pn+1
i,j,k
∆zi,j,k−1
+ γn
i,j,k−1
zi,j,k−1 − zi,j,k
∆zi,j,k−1
) − e
qi,j,k
We then multiply each side by the volume of a grid block, Vi,j,k, where
3

4. Volume of each grid block (ft3
) : Vi,j,k = ∆x∆y∆z
Flow rate : qi,j,k = e
qi,j,kVi,j,k
The FDE then becomes
Vi,j,k
∂
∂t
(
φ
B
)|i,j,k = ∆y∆zλn
i+1,j,k(
Pn+1
i+1,j,k − Pn+1
i,j,k
∆xi+1,j,k
+ γn
i+1,j,k
zi+1,j,k − zi,j,k
∆xi+1,j,k
)
+ ∆y∆zλn
i−1,j,k(
Pn+1
i−1,j,k − Pn+1
i,j,k
∆xi−1,j,k
+ γn
i−1,j,k
zi−1,j,k − zi,j,k
∆xi−1,j,k
)
+ ∆x∆zλn
i,j+1,k(
Pn+1
i,j+1,k − Pn+1
i,j,k
∆yi,j+1,k
+ γn
i,j+1,k
zi,j+1,k − zi,j,k
∆xi,j+1,k
)
+ ∆x∆zλn
i,j−1,k(
Pn+1
i,j−1,k − Pn+1
i,j,k
∆yi,j−1,k
+ γn
i,j−1,k
zi,j−1,k − zi,j,k
∆yi,j−1,k
)
+ ∆x∆yλn
i,j,k+1(
Pn+1
i,j,k+1 − Pn+1
i,j,k
∆zi,j,k+1
+ γn
i,j,k+1
zi,j,k+1 − zi,j,k
∆zi,j,k+1
)
+ ∆x∆yλn
i,j,k−1(
Pn+1
i,j,k−1 − Pn+1
i,j,k
∆zi,j,k−1
+ γn
i,j,k−1
zi,j,k−1 − zi,j,k
∆zi,j,k−1
) − qi,j,k
We will now define our total transmissibility term, T. The total transmissibility is made
up of two terms; geometric transmissibility (Tg) and flow transmissibility (Tf ). This is
expressed as follows:
T = 0.00633TgTf
1. In x-direction
Ti+1,j,k = 0.00633
∆y∆z
∆xi+1,j,k
λi+1,j,k
Ti−1,j,k = 0.00633
∆y∆z
∆xi−1,j,k
λi−1,j,k
2. In y-direction
Ti,j+1,k = 0.00633
∆x∆z
∆yi,j+1,k
λi,j+1,k
Ti,j−1,k = 0.00633
∆x∆z
∆yi,j−1,k
λi,j−1,k
3. In z-direction
Ti,j,k+1 = 0.00633
∆x∆y
∆zi,j,k+1
λi,j,k+1
4

