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FLOW THROUGH FRACTURES
Submitted to: Dr. S.K. NANDA
HOD, Earth Sciences
Akhilesh Kumar Maury
B.Tech (APE), Gas Specailisation
Email id: email@example.com
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FLOW THROUGH FRACTURES
In highly consolidated rocks where grain packing is relatively tight like carbonates rock, natural fractures
play a vital role in enhancing production rate. There are lot of uncertainties involved in understanding the
architecture and properties of fractures which lead to reservoir modeling. Very often reservoir simulation
software don’t account for complex geometry of real fracture system. The reason is very simple that no
such technology has been developed to image the micro fractures. Moreover no such reservoir modeling
software uses micro-scale flow equations to model the change in flow variables. Fractures are considered
to be highly conductive channels foe flow among all type of porous- permeable formations. Flow mainly
depends upon constrains like fracture aperture and densities etc.
The sole objective of this paper is to analyze the fluid flow in fractures and effects of fracture properties
on flow. To create an outline for the mathematical modeling and reservoir stimulation, advantages and
Akhilesh Kumar Maury
B.tech, Applied Petroleum Engineering,
Under the guidance of:
Dr. S.K. Nanda,
HOD, Earth Sciences.
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A naturally fractured reservoir can be defined as a reservoir that contains fractures (planar discontinuities) created
by natural processes like diastrophism and volume shrinkage, distributed as a consistent connected network
throughout the reservoir. It is undeniable that reservoir characterization, modeling and simulation of naturally
fractured reservoirs present unique challenges that differentiate them from conventional, single porosity reservoirs.
Not only do the intrinsic characteristics of the fractures, as well as the matrix, have to be characterized, but the
interaction between matrix and fractures must also be modeled accurately. Fractured petroleum reservoirs represent
over 20% of the world's oil and gas reserves, but are however among the most complicated class of reservoirs to
produce efficiently. Most of the major naturally fractured reservoirs have active aquifers associated with them, or
would eventually resort to some kind of secondary recovery process such as water flooding, implying that it is
essential to have a good understanding of the physics of multiphase flow for such reservoirs.
This complexity of naturally fractured reservoirs necessitates the need for their accurate representation from a
modeling and simulation perspective, such that production and recovery from such reservoirs be maximized.
Flow in porous media is a very complex phenomenon and as such cannot be described as explicitly as flow through
pipes or conduits. It is rather easy to measure the length and diameter of a pipe and compute its flow capacity as a
function of pressure; in porous media, however, flow is different in that there are no clear-cut flow paths which lend
themselves to measurement. The analysis of fluid flow in porous media has evolved throughout the years along two
fronts—the experimental and the analytical. Physicists, engineers, hydrologists, and the like have examined
experimentally the behavior of various fluids as they flow through porous media ranging from sand packs to fused
Pyrex glass. On the basis of their analyses, they have attempted to formulate laws and correlations that can then be
utilized to make analytical predictions for similar systems. The main objective of this chapter is to present the
mathematical relationships that are designed to describe the flow behavior of the reservoir fluids. The mathematical
forms of these relationships will vary depending upon the characteristics of the reservoir. The primary reservoir
characteristics that must be considered include:
Types of fluids in the reservoir
Number of flowing fluids in the reservoir
There are basically three types of flow regimes that must be recognized in order to describe the fluid flow behavior
and reservoir pressure distribution as a function of time. There are three flow regimes:
The flow regime is identified as a steady-state flow if the pressure at every location in the reservoir remains
constant, i.e., does not change with time. In reservoirs, the steady-state flow condition can only occur when the
reservoir is completely recharged and supported by strong aquifer or pressure maintenance operations.
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The unsteady-state flow (frequently called transient flow) is defined as the fluid flowing condition at which the rate
of change of pressure with respect to time at any position in the reservoir is not zero or constant. This definition
suggests that the pressure derivative with respect to time is essentially a function of both position i and time t.
When the pressure at different locations in the reservoir is declining linearly as a function of time, i.e., at a constant
declining rate, the flowing condition is characterized as the pseudo-steady-state flow. Mathematically, this definition
states that the rate of change of pressure with respect to time at every position is constant, or It should be pointed out
that the pseudo-steady-state flow is commonly referred to as semi-steady-state flow and quasi-steady-state flow.
