radial flow
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Solution to a radial diffusion model with polynomialdecaying flux at the boundary
Yarith del Angel, Mayra Nunez-Lopez and Jorge X. Velasco-Hernandez
Instituto Mexicano del PetroleoPrograma de Matematicas Aplicadas y Computacion
March 13, 2012
Yarith del Angel, Mayra Nunez-Lopez and Jorge X. Velasco-Hernandez (IMP) March 13, 2012 1 / 16
Introduction
In order to plan a successful secondary recovery project sufficient reliableinformation concerning the nature of the reservoir-fluid system must beavailable.Heterogeneous medium:
ω∂2p2D
∂tD=
∂2p2D
∂r2D
+1
rD
∂p2
∂rD− (1− ω)
∂p1D
∂tD(1)
(1− ω)∂p1D
∂tD= λ(p2D − p1D) (2)
Figure: Heterogeneous porous moYarith del Angel, Mayra Nunez-Lopez and Jorge X. Velasco-Hernandez (IMP) March 13, 2012 2 / 16
Introduction
Diffusion in fractal medium:
c∂p
∂t=
k
µ
1
rDf−1
∂
∂r
(rDf−1−θ ∂p
∂r
)(3)
where k = k0r−θ.
Homogeneous medium:
∂2p
∂r2+
1
r
∂p
∂r=φµctk
∂p
∂t, (4)
Yarith del Angel, Mayra Nunez-Lopez and Jorge X. Velasco-Hernandez (IMP) March 13, 2012 3 / 16
Chicontepec
Chicontepec shows low production rates
Chicontepec is characterized not so much by a fractal network thatextends along particular formations, but rather by an extremecompartmentalization of clusters of the network.In Chicontepec, each cluster is embedded in a particular body ofsandstone which in itself is surrounded by very low permeabilitystructures.Whenever oil is extracted from a sandstone, it therefore depletes soondue to the relative limited volume that the sandstone body can hold.
Figure: pD vs tD for a closed boundary reservoir located at RD = 10.
Yarith del Angel, Mayra Nunez-Lopez and Jorge X. Velasco-Hernandez (IMP) March 13, 2012 4 / 16
The basic model
The observed behavior in the flow-rate production or in the pressure canbe attributed to many different factors other than fractures such as faults,stratification, and heterogeneities.Fluid flow:
∂2p
∂r2+
1
r
∂p
∂r=φµctk
∂p
∂t, (5)
rD =r
rw,
tD =kt
φµctr2w
,
pD = (pi − p)2πhk
qBµ,
Yarith del Angel, Mayra Nunez-Lopez and Jorge X. Velasco-Hernandez (IMP) March 13, 2012 5 / 16
The basic model
With this variables, we have rD = 1 and our main equation becomes
∂2pD∂r2
D
+1
rD
∂pD∂rD
=∂pD∂tD
. (6)
We consider either of the following boundary conditions in the well radius
pD(1, tD) = 1 production at constant pressure (7)
∂pD∂rD
∣∣∣∣rD=1
= −1 production at constant rate (8)
Usually this condition is expressed as
∂pD∂rD
∣∣∣∣rD=RD
= 0,
Yarith del Angel, Mayra Nunez-Lopez and Jorge X. Velasco-Hernandez (IMP) March 13, 2012 6 / 16
The basic model
Chicontepec:The oil depletes before the fluxes equilibrate albeit with the flux at theboundary being very small but non-zero.Boundary approximates the waiting times of a sub-diffusive flow outsidethe sandstone:
∂pD∂rD
∣∣∣∣rD=RD
= −ε(1− t−αD ), (9)
Finally, letpD(rD , 0) = 0. (10)
Yarith del Angel, Mayra Nunez-Lopez and Jorge X. Velasco-Hernandez (IMP) March 13, 2012 7 / 16
Closed reservoir
Dimensionless Time
Dim
ensi
onle
ss P
ress
ure
0 5 10 15 20 25 30
1.0
1.5
2.0
Figure: pD vs tD for a closed boundary reservoir located at RD = 10.
pD(1, tD) =2
R2 − 1tD +
R2D
R2D − 1
ln (RD)− 1
2
+2∞∑i=1
e−u2i tD
u2i
J21 (RDui )(
J21 (RDui )− J2
1 (ui )) . (11)
Yarith del Angel, Mayra Nunez-Lopez and Jorge X. Velasco-Hernandez (IMP) March 13, 2012 8 / 16
A model with variable flux at the boundary
Interior boundary condition(∂pD∂rD
)rD=1
= −1 , tD > 0 .
and exterior boundary condition(∂pD∂rD
)rD=RD
= H(tD − 1)ε(1− t−αD )
(∂pD∂rD
)rD=RD
=
{0, tD ≤ 1
ε(1− t−αD ), tD > 1(12)
pD(rD , s) =K1(RD
√s)I0(rD
√s) + I1(RD
√s)K0(rD
√s)
s32
[I1(RD
√s)K1(
√s)− I1(
√s)K1(RD
√s)] + f (s), (13)
Yarith del Angel, Mayra Nunez-Lopez and Jorge X. Velasco-Hernandez (IMP) March 13, 2012 9 / 16
A model with variable flux at the boundary
The production rate decreases very fast and the reservoir gradually fills
Dimensionless Time
Dim
ensi
onle
ss P
ress
ure
---- Ε=0
---- Ε=1.5
- - Ε=2.5
---- Ε=0.5
---- Ε=0.85
2000 4000 6000 8000 10 000
4.0
4.5
5.0
Figure: The influence of ε on the radial model with flux at the boundary can beseen in the curve pD vs tD and external radius RD = 200.
