Post on 23-Jan-2022
Peaceman’s Numerical Productivity Index for Non-Linear Flows in Porous Media
by
Dahwei Chang, Ph.D.
A Thesis
In
MATHEMATICS
Submitted to the Graduate Faculty of Texas Tech University in
Partial Fulfillment of the Requirements for
the Degree of
MASTER OF SCIENCE
IN
MATHEMATICS
Approved
Dr. Eugenio Aulisa Chairperson of the Committee
Dr. Magdalena Toda
Dr. Victorie Howle
Fred Hartmeister Dean of the Graduate School
August, 2009
Texas Tech University, Dahwei Chang, August 2009
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ACKNOWLEDGMENTS
It is with Dr. Aulisa’s guidance that I am able to finish this work. At my age, I
thought I have learned to have good patience. He definitely has more patience than I
do. This research is a small portion of his master plan to find solutions for non-linear
flows in porous media. I am glad to have participated in it.
My committee members: Dr. Toda and Dr. Howle are very nice to me. They
are very patient and knowledgeable. Thank you very much!
Texas Tech University, Dahwei Chang, August 2009
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TABLE OF CONTENTS
ACKNOWLEDGMENTS ...…………………………….………………ii
ABSTRACT ……………………………………………………………..v
LIST OF TABLES ……………………………………………………....vi
LIST OF FIGURES ..…………………………………………………...vii
CHAPTER
I. INTRODUCTION ……………………………..…………………..1
II. THEORY ……………………………………….............................3
2.1 Darcy’’s Law ……………………………………..3
2.2 Formulation of the Problem ………………………4
2.3 Set up the Initial Boundary Value Problem ……...7
2.4 Initial Boundary Value Problem ……………..…..9
2.5 Auxiliary Boundary Value Problem …………….10
2.6 Summary of the PSS Regime ………………..….12
2.7 Geometry of the Simulation ……………………..13
III. PEACEMAN NUMERICAL PRODUCTIVITY INDEX FOR
DARCY (LINEAR) FLOWS ……………………….............................14
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IV. PEACEMAN BUMERICAL PRODUCTIVITY INDEX FOR
NON-DARCY (NON-LINEAR) FLOWS .….............................17
V. RELATIONSHIP BETWEEN ACTUAL WELL AND THE WELL
BLOCK …………………………………………………………..19
VI. DATA AND RESULTS …………………………………….........21
6.1 The Block Invariant c …………..………………..21
6.2 Application of the Block Invariant c.……………..22
6.3 Radii ro and r1 ………………………………………24
6.4 The Degrees of Freedom ………………………….25
VII. CONCLUSION ….………………………………………………. 26
BIBLIBIOGRAPHY ……………………………………………......... 27
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ABSTRACT
From Darcy’s law to the Darcy-Forchheimer equation, there has been a lot of
effort put into finding solutions for flows in porous media. Peaceman used a system
of well blocks to replace the well bore in finding numerical solutions for linear flows.
Our work uses a single well block to find the pressure distribution throughout the well
for non-linear flows. In the process we found a block invariant which can be used to
build the pressure distribution formula. From it, we can find the productivity index,
one of the important factors in petroleum engineering.
Theoretical derivation and numerical data are also presented in this report.
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LIST OF TABLES
6.1 Values of Block Invariant c for a Rectangular Reservoir ……………21 6.2 Values of Block Invariant c for a Circular Reservoir ……..…………21 6.3 Comparison of PI between Actual Well and Different Well Block Sizes for a Rectangular Reservoir ..……………………....………….22 6.4 Comparison of PI between Actual Well and Different Well Block Sizes for a Circular Reservoir ..……..…………………....………….22 6.5 Comparison of Pressure at the Well Bore between Actual Well and Different Well Block Sizes for a Rectangular Reservoir ....…………23 6.6 Comparison of Pressure at the Well Bore between Actual Well and Different Well Block Sizes for a Circular Reservoir ………………..23 6.7 Ratio of Radius ro/ Well Block Size for a Rectangular Reservoir…..24 6.8 Ratio of Radius ro/ Well Block Size for a Circular Reservoir…….....24 6.9 Degrees of Freedom …………………………………………………25
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LIST OF FIGURES
2.1 Linear Flow ………………………………………………………….3 2.2 Relationship between Total Discharge and Pressure Drawdown for Darcy and non-Darcy Flows …………………………………………6 2.3 Geometry of the Reservoir and the Well Bore ………………………7 2.4 Production Rate Decreases in Time …………………………………10 2.5 Well Pressure Distribution in Time ………………………………….12 2.6 Reservoir and a Well Block ………………………………………….13 2.7 Reservoir and an Actual Well ………………………………………..13 5.1 Pressure in a Well Block …………………………………….………19 5.2 Pressure in an Actual Well …………………………………………...19 5.3 Pressure Distribution …………………………………………………20
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CHAPTER I
INTRODUCTION
There have been numerous studies on fluid filtration in porous media because
of its wide applications. One of its major applications is on flows in oil reservoirs. In
1856, Darcy 1 proposed a linear differential equation between the pressure drop and
the production rate. Later Forchheimer (1901)2 introduced a nonlinear term for high
velocity flows. Accurate numerical solutions for this Darcy-Forchheimer equation
experience difficulty in finding the pressure near the well bore due to the drastic
change in pressure. This is generally accomplished by using high resolution meshes
around the wellbore.