5. Ti,j,k−1 = 0.00633
∆x∆y
∆zi,j,k−1
λi,j,k−1
The flow transmissibility term is defined by the mobility (λ) while the geometric term
is defined by the other grid block dimensions.
The harmonic average of the flow transmissibility for each grid block is used, in order
to obtain more accurate results. This is given by:
1. In x- direction
Tf i+1,j,k =
Tf i+1,j,kTf i,j,k(
∆xi,j,k
2
+
∆xi+1,j,k
2
)
Tf i+1,j,k
∆xi,j,k
2
+
Tf i,j,k
∆xi+1,j,k
2
Tf i+1,j,k =
Tf i+1,j,kTf i,j,k(∆xi,j,k + ∆xi+1,j,k)
Tf i+1,j,k∆xi,j,k + Tf i,j,k∆xi+1,j,k
Tf i−1,j,k =
Tf i−1,j,kTf i,j,k(
∆xi,j,k
2
+
∆xi−1,j,k
2
)
Tf i−1,j,k
∆xi,j,k
2
+
Tf i,j,k
∆xi−1,j,k
2
Tf i−1,j,k =
Tf i−1,j,kTf i,j,k(∆xi,j,k + ∆xi−1,j,k)
Tf i−1,j,k∆xi,j,k + Tf i,j,k∆xi−1,j,k
2. In y-direction
Tf i,j+1,k =
Tf i,j+1,kTf i,j,k(
∆yi,j,k
2
+
∆yi,j+1,k
2
)
Tf i,j+1,k
∆yi,j,k
2
+
Tf i,j,k
∆yi,j+1,k
2
Tf i,j+1,k =
Tf i,j+1,kTf i,j,k(∆yi,j,k + ∆yi,j+1,k)
Tf i,j+1,k∆yi,j,k + Tf i,j,k∆yi,j+1,k
Tf i,j−1,k =
Tf i,j−1,kTf i,j,k(
∆yi,j,k
2
+
∆yi,j−1,k
2
)
Tf i,j−1,k
∆yi,j,k
2
+
Tf i,j,k
∆yi,j−1,k
2
Tf i,j−1,k =
Tf i,j−1,kTf i,j,k(∆yi,j,k + ∆yi,j−1,k)
Tf i,j−1,k∆yi,j,k + Tf i,j,k∆yi,j−1,k
3. In z-direction
Tf i,j,k+1 =
Tf i,j,k+1Tf i,j,k(
∆zi,j,k
2
+
∆zi,j,k+1
2
)
Tf i,j,k+1
∆zi,j,k
2
+
Tf i,j,k
∆zi,j,k+1
2
Tf i,j,k+1 =
Tf i,j,k+1Tf i,j,k(∆zi,j,k + ∆zi,j,k+1)
Tf i,j,k+1∆zi,j,k + Tf i,j,k∆zi,j,k+1
5

6. Tf i,j,k−1 =
Tf i,j,k−1Tf i,j,k(
∆zi,j,k
2
+
∆zi,j,k−1
2
)
Tf i,j,k−1
∆zi,j,k
2
+
Tf i,j,k
∆zi,j,k−1
2
Tf i,j,k−1 =
Tf i,j,k−1Tf i,j,k(∆zi,j,k + ∆zi,j,k−1)
Tf i,j,k−1∆zi,j,k + Tf i,j,k∆zi,j,k−1
So now we define the RHS of our FDE in terms of transmissibility which becomes:
Vi,j,k
∂
∂t
(
φ
B
)|i,j,k = Tn
i+1,j,k(Pn+1
i+1,j,k − Pn+1
i,j,k ) + γn
i+1,j,kTn
i+1,j,k(zi+1,j,k − zi,j,k)
+ Tn
i−1,j,k(Pn+1
i−1,j,k − Pn+1
i,j,k ) + γn
i−1,j,kTn
i−1,j,k(zi−1,j,k − zi,j,k)
+ Tn
i,j+1,k(Pn+1
i,j+1,k − Pn+1
i,j,k ) + γn
i,j+1,kTn
i,j+1,k(zi,j+1,k − zi,j,k)
+ Tn
i,j−1,k(Pn+1
i,j−1,k − Pn+1
i,j,k ) + γn
i,j−1,kTn
i,j−1,k(zi,j−1,k − zi,j,k)
+ Tn
i,j,k+1(Pn+1
i,j,k+1 − Pn+1
i,j,k ) + γn
i,j,k+1Tn
i,j,k+1(zi,j,k+1 − zi,j,k)
+ Tn
i,j,k−1(Pn+1
i,j,k−1 − Pn+1
i,j,k ) + γn
i,j,k−1Tn
i,j,k−1(zi,j,k−1 − zi,j,k) − qi,j,k
Now we tackle the LHS of the FDE which deals with temporal discretization. Since oil
is a slightly compressible fluid, the rate of accumulation term containing our formation
volume factor and porosity becomes
φ
B
=
Bo
exp[−Co(P − Po
)]
φoexp(−Cr(P − Po)]
=
φo
Bo
exp[Cr(P − Po
) + Co(P − Po
)]
Using a taylor expansion series, the above equation simplifies to
φ
B
=
φo
Bo
exp[1 + Cr(P − Po
) + Co(P − Po
)]
∂
∂t
(
φ
B
) =
φo
Bo
(Cr
∂P
∂t
+ Co
∂P
∂t
) =
φo
Bo
(Ct
∂P
∂t
)
Now our FDE becomes
6

