Slightly Compressible Fluid
Slightly compressible fluid (and equation of state used to describe it)
From: Unconventional Reservoir Rate-Transient Analysis, 2021
Related terms:
Viscosity, Compressibility, Incompressible Fluid, Pressure Distribution, Compressible Fluid, Compressible Fluid Flow, Flow Equation, Material Balance, Radial Flow
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Single-phase flow equation for various
fluids
Jamal H. Abou-Kassem, ... S.M. Farouq Ali, in Petroleum Reservoir Simulation
(Second Edition), 2020
7.2.2 Slightly compressible fluid
A slightly compressible fluid has a small but constant compressibility (c) that usually
ranges from 10− 5 to 10− 6 psi− 1. Gas-free oil, water, and oil above bubble-point
pressure are examples of slightly compressible fluids. The pressure dependence of
the density, FVF, and viscosity for slightly compressible fluids is expressed as
(7.5)
(7.6)
and
(7.7)
where °, B°, and μ° are fluid density, FVF, and viscosity, respectively, at reference
pressure (p°) and reservoir temperature and cμ is the fractional change of viscosity
with pressure change. Oil above its bubble-point pressure can be treated as a slightly
compressible fluid with the reference pressure being the oil bubble-point pressure,
and in this case, °, B°, and μ° are the oil-saturated properties at the oil bubble-point
pressure.
> Read full chapter
Fundamentals of Reservoir Fluid Flow
Tarek Ahmed, in Reservoir Engineering Handbook (Fourth Edition), 2010
Solution
For a slightly compressible fluid, the oil flow rate can be calculated by applying
Equation 6-33:
Assuming an incompressible fluid, the flow rate can be estimated by applying Darcy's
equation, i.e., Equation 6-27:
> Read full chapter
Fundamentals of Reservoir Fluid Flow
Tarek Ahmed, in Reservoir Engineering Handbook (Fifth Edition), 2019
Radial Flow of Slightly Compressible Fluids
Craft et al. (1990) used Equation 6-18 to express the dependency of the flow rate on
pressure for slightly compressible fluids. If this equation is substituted into the radial
form of Darcy’s Law, the following is obtained:
where qref is the flow rate at some reference pressure pref.
Separating the variables in the above equation and integrating over the length of the
porous medium gives:
or:
where qref is oil flow rate at a reference pressure pref. Choosing the bottom-hole flow
pressure pwf as the reference pressure and expressing the flow rate in STB/day gives:
(6-33)
Where:
co = isothermal compressibility coefficient, psi–1
Qo = oil flow rate, STB/day
k = permeability, md
Example 6-6
The following data are available on a well in the Red River Field:
Assuming a slightly compressible fluid, calculate the oil flow rate. Compare the result
with that of incompressible fluid.
Solution
For a slightly compressible fluid, the oil flow rate can be calculated by applying
Equation 6-33:
Assuming an incompressible fluid, the flow rate can be estimated by applying Darcy’s
equation, i.e., Equation 6-27:
> Read full chapter
Well Testing Analysis
Tarek Ahmed, D. Nathan Meehan, in Advanced Reservoir Management and Engineering (Second Edition), 2012
Slightly Compressible Fluids
These “slightly” compressible fluids exhibit small changes in volume, or density, with
changes in pressure. Knowing the volume Vref of a slightly compressible liquid at a
reference (initial) pressure pref, the changes in the volumetric behavior of such fluids
as a function of pressure p can be mathematically described by integrating Eq. (1.1),
to give:
(1.3)
where
p=pressure, psia
V=volume at pressure p, ft3
pref=initial (reference) pressure, psia
Vref=fluid volume at initial (reference) pressure, psia
The exponential ex may be represented by a series expansion as:
(1.4)
Because the exponent x (which represents the term c(pref−p)) is very small, the ex term
can be approximated by truncating Eq. (1.4) to:
(1.5)
Combining Eq. (1.5) with (1.3) gives:
(1.6)
A similar derivation is applied to Eq. (1.2), to give:
(1.7)
where
V=volume at pressure p
=density at pressure p
Vref=volume at initial (reference) pressure pref
ref=density at initial (reference) pressure pref
It should be pointed out that many crude oil and water systems fit into this category.
> Read full chapter
Decline Curves Analysis of Long Linear
Flow
In Advanced Production Decline Analysis and Application, 2015
8.2.1 Physical Model
We consider the flow of a slightly compressible fluid of constant viscosity in a
rectangular reservoir in which the outer boundaries 2xe and 2ye are closed. The well
is located at the center of the reservoir and fluid is produced at a constant rate
q (or constant BHFP pwf). The reservoir thickness is h, the fracture half-length is
xf ≤ xe, the distance from the well to the boundaries is yw = ye, and the flow area is
Ac = 4xfh, as shown in Figure 8.5. Initially, the pressure is uniform throughout the
reservoir. The skin and wellbore storage effect is not considered. The initial reservoir
pressure, wellbore radius, porosity, total compressibility, permeability, fluid viscosity,
and volume factor are represented by pi, rw, , Ct, K, μ, and B respectively.