In an alternative approach, numerical solutions have been found by replacing
the wellbore with a grid block that is generally larger than the wellbore size. By using
the well-block finite difference numerical solution, Peaceman (1978)3 found a semi-
analytical formula for reconstructing the real pressure distribution in the wellbore
proximity and for evaluating the well productivity index for the linear Darcy case.
Aulisa and others (2007)4 introduced a generalized non-linear equation which
relates the velocity vector field of filtration to the gradient of pressure in a non-linear
way. The system formed by the latter equation, the continuity equation, the equation
of state for slightly compressible fluids and the appropriate initial and boundary
conditions defines an initial boundary value problem for the pressure only. Further
analysis showed the existence of an auxiliary boundary value problem whose solution
could be used to build a “time-invariant” solution of the initial boundary value
problem.
In this work, starting from these results, we extend Peaceman’s work to non-
linear flows by using well block finite element solutions. We show that for any well
block size there exists an invariant coefficient that allows us to reconstruct accurately
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the actual pressure profile, even for high non-linearity. All numerical simulations
have been done by using COMSOL Multiphysics.
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CHAPTER II
THEORY
2.1 Darcy’s Law
Darcy’s law states that the discharge rate Q is proportional to the pressure drop
over a given distance and is defined in Eq. 2.1 (see figure 1). The negative sign is
needed because fluids flow from high pressure to low pressure. Dividing both sides of
Eq. 2.1 by the area, A, and we got Eq. 2.2 which defines the Darcy flux, q.
Figure 1.1 View from my window
L) P(P A -
Q ab
µκ −= (2.1)
µκ P -
q∇= (2.2)
The pore velocity, v, is defined by the ratio of Darcy flux and the porosity of
the fluid, φ. It shows the higher the porosity, the higher the flux.
ϕq
v =→
(2.3)
Units: Q: total discharge, m3/s κ: permeability, m2 A: cross section area, m2 P: pressure, Newton/ m2 = kg/(m.s2) Q: Darcy flux, m/s L: distance, m
b A a
L Q
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v: velocity, m/s φ: fraction µ: fluid viscosity, kg/(m.s)
If we substitute q from Eq. 2.2 into Eq. 2.3 and drop φ, we get
P v ∇=−→
κµ
(2.4)
For a fixed pressure difference, the higher the viscosity, the slower the
velocity; the higher the permeability, the higher the fluid velocity.
2.2 Formulation of the Problem
Fluid flows, which deviate from Darcy’s law, are typically observed in high-
rate well production. In non-Darcy flow, the fluid converging to the well bore reaches
velocities exceeding the lamina Reynolds number, resulting in a turbulent regime.
There are different approaches for modeling non-Darcy phenomena. It seems that the
most appropriate is derived from the general Brinkman-Forchheimer equation. The
derivation of this equation by Aulisa et al. (2009)5 is as follows.
Let ∇ and ∆ denote the gradient and the Laplace operator. The time dependent
Brinkman-Forchheimer equation describing the velocity vector field →v and the
pressure P in porous media can be written in the form
a
v c - v v || v || v - P
t
µρ µ βκ
→→ → → ∂ ∆ + + = ∇
∂
, (2.5)
with 1/2
F
κϕρβ = ,
Here ca is the acceleration coefficient, F is the Forchheimer coefficient, φ is the
porosity, κ is the permeability, µ is the viscosity and ρ is the density of the fluid.
The continuity equation can be written in the form
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0 )v( =⋅∇+∂∂ →
ρρt
(2.6)
In order to solve this system of equations, we assume that the first two terms in
Eq. 2.5 can be neglected. We also assume the fluid is slightly compressible and
satisfies the state equation
ργρ -1' = ( )e )P - (P 0
o-1γρρ = , (2.7)
where the γ – constant characterizes the compressibility of the fluid.