7. Vi,j,k
φo
Bo
(Ct
∂P
∂t
) = Tn
i+1,j,k(Pn+1
i+1,j,k − Pn+1
i,j,k ) + γn
i+1,j,kTn
i+1,j,k(zi+1,j,k − zi,j,k)
+ Tn
i−1,j,k(Pn+1
i−1,j,k − Pn+1
i,j,k ) + γn
i−1,j,kTn
i−1,j,k(zi−1,j,k − zi,j,k)
+ Tn
i,j+1,k(Pn+1
i,j+1,k − Pn+1
i,j,k ) + γn
i,j+1,kTn
i,j+1,k(zi,j+1,k − zi,j,k)
+ Tn
i,j−1,k(Pn+1
i,j−1,k − Pn+1
i,j,k ) + γn
i,j−1,kTn
i,j−1,k(zi,j−1,k − zi,j,k)
+ Tn
i,j,k+1(Pn+1
i,j,k+1 − Pn+1
i,j,k ) + γn
i,j,k+1Tn
i,j,k+1(zi,j,k+1 − zi,j,k)
+ Tn
i,j,k−1(Pn+1
i,j,k−1 − Pn+1
i,j,k ) + γn
i,j,k−1Tn
i,j,k−1(zi,j,k−1 − zi,j,k) − qi,j,k
Now the LHS is ready to be discretized which becomes
Vi,j,k
φo
Bo
(Ct
Pn+1
i,j,k − Pn
i,j,k
∆t
) = Tn
i+1,j,k(Pn+1
i+1,j,k − Pn+1
i,j,k ) + γn
i+1,j,kTn
i+1,j,k(zi+1,j,k − zi,j,k)
+ Tn
i−1,j,k(Pn+1
i−1,j,k − Pn+1
i,j,k ) + γn
i−1,j,kTn
i−1,j,k(zi−1,j,k − zi,j,k)
+ Tn
i,j+1,k(Pn+1
i,j+1,k − Pn+1
i,j,k ) + γn
i,j+1,kTn
i,j+1,k(zi,j+1,k − zi,j,k)
+ Tn
i,j−1,k(Pn+1
i,j−1,k − Pn+1
i,j,k ) + γn
i,j−1,kTn
i,j−1,k(zi,j−1,k − zi,j,k)
+ Tn
i,j,k+1(Pn+1
i,j,k+1 − Pn+1
i,j,k ) + γn
i,j,k+1Tn
i,j,k+1(zi,j,k+1 − zi,j,k)
+ Tn
i,j,k−1(Pn+1
i,j,k−1 − Pn+1
i,j,k ) + γn
i,j,k−1Tn
i,j,k−1(zi,j,k−1 − zi,j,k) − qi,j,k
In the x and y direction, ∆Z = 0, therefore
Vi,j,k
φo
Bo
(Ct
Pn+1
i,j,k − Pn
i,j,k
∆t
) = Tn
i+1,j,k(Pn+1
i+1,j,k − Pn+1
i,j,k ) + Tn
i−1,j,k(Pn+1
i−1,j,k − Pn+1
i,j,k )
+ Tn
i,j+1,k(Pn+1
i,j+1,k − Pn+1
i,j,k ) + Tn
i,j−1,k(Pn+1
i,j−1,k − Pn+1
i,j,k )
+ Tn
i,j,k+1(Pn+1
i,j,k+1 − Pn+1
i,j,k ) + γn
i,j,k+1Tn
i,j,k+1(zi,j,k+1 − zi,j,k)
+ Tn
i,j,k−1(Pn+1
i,j,k−1 − Pn+1
i,j,k ) + γn
i,j,k−1Tn
i,j,k−1(zi,j,k−1 − zi,j,k) − qi,j,k
where
Dn
= Vi,j,k
φo
Bo
Ct
Gn
= Σγn
Tn
∆Z
For the pressure gradient for each block we use an arithmetic average for more accurate
results. Therefore
γi,j,k−1 =
γi,j,k−1 + γi,j,k
2
γi,j,k+1 =
γi,j,k+1 + γi,j,k
2
The final general form of our FDE with the lagging coefficients becomes
ΣTn
(∆Pn+1
) = Dn
Pn+1
i,j,k − Pn
i,j,k
∆t
+ Gn
− qi,j,k
7