Fluid Flow through Fractured Reservoirs
One unique feature of naturally fractured reservoirs developed by water encroachment from aquifer or water
flooding is that incremental oil production by infill drilling wells is very little.
This implies that the dominant driving force in water advancement is capillary pressure. Including capillary pressure
under these circumstances is a must.
A model using capillary pressure and relative permeability is presented in this work revealing a linear relationship
between the oil production rate and the reciprocal of oil recovery or the accumulate production.
In addition, it is presented a material balance equation for naturally fractured reservoirs that considers an initially
black oil fluid in a porous medium composed of interdependent matrix and fracture systems.
The equation leads to an improved method of modeling naturally fractured reservoirs by considering the
compressibility difference between fracture and matrix systems.
Modeling separate estimates of oil accumulation have significant economic implications. Poor fracture-matrix
communication will give initially high oil rates that drop quickly because oil is basically produced from the fracture
network. Pore pressure reduction due to production will tend to close fractures leaving behind considerable oil
reserves in the matrix system.
Through this work, an attempt has been made to improve our understanding of the flow behavior of naturally
Single Phase Flow:
It is seen that although the form of the existing transfer function is correct for single-phase flow, most of the existing
shape factors are only valid for parallelepiped matrix blocks and pseudo-steady state flow. It is shown that this might
not always be a good approximation for certain naturally fractured reservoirs, where transient flow and the non-
orthogonally effects are dominant. The effect of transient flow and non-orthogonally of the fracture system was
verified mathematically. It was seen that a time dependent shape factor is required to model transient flow. It is also
acknowledged that the rate of mass transfer can vary quite significantly as a function of the non-orthogonally or the
With these in mind, a general numerical technique to calculate the shape factor for any arbitrary shape of the matrix
(i.e. non-orthogonal fractures) is proposed.
This technique also accounts for both transient and pseudo-steady state pressure behavior. The results were verified
against fine-grid single porosity models and were found to be in excellent agreement.
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Two Phase Flow:
Mechanisms for two-phase mass transfer are studied and it is shown that fluid expansion and imbibition are the main
driving forces governing dual porosity mass transfer.
However, the existing multiphase transfer function is a direct generalization of the single-phase transfer function,
and since the only mechanism governing single-phase flow is fluid expansion, this generalization is not accurate.
That is why a modified transfer function is derived that accurately accounts for fluid expansion and imbibition.
The new transfer function separates the effects of fluid expansion and imbibition into two different terms.
It was seen that the term for fluid expansion is the same as the existing transfer function, and the shape factor
derived for fluid expansion is the same as that obtained for single-phase flow.
The term for imbibition requires a new shape factor and it is seen that this shape factor is a function of time. It is
observed that while prediction of wetting phase imbibition is inaccurate with the existing transfer function, the new
transfer function matches fine grid simulations very well, verifying its validity.
Immiscible Two-Phase Flow:
It is seen that the new transfer function derived for immiscible two-phase flow can be easily generalized to three-
phase compositional flow. This is based on the observation that addition of a third phase does not add any new
mechanisms governing dual porosity mass transfer.
The effect of gravity segregation is also added by assuming that linear superposition of the three mechanisms is
applicable. However, the validity of the above assumption is not verified. Once over all, the present work bring
together valuable information regarding the investigation and evaluation of the naturally fractured reservoirs,
emphasizing the contribution of different disciplines like geophysics, petro-physics, geology, reservoir and
petroleum engineering, the novelty being for reservoir modeling and simulation through a new transfer function
which address the multiphase flow for naturally encountered shapes of matrixes.
The flow of fluids through fractures in rocks is a process that has importance for many areas of the geosciences,
ranging from ground-water hydrology to oil. Research on fluid flow in fractures and in fractured porous media has a
history that spans nearly four decades.
Several conceptual models have been developed for describing fluid flow in fractured porous media. Fundamentally,
each method can be distinguished on the basis of the storage and flow capabilities of the porous medium and the
fracture. The storage characteristics are associated with porosity, and the flow characteristics are associated with
permeability. Four conceptual models have dominated the research: 1) explicit discrete fracture, 2) dual
continuum, 