Then the three governing equations for non-Darcy flows becomes
P, v - )v( - t
P '' ∇⋅⋅∇=∂∂ →→
ρρρ (2.8)
0 v ||v|| - v - P =∇−→→→
βκµ
, (2.9)
ργρ -1' = , (2.10)
For slightly compressible liquids coefficient -1 γ is of the order of 10-8,
therefore, following the engineering tradition, the second term in Eq. 2.8, P v ' ∇⋅→
ρ
(small compare to the first term) will be neglected. Finally these three equations can
be rewritten as
, )v( - t
P '
→⋅∇=
∂∂ ρρ (2.11)
0 v ||v|| - v - P =∇−→→→
βκµ
. (2.12)
Equation 11 is referred as the continuity equation. Equation 2.12 is referred as
the Darcy-Forchheimer equation and Eq. 2.10 as the equation of state for slightly
compressible liquids. Equation 2.11 and 2.12 describe flows in porous media.
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Figure 2.2 shows the relationship between total discharge and pressure
drawdown for Darcy and non-Darcy flows.
Darcy flow: P v ∇=−→
κµ
(2.4)
Non-Darcy flow: 0 v ||v|| - v - P =∇−→→→
βκµ
(2.12)
Figure 2.2 Relationship between Total Discharge and Pressure Drawdown for Darcy and non-Darcy Flows
The velocity vector field t)(x, v→
as a dependent variable can be uniquely
represented as a function of the pressure gradient. Let us assume the following
approximation of the Darcy-Forchheimer equation (Eq. 2.12)
v - (|| P||) Pf→
= ∇ ∇ . (2.13)
The length of the vector ||v||→
is defined correspondingly as
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|| v || (|| P||) || P||f→
= ∇ ∇ . (2.14)
It can be proved that →v defined in Eq. 2.13 solves Eq. 2.12, where f is defined
by the following,
2
2f
4 || P|| α α β=
+ + ∇ 2.3 Set up the Initial Boundary Value Problem
In our model the exterior boundary of the reservoir is considered impermeable.
The boundary condition on the well, the condition of non-flux on the exterior
boundary of the reservoir, and the prescribed initial pressure distribution together form
the initial boundary value problem (IBVP) for system Eq. 2.7 – 2.9. Figure 2.3 shows
the geometry of the reservoir and the well bore.
Let U ⊂ Rn, n = 1, 2, or 3 be the reservoir domain bounded by the exterior
impermeable boundary, and the well interface
U
W
ΓU
Γw
Figure 2.3 Geometry of the Reservoir and the Well Bore
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compact. and with, U UWUWU,W ΓΓ∅=Γ∩ΓΓ∪Γ=∂
The part ΓU is associated with the exterior boundary of the reservoir U, and the
part ΓW is associated with the boundary of the well. Let P(x, to) = Po(x) be the given
initial pressure distribution in the reservoir.
We define the following notation. If P is a function defined on Ū, then let UP
and WP denote the average of P on U and ΓW respectively, defined by
W
W
1P ( ) = P(x, t) d s
| W| xtΓ∫ - average value of the pressure on the well;
U
U
1P ( ) = P(x, t) d x
| U|t
Γ∫ - average value of the pressure in the reservoir;
x = Rn - spatial independent variable, n = 1, 2, 3;
|W| = mesn-1 U - area of the well;
|U| = mesn U - volume of the reservoir. Definition (Diffusive Capacity)
Let the pressure function P(x, t) and the vector velocity v→
(x, t) form the
solution of system (Eq. 2.7 - 2.9) in the bound domain U, with boundary condition
Uv (x, t) n | = 0.→ →
Γ⋅ Assume for any t > 0, P(x, t) to be such that UP (t) > WP (t).
The diffusive capacity of ΓW with respect to ΓU corresponding to the solution P(x, t)
and v→
(x, t) is the ratio
W
x
U W
v(x, t) n d s
J (P, v, t) = .P (t) - P (t)
→ →
→Γ
⋅∫ (2.15)
Here n→
is the external normal on the piecewise smooth surface ΓW.
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Definition (PSS regime)
Let the well production rate Q be time independent: W
xv(x, t) n d s = Q.→ →
Γ
⋅∫
We will call the system a “pseudo-steady state regime” (PSS), if the
corresponding pressure drawdown (difference between the average of the pressure on
the well surface and the average of the pressure in the reservoir) is constant. For the
PSS regime the diffusive capacity /PI is time invariant.
The diffusive capacity is not unique and depends not only on the boundary
condition on ΓW but also on the class of admissible functions, on which the functional
J (P, v→
, t) is defined. The main reason why diffusive capacity is introduced in this
way is to reflect the major engineering idea behind the Productivity Index. The other
reason is because of its generality.
2.4 Initial Boundary Value Problem
Let the domain U, its boundaries, and the initial pressure function have the
same meaning as in section 2-4. Assume that the well is operating under the condition
of time independent constant production rate Q; the initial reservoir pressure is known.