8. The lagging coefficients are Dn
, Tn
and Gn
because they are computed using the cur-
rent pressure values (Pn
) at a particular time step and serve as coefficients for calculat-
ing our unknown pressure values (Pn+1
). In this study we consider Dn
to be constant
since compressibility and porosity are not pressure dependent.
For a grid block containing from which a well is not producing from, q = 0, however,
for a grid block from which a well is producing from, the flow rate for constant well
pressure production as is the case in this simulation study is defined as follows:
q(
ft3
day
) = 0.00633WIi,j,kλn
i,j,k ∗ (Pn+1
i,j,k − Pwf )
Well Index : WI =
2π∆Z
ln( re
rw
) + skin
For an isotropic reservoir, drainage radius is defined as:
re = 0.208∆X
Since this is radial flow to a vertical well,
λn
i,j,k = λn
xi,j,k
6 Simulation Studies Conducted
We conduct three simulation studies which variable skin values to assess skin damage
on oil production and the necessity for well stimulation
6.1 Standard Case: No Damage or Stimulation
In the first simulation study, we consider the case where there is no damage (skin = 0).
In the figure below we have our production rate vs time.
8

9. 0 20 40 60 80 100 120
Time,days
10-2
10-1
100
101
102
103
104
Production
Rate,
ft
3
/day
Standard Case: Oil Production Rate vs. Time
We also have the reservoir pressure distribution at 30, 60, 90 and 120 days of produc-
tion in the following figures.
9

10. 10

11. 11

12. 6.2 Damaged Well Case
In the second simulation study, we consider the case where there is damage (skin = 3).
In the figure below we have our production rate vs time.
0 20 40 60 80 100 120
Time,days
10-1
100
101
102
103
104
Production
Rate,
ft
3
/day
Damaged Well Case: Oil Production Rate vs. Time
We also have the reservoir pressure distribution at 30, 60, 90 and 120 days of produc-
tion in the following figures.
12

13. 13

14. 6.3 Stimulated Well Case
In the third simulation study, we consider the case where with well stimulation (skin =
-1).
In the figure below we have our production rate vs time.
0 20 40 60 80 100 120
Time,days
10-2
10-1
100
101
102
103
104
105
Production
Rate,
ft
3
/day
Stimulated Well Case: Oil Production Rate vs. Time
14

15. We also have the reservoir pressure distribution at 30, 60, 90 and 120 days of produc-
tion in the following figures.
15

16. 16

17. 7 Conclusion
The stimulated well has the highest initial production rate of 13456 ft3
day
followed by the
well with no damage, with an initial production rate of 8785 ft3
day
and then the damaged
well, with an initial production rate of 4303 ft3
day
. The order is in line with our expecta-
tion.
With the initial production rate of the stimulated well approximately 53% greater than
the well with no damage and approximately 212% greater than the well with damage,
well stimulation is justified. Also we note that skin damage can have a significant im-
pact on our well production.
17