Figure 8.5. Vertically fractured well in a rectangular dual-porosity reservoir
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Linearization of flow equations
Jamal H. Abou-Kassem, ... S.M. Farouq Ali, in Petroleum Reservoir Simulation
(Second Edition), 2020
8.3.2 Nonlinearity of the slightly compressible fluid flow equation
The implicit flow equation for a slightly compressible fluid is expressed as Eq. (7.81a):
(8.5)
where the FVF, viscosity, and density are described by Eqs. (7.5) through (7.7):
(8.6)
(8.7)
and
(8.8)
The numerical values of c and cμ for slightly compressible fluids are in the order of
magnitude of 10− 6 to 10− 5. Consequently, the effect of pressure variation on the
FVF, viscosity, and gravity can be neglected without introducing noticeable errors.
Simply stated B B , μ μ , and
, and in turn, transmissibilities and gravity
are independent of pressure (i.e., Tl,nn + 1 Tl,n and l,nn
l,n). Therefore, Eq. (8.5)
simplifies to
(8.9)
Eq. (8.9) is a linear algebraic equation because the coefficients of the unknown
pressures at time level n + 1 are independent of pressure.
The 1-D flow equation in the x-direction for a slightly compressible fluid is obtained
from Eq. (8.9) in the same way that was described in the previous section.
For gridblock 1,
(8.10a)
For gridblock i = 2,3,…nx − 1,
(8.10b)
For gridblock nx,
(8.10c)
In the aforementioned equation, Txi
defined by Eqs. (8.3a) and (8.4):
1/2
and Gxi
1/2
for a block-centered grid are
(8.3a)
and
(8.4)
Here again, the well production rate (qscin + 1) and fictitious well rates are handled
in exactly the same way as discussed in the previous section. The resulting set of nx
linear algebraic equations can be solved for the unknown pressures (p1n + 1, p2n + 1,
p3n + 1, … pnxn + 1) by the algorithm presented in Section 7.3.2.2.
Although each of Eqs. (8.2) and (8.10) represents a set of linear algebraic equations,
there is a basic difference between them. In Eq. (8.2), the reservoir pressure depends
on space (location) only, whereas in Eq. (8.10), reservoir pressure depends on both
space and time. The implication of this difference is that the flow equation for an
incompressible fluid (Eq. 8.2) has a steady-state solution (i.e., a solution that is
independent of time), whereas the flow equation for a slightly compressible fluid
(Eq. 8.10) has an unsteady-state solution (i.e., a solution that is dependent on time).
It should be mentioned that the pressure solution for Eq. (8.10) at any time step is
obtained without iteration because the equation is linear.
We must reiterate that the linearity of Eq. (8.9) is the result of neglecting the pressure
dependence of FVF and viscosity in transmissibility, the well production rate, and
the fictitious well rates on the LHS of Eq. (8.5). If Eqs. (8.6) and (8.7) are used to
reflect such pressure dependence, the resulting flow equation becomes nonlinear.
In conclusion, understanding the behavior of fluid properties has led to devising a
practical way of linearizing the flow equation for a slightly compressible fluid.
> Read full chapter
Single-Phase Fluid Flow in a Two-Layer
Reservoir with Significant Crossflow
Chengtai Gao, Hedong Sun, in Well Test Analysis for Multilayered Reservoirs with
Formation Crossflow, 2017
Abstract
Single-phase flow of a slightly compressible fluid is considered in a two-layer, infinite
reservoir. The layers are separated everywhere by a semipermeable barrier, which
allows significant crossflow. Solutions are obtained for pressure response and
crossflow during drawdown and buildup tests when both layers are perforated and
when only one layer is completed.
The numerical results provided an asymptotic analytical solution for pressure and
crossflow, in terms of a modified Boltzmann variable. Analytical approximations were
discovered for crossflow, which apply during “middle” times (the time before area
crossflow rate reaches it's stationary limit). The results obtained are in a form that
can be extended to the multilayer problem.
A well test procedure and interpretation method is proposed for determining the parameters of the individual layers and the semipermeability. The procedure requires
the measurement of wellbore pressure in the two isolated layers, as in a vertical
permeability test. If separate drawdown tests can be run in each layer, the tests need
not be continued into the second linear part of the pressure versus the log time plot.
The resistance of intervening shale between layers can be determined, even if it is
quite high.
A new model of a two-layer reservoir is developed by the application of the maximum
effective hole-diameter concept. The new model is numerically stable when a skin
factor is negative.
> Read full chapter
Decline Curves of Complex Reservoir
In Advanced Production Decline Analysis and Application, 2015
11.4.4.1 Building of mathematical model
We consider the flow of a slightly compressible fluid of constant viscosity in a
circular triple-porosity reservoir in which the outer boundary re is closed. The well is
located at the center of the reservoir and fluid is produced at a constant rate of q;
Matrix 1 is the reservoir space, matrix 2, and fracture are the filtration channel; fluid
crossflows from matrix 1 to fracture and matrix 2 and from matrix 2 toward fracture
in a pseudo-steady state as shown in Figure 11.8(c), and the other conditions are
assumed to be the same as those for the triporosity and monopermeability model.
The basic partial differential equations of decline analysis describing the flow of the
above triporosity and dual-permeability nesting system include
(11.122)
(11.123)
(11.124)
Initial condition is
(11.125)
Inner boundary condition is
(11.126)
Outer boundary condition is
(11.127)
The subscript f represents the fracture system, while m1 and m2 represent the matrix
system, where
The dimensionless variables are defined as
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