Then under assumptions in the previous section the initial boundary value problem
(IBVP) modeling the filtration process can be formulated as follows
' P = - ( v),
tρ ρ
→∂ ∇ ⋅∂
(2.16)
P - v - || v || v = 0,µ βκ
→ → →− ∇ (2.17)
W
v n d s = Q ,→ →
Γ
⋅∫ (2.18)
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Uv n | = 0 ,→ →
Γ⋅ (2.19)
0P(x, 0) = P (x) . (2.20)
In addition, by using assumption that β is constant and the fluid is
incompressible, Eq. 2.16 – 2.20 reduces to
1 P = ( (f(|| P||)) P) ,
tγ − ∂ ∇ ⋅ ∇ ∇
∂ (2.21)
W
Pf (|| P||) d s = - Q ,
n→
Γ
∂∇∂
∫ (2.22)
U
P | = 0 ,
nΓ→
∂
∂ (2.23)
0P(x, 0) = P (x) . (2.24)
The IBVP (Eq. 2.21 – 2.24) is ill-posed: it has an infinite number of solutions.
Remember that the Productivity Index is modeled as an integral characteristic of the
solution, hence, the lack of uniqueness in the definition. Later we will restrict the
solution by introducing a class of functions for which it is unique up to an additive
constant.
2.5 Auxiliary Boundary Value Problem
In Eq. 2.21, pressure P is a function of time and position. In a real well, once
the well starts producing, the production rate in the well decreases and soon settles
down to a constant value as in figure 2.4. The pressure distribution throughout the
reservoir is also settles down to a constant distribution.
Q
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Figure 2.4 Production Rate Decreases in Time
We are interested in finding this pressure distribution. Therefore, we suggest
the following expression for the pressure. In this expression, the factor of time and
position are separated.
B. t |U|
Q - (x)W t)(x, P += γ (2.25)
If we substitute Eq. 2.25 into Eq. 2.21 – 2.24, then the IBVP becomes
W), ||)W(|| f ( A - |U|
Q ∇∇⋅∇==− (2.26)
0, |W
W=Γ (2.27)
W
| = 0 ,n
UΓ→∂
∂ (2.28)
The divergence theorem implies
W
Wf (|| W||) d s = - Q .
n→
Γ
∂∇∂
∫ (2.29)
Time
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The equation system Eq. 2.26 – 2.29 is the auxiliary boundary value problem.
Eq. 2.27 describes a Dirichlet boundary condition for W. Eq. 2.28 describes the
Neumann boundary condition for W, i.e. no flux out of the reservoir. The solution of
this BVP exists and is unique in an appropriate functional space. Our simulation is set
to solve these equations. We assume the steady state production rate Q and define A =
Q/|U|, A is a constant. We also assume that the boundary U∂ is smooth. Figure 2.5
shows the relationship between P(x, t) and W(x).
Figure 2.5 Well Pressure Distribution in Time 2.6 Summary of PSS regime
Let assumptions in section 2-2 hold. If the initial function P0(x, 0) = W(x) in
Eq. 2.25 is a solution of auxiliary BVP (Eq. 2.26 – 2.29), then the diffusive capacity is
steady state invariant over the solution of IBVP (Eq. 2.21 – 2.24), and can be
computed by using the formula
B
P(x,t)
t
x
W(x)
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WWU U
Q QJ (P, v, t) = J (Q, t) = = = J (Q) ,
1 1 1P dx - P dx W(x) dx
|U| | | |U|
→
ΓΓ∫ ∫ ∫
Although P is a function of time (so is its average), the pressure difference is
independent of time. Therefore J (Q) is time invariant and depends on α, β, Q, and on
the geometry of the domain U.
2.7 Geometry of the Simulation
In our simulation, as in figure 2.6, we follow Peaceman’s idea in which a well
block is used to replace the well bore and set no boundary conditions on the block.
Because there is no boundary condition on the block, less number of elements are
needed when equation system is solved based on the finite element method. Hence,
less number of mesh points and degrees of freedom are needed. This advantage may
become useful when the solution is extended to include time factor or for compressible
fluid.
Figure 2.6 Reservoir and a Well Block Figure 2.7 Reservoir and an Actual Well
U2
U1
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For our geometry, the auxiliary BVP of Eq. 2.26 – 2.29 are transformed into Eq. 2.30 – 2.33.