Yarith del Angel, Mayra Nunez-Lopez and Jorge X. Velasco-Hernandez (IMP) March 13, 2012 10 / 16
Numerical solution
Let pki the value of the pressure drop at node (ri , tk), approximating theequation (2) using a explicit finite difference scheme we have
pk+1i = (g − fi )p
ki−1 + (1− 2g)pki + (fi + g)pki+1
where fi = ∆t2∆r(1+i∆r) , g = ∆t
∆r2 and pki ≈ p(ri , tk),
0
1
2
3
4
5
6
0 2000 4000 6000 8000 10000
Dim
en
sio
nle
ss
Pre
ssu
re
Dimensionless Time
0
1
2
3
4
5
6
0 50 100 150 200
Dim
en
sio
nle
ss
Pre
ssu
re
Radius
(a) (b)
Figure: Numerical solution of the radial diffusion with parameters ε=0.3 andα=0.4. (a) Plot of pD vs tD at rD = 1. (b) Plot of pD vs rD at t = T .
Yarith del Angel, Mayra Nunez-Lopez and Jorge X. Velasco-Hernandez (IMP) March 13, 2012 11 / 16
Numerical solution
0
1
2
3
4
5
6
0 2000 4000 6000 8000 10000
Dim
en
sio
nle
ss P
ress
ure
Dimensionless Time
epsilon 0.1
epsilon 0.3
epsilon 0.8
epsilon 1
-1
0
1
2
3
4
5
6
0 50 100 150 200 250
Dim
en
sio
nle
ss P
ress
ure
Radius
epsilon 0.1
epsilon 0.3
epsilon 0.8
epsilon 1.0
(a) (b)
Figure: Numerical solution for different permeabilities for α=0.4.
Yarith del Angel, Mayra Nunez-Lopez and Jorge X. Velasco-Hernandez (IMP) March 13, 2012 12 / 16
Conservation de the diffusion equation
To validate the numerical solution we calculated the analytical integralover tD and rD and verify the conservation of flux in the systemFinally we integrate respect to time, where T is the final time of theproduction reservoir and the total flow in the system is∫ T
0
d
dt
∫ rD=RD
rD=1p(rD , tD)rDdrD = (1− RDε)T − αRDεT
1−α.
Figure: Numerical approximation for different mesh sizes where ε=0.15 andα=0.4.
0
2
4
6
8
10
12
14
16
0 2000 4000 6000 8000 10000
Pressure
Time
100
500
1000
2000
4000
5000
6000
Yarith del Angel, Mayra Nunez-Lopez and Jorge X. Velasco-Hernandez (IMP) March 13, 2012 13 / 16
Conservation of the diffusion equation
We compare the numerical solution with different mesh size with thefollowing parameters: RD = 200, T = 10000, ε=0.15 and α=0.4, where∆t =0.0005.
Mesh size Numerical integral Relative error100 37609.10 3.43500 14450.80 0.701000 11512.00 0.352000 10038.60 0.184000 9300.90 0.095000 9153.27 0.076000 9054.84 0.067
Table: Relative errors with different mesh size with ∆t=0.0005.
Analytical integral: 8484.0928
Yarith del Angel, Mayra Nunez-Lopez and Jorge X. Velasco-Hernandez (IMP) March 13, 2012 14 / 16
Concluding remarks
We obtain a radial diffusion model for variable flux at the boundary torepresent the behavior of Chicontepec.Future work:
Obtain the analytical solution for different values of rD and comparewith the numerical solution
Calculate the semi-log derivative for long times to apply the model totest pressure .
Yarith del Angel, Mayra Nunez-Lopez and Jorge X. Velasco-Hernandez (IMP) March 13, 2012 15 / 16
References
[1] Van Everdingen, A.F., and Hurst W.: The Application of the LaplaceTransformation to Flow Problems in Reservoirs. Trans. AIME, Vol.186. (1949).
[2] Klins, M.A., Bouchard, A.J. and Cable, C.L.: A Polynomial Approachto the van Everdingen-Hurst Dimensionless Variables for WaterEncroachment. SPE, 15433. 320-326, (1988).
[3] Bourdarot: Well Testing: Interpretation Methods. ditions Technip,(1996).
[4] Carslaw H.S. and Jaeger J.C.: Operational Methods in AppliedMathematics. Dover, (1963).
[5] Morton K.W. and Mayers D.F.: Numerical solutions of PartialDifferential Equations. Cambridge, (1994).
Yarith del Angel, Mayra Nunez-Lopez and Jorge X. Velasco-Hernandez (IMP) March 13, 2012 16 / 16
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