W) ||)W(|| f ( A - |U|
Q ∇∇⋅∇==− u) ||)u(|| f (
|U|Q
1
∇∇⋅∇=− on U1 (2.30)
0, |WW
=Γ u) ||)u(|| f ( |U|
Q
2
∇∇⋅∇=− on U2 (2.31)
W
| = 0 ,n
UΓ→∂
∂ 0
n
W |U
=Γ∂
∂→ (2.32)
W
Wf (|| W||) d s = - Q .
n→
Γ
∂∇∂
∫ Q - s d n
u ||)u(|| f
2
=∂
∂∇∫Γ
→ (2.33)
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CHAPTER III
PEACEMAN NUMERICAL PRODUCTIVITY INDEX FOR DARCY
(LINEAR) FLOWS
One of the important factors in petroleum engineering is the productivity index
(PI) which is defined by Darcy in the following equation.
)) P(r(P 2q
PIwaverage −
=π
α,
where q is the production rate is defined as the specific discharge per unit
cross-sectional area normal to the direction of the flow,κµα = , µ is viscosity and κ is
the permeability of the fluid. Paverage is the average pressure of the reservoir, and P(rw)
is the pressure at the well bore.
In extrapolating the value of P(rw), Peaceman replaced the well bore with a
system of well blocks. In our work, the well bore is replaced by a single well block.
Darcy’s law also states that the filtration velocity into the well bore is
proportional to the pressure difference between the well bore and the reservoir, i.e.
P - v ∇=→
α In the radial case, this equation becomes
r d
Pd - (r) v =α
In the radial case, the relationship between the this velocity and its production
rate is described as
r 2
Q - v(r)
π= (3.1)
If we combine these two equations and integrate over the radius of the well
block, we obtain
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∫∫ =
P
P
r
r oo
Pd r d r 2
Q-
πα
and
oo r
r ln
2Q
) P(r P(r)π
α+= (3.2)
Eq. 3.2 represents the continuous pressure distribution in a reservoir. P(ro) is
the pressure at ro.
We use the following procedures to reconstruct this equation.
a) First we assume pressure at some radius r1 is equal to c where c is a block
invariant and ∆x is the size of the well block.
b) Find ro for a certain well size using a value of actual well simulation. We use
oow r
30 ln
2Q
) P(r 30)P(rπ
α+== (3.3)
where P(rw = 30) is the pressure of at the well bore. From this equation, ro can be solved by
30))) P(r- (P Q 2
( exp 30 r woo ==απ
c) From our simulation, we found P(r1) by averaging the pressures from the four
corners of the rectangular well block and rewrite Eq. 3.3 as follows
oo1 r
x c ln
2
Q ) P(r )P(r
∆+=π
α (3.4)
from which c can be found as
))) P(r- )(r (P Q
2( exp
x
r c o1
o
απ
∆= .
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Now for any given well block, size ∆x, hence r1 = c ∆x, we can find ro from
Eq. (3.5). Then P(rw) can be solved from the pressure distribution formula Eq. 3.1,
hence the product index.
))) P(r- )(P(r Q
2( expx c r 1oo α
π∆= (3.5)
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CHAPTER IV
PEACEMAN NUMERICAL PRODUCTIVITY INDEX FOR NON-
DARCY (NON-LINEAR) FLOWS
In the real world, flows in the porous medium are not ideally linear. Many
have added nonlinear terms to the Darcy equation. One of them is Darcy-Forchheimer
equation in which one nonlinear term that is proportional to the square of the velocity,
→v vβ , is added to the Darcy’s equation. Here β is the Forchheimer factor and is
proportional to the density of the fluid.
Darcy-Forchheimer equation
P - v ∇=+→→vv αβ (4.1)
In the radial case, this equation becomes
dr
dP - v v2 =+ αβ (4.2)
If we combine Eq. 3.1 and 4.2, integrate over the radius, we have
∫∫ =
+P
P
r
r 22
2
oo
Pd r d r 2
Q- r d
r 4Q
π
απ
β
++=
r
1 -
r
1
4
Q
r
r ln
2
Q ) P(r P(r)
o2
2
oo π
βπ
α (4.3)
The third term on the right hand side is the non-linear term. Eq. 4.3 is the
pressure distribution equation for the non-linear flows.
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If we follow the procedures in the linear case, first finding ro by using Eq. 4.3
with P(rw), in our case rw = 30, from a actual well. Then assume r1 = c ∆ x, where c is
the block invariant and it is solved by using Eq. 4.5.
++=
30
1 -
r
1
4
Q
r
30 ln
2
Q ) P(r P(30)
o2
2
oo π
βπ
α (4.4)
∆+∆+=
x c
1 -
r
1
4
Q
r
x c ln
2
Q ) P(r )P(r
o2
2
oo1 π
βπ
α (4.5)
Now for any given well block, size ∆x, hence r1 = c ∆x, we can find ro from
Eq. 4.5. Then P(rw) can be solved from the pressure distribution formula Eq. 4.3.
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CHAPTER V
RELATIONSHIP BETWEEN ACTUAL WELL AND THE WELL
BLOCK
Figure 5.1 shows the geometry of rectangular reservoir of 4000 x 8000 and a
well of radius, rw, equals to 30. Figure 5.2 shows the geometry of our simulation for
the same reservoir but with a square well block of ∆x (= 50 – 200). We have also
applied our simulation to a circular reservoir of radius 10000.
Figure 5.1 Pressure in a Well Block Figure 5.2 Pressure in an Actual Well In our finding, the pressure distribution in the reservoir is
++=
r
1 -
r
1
4
Q
r
r ln
2
Q ) P(r P(r)
o2
2
oo π
βπ
α
The radius of the well bore is 30 and the pressure at the well bore P(rw = 30) is
++==
30
1 -
r
1
4
Q
r
30 ln
2
Q ) P(r 30)P(r
o2
2
oow
πβ
πα
P1 = P(r1) is referring to the pressure at the corner of the rectangular well block and P0
= P(r0) is for the pressure at the center of the well block.
∆+∆+=
x c
1 -
r
1
4
Q
r
x c ln
2
Q ) P(r )P(r
o2
2
oo1 π
βπ
α
∆x
P1
Po P(30)
rw
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In the pressure distribution formula Eq. 4.3, ro is a radius measured from the
center of the well block. We found ro = (0.357 ~ 0.439) ∆x, proportional to the
increase of the non-linear Forchheimer factor (See Table 6.7 and 6.8). The radius r1
that is used to find block invariant is equal to 0.686 ∆x.
Figure 5.3 is the pressure distribution around the well block from r = 0 to 150.
This figure is for the case of well block size equals to 100, ro = 39.66 = 0.39 ∆x, and r1
= 68.57 = 0.686 ∆x = c ∆x. The Forchheimer factor of the non-linear term β equals to
24.
Figure 5.3 Pressure Distribution
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CHAPTER VI
DATA AND RESULTS
6.1 The Block Invariant c
In order to see whether c is invariant in the pressure distribution formula, we
have completed simulations for well block sizes 50, 100, 150, 200. Our simulations
also include various strengths in the non-linear term and two production rates. Finally
all the calculations have been done for both rectangular (dimension = 4000 x 8000)
and circular (radius = 10000) reservoirs for comparison. These values of c are in
Table 6.1 and 6.2.
The overall average of c is equal to 0.686 with a standard deviation of 0.0028. Table 6.1 Values of Block Invariant c for a Rectangular Reservoir β = Forchheimer Factor ∆x = Well Block Size PI = Product Index
q 10 100 β 0 2.4 24 240 2400 0 2.4 24 240 2400
∆x/50 0.687 0.687 0.687 0.687 0.686 0.687 0.687 0.687 0.686 0.686
100 0.685 0.683 0.683 0.683 0.682 0.683 0.683 0.683 0.682 0.681
150 0.685 0.685 0.686 0.687 0.686 0.685 0.686 0.687 0.686 0.686
200 0.680 0.680 0.681 0.683 0.683 0.680 0.681 0.683 0.683 0.682
Average 0.684
Table 6.2 Values of Block Invariant c for a Circular Reservoir
q 10 100 β 0 2.4 24 240 2400 0 2.4 24 240 2400
∆x/ 50 0.686 0.686 0.685 0.684 0.683 0.686 0.685 0.684 0.683 0.682
100 0.688 0.688 0.688 0.688 0.687 0.688 0.688 0.688 0.687 0.686
150 0.687 0.687 0.687 0.687 0.688 0.687 0.687 0.687 0.688 0.688
200 0.689 0.689 0.690 0.690 0.692 0.689 0.690 0.690 0.682 0.693
Average 0.687
Texas Tech University, Dahwei Chang, August 2009
23
6.2 Application of the Block Invariant c
This part of calculation is to use the value of c = 0.686 to find ro of a fixed
well block. Then use the pressure distribution formula to find pressure at the well
bore P(rw). With it we can solve for the productivity index (PI) and compare them
to the corresponding values of P(rw) and PI values from an actual well (see figure
6.3 – 6.6). The comparison show pretty good results.
Table 6.3 Comparison of PI between Actual Well and Different Well Block Sizes
for a Rectangular Reservoir
β = Forchheimer Factor ∆x = Well Block Size PI = Productivity Index
q = Production Rate P = Pressure
Values from actual well are in shade.
q = 10 q = 100
β 0 2.4 24 240 2400 0 2.4 24 240 2400 PI 0.1484 0.1480 0.1453 0.1226 0.0481 0.1484 0.1453 0.1226 0.0481 0.0068 Δx
50 0.1484 0.1481 0.1454 0.1226 0.0481 0.1484 0.1454 0.1226 0.0481 0.0068 100 0.1483 0.1480 0.1452 0.1225 0.0480 0.1483 0.1452 0.1225 0.0480 0.0068 150 0.1484 0.1481 0.1453 0.1226 0.0481 0.1484 0.1453 0.1226 0.0481 0.0068 200 0.1483 0.1479 0.1452 0.1225 0.0480 0.1483 0.1452 0.1225 0.0480 0.0068
Table 6.4 Comparison of PI between Actual Well and Different Well Block Sizes
for a Circular Reservoir
q = 10 q = 100
β 0 2.4 24 240 2400 0 2.4 24 240 2400 PI 0.1977 0.1972 0.1929 0.1582 0.0566 0.1977 0.1929 0.1582 0.0566 0.0076 Δx
50 0.1977 0.1972 0.1928 0.1581 0.0564 0.1977 0.1928 0.1581 0.0564 0.0076 100 0.1978 0.1973 0.1930 0.1583 0.0566 0.1978 0.1930 0.1583 0.0566 0.0076 150 0.1979 0.1974 0.1931 0.1584 0.0567 0.1979 0.1931 0.1584 0.0567 0.0076 200 0.1981 0.1976 0.1933 0.1586 0.0567 0.1981 0.1933 0.1586 0.0567 0.0076
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Tab
le 6
.5
Co
mpa
riso
n o
f Pre
ssur
e at
the
We
ll B
ore
bet
wee
n an
Act
ual W
ell
and
Diff
ere
nt W
ell
Blo
ck S
ize
s fo
r a
Rec
tang
ula
r R
eser
voir
q
= 1
0 q
= 1
00
β
0 2.
4 24
24
0 24
00
0 2.
4 24
24
0 24
00
P
-140
.2
-140
.5
-142
.6
-163
.8
-374
.3
-140
2.2
-142
5.9
-163
8.3
-374
2.5
-247
19.9
Δ
x
50
-1
40.2
-1
40.4
-1
42.6
-1
63.8
-3
74.1
-1
401.
9 -1
425.
5 -1
637.
7 -3
740.
9 -2
4707
.7
100
-140
.2
-140
.5
-142
.6
-163
.9
-374
.9
-140
2.8
-142
6.5
-163
9.2
-374
8.8
-247
81.0
15
0 -1
40.2
-1
40.5
-1
42.6
-1
63.8
-3
74.2
-1
402.
3 -1
425.
9 -1
637.
9 -3
742.
1 -2
4713
.0
200
-140
.2
-140
.6
-142
.7
-163
.9
-374
.5
-140
3.4
-142
7.0
-163
9.1
-374
5.3
-247
45.0
T
able
6.6
C
om
pari
son
of P
ress
ure
at t
he W
ell
Bo
re b
etw
een
an A
ctua
l We
ll a
nd D
iffer
ent
We
ll B
lock
Siz
es
for
a C
ircu
lar
Res
ervo
ir
q
= 1
0 q
= 1
00
β
0 2.
4 24
24
0 24
00
0 2.
4 24
24
0 24
00
P
-84.
498
-84.
699
-86.
509
-104
.60
-285
.53
-844
.98
-865
.09
-104
6.01
-2
855.
3 -2
0947
.8
Δx
50
-84.
501
-84.
702
-86.
516
-104
.68
-286
.39
-845
.01
-865
.16
-104
6.78
-2
863.
9 -2
1036
.2
100
-84.
438
-84.
639
-86.
446
-104
.53
-285
.42
-844
.38
-864
.46
-104
5.29
-2
854.
2 -2
0943
.2
150
-84.
470
-84.
671
-86.
478
-104
.55
-285
.29
-844
.70
-864
.78
-104
5.49
-2
852.
9 -2
0926
.1
200
-84.
413
-84.
614
-86.
419
-104
.47
-284
.99
-844
.13
-864
.19
-104
4.70
-2
849.
9 -2
0899
.7
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6.3
Rad
ii r o
and
r1
In f
igur
e 2.
10 w
e sa
w t
he r
elat
ions
hip
betw
een
ro
and
r 1, a
nd t
heir
rela
tions
hip
with
the
act
ual w
ell.
Alth
oug
h th
e ra
tio (
r 1 /
wel
l bl
ock
siz
e) =
c,
is a
blo
ck in
vari
ant
, w
e no
tice
that
the
rat
io o
f (r o
/ w
ell b
lock
siz
e) in
crea
se a
s β in
cre
ases
. H
ere
are
the
data
an
d w
e ar
e st
ill lo
oki
ng fo
r th
e ex
pla
natio
n fo
r th
is e
ffect
. S
ee fi
gure
6.7
- 6.
8.
T
able
6.7
R
atio
Rat
io o
f Rad
ius
ro/
We
ll B
lock
Siz
e fo
r a
Rec
tang
ula
r R
eser
voir
β =
Fo
rchh
eim
er F
acto
r ∆x
= W
ell
Blo
ck S
ize
q 10
10
0 β
0 2.
4 24
24
0 24
00
q 2.
4 24
24
0 24
00
∆x
0.35
5 0.
356
0.36
8 0.
408
0.43
3 0.
355
0.36
8 0.
408
0.43
3 0.
437
50
0.35
6 0.
356
0.36
3 0.
395
0.43
1 0.
356
0.36
3 0.
395
0.43
1 0.
438
100
0.35
4 0.
353
0.35
8 0.
384
0.42
4 0.
353
0.35
8 0.
384
0.42
4 0.
434
150
0.35
2 0.
352
0.35
6 0.
379
0.42
1 0.
352
0.35
6 0.
379
0.42
1 0.
434
200
0.35
4 0.
355
0.36
1 0.
392
0.42
7 0.
354
0.36
1 0.
392
0.42
7 0.
436
Ave
rage
0.
355
0.35
6 0.
368
0.40
8 0.
433
0.35
5 0.
368
0.40
8 0.
433
0.43
7
Tab
le 6
.8
Rat
io o
f Rad
ius
ro
/ W
ell
Blo
ck S
ize
for
a C
ircu
lar
Res
ervo
ir
q 10
10
0 β
0 2.
4 24
24
0 24
00
q 2.
4 24
24
0 24
00
∆x
0.35
8 0.
360
0.37
1 0.
412
0.43
7 0.
358
0.37
1 0.
412
0.43
7 0.
441
50
0.35
7 0.
358
0.36
4 0.
397
0.43
2 0.
357
0.36
4 0.
397
0.43
2 0.
440
100
0.35
4 0.
354
0.35
9 0.
385
0.42
5 0.
354
0.35
9 0.
385
0.42
5 0.
435
150
0.35
4 0.
354
0.35
7 0.
379
0.42
1 0.
354
0.35
7 0.
379
0.42
1 0.
434
200
0.35
6 0.
356
0.36
3 0.
393
0.42
9 0.
356
0.36
3 0.
393
0.42
9 0.
438
Ave
rage
0.
358
0.36
0 0.
371
0.41
2 0.
437
0.35
8 0.
371
0.41
2 0.
437
0.44
1
6.4 Degrees of Freedom Table 6.9 shows the degrees of freedom in our simulations. The circular reservoir has radius equal to 10000 which is larger than the rectangular reservoir of dimension 4000 x 8000. This is why our simulations for circular reservoir used more elements in the mesh, hence more degrees of freedom.
Degrees of Freedom Actual Well Simulation Rectangular Reservoir Circular Reservoir
3107 3412 Our Simulation Well Block Size
50 1699 2677 100 1346 2313 150 1219 2157 200 1176 2009
Texas Tech University, Dahwei Chang, August 2009
26
Texas Tech University, Dahwei Chang, August, 2009
27
CHAPTER VII
COMCLUSION
This work is for time independent. It may extend to time dependent initial
boundary value problems. The compressibility γ, usually the order of 10 -8, is ignored
in our calculation, this work may also be extended to include the compressibility in the
equation of state.
Another possibility for future work is to extend it to 3-D. The degrees of
freedom in our simulation are in the order 1000 – 2000. It only takes less than a
second to process on a modern computer. This may reduce the time when we extend it
to 3-D.
Texas Tech University, Dahwei Chang, August, 2009
28
BIBLIOGRAPY [1] Darcy, H. Les Fontaines Publiques de la Ville de Dijon, Dalmont, Paris, 1856. [2] Forchheimer, P. Wasserbewegung durch Boden Zeit, Ver. Deut. Ing. 45, 1901. [3] Peaceman, D. “Interpretation of Well-Block Pressures in Numerical Reservoir Simulation”, Society of Petroleum Engineers Journal of AIME, (Feb. 1978): 183 – 192. [4] Aulisa, E., Ibragimov, A., and Toda, M. “Geometric Framework for Modeling Nonlinear Flows in Porous Media, and Its Applications to Engineering, Nonlinear Analysis - Real World Applications”, Elsevier, Appear in 2009. [5] Aulisa, E. Ibragimov, A., Valko, P., and Walton, J. “Mathematical Framework of the Well Productivity Index for Fast Forchheimer (Non-Darcy) Flows in Porous Media”, COMSOL Users Conference 2006 Boston, Proceedings, http://www.comsol.com/academic/papers/1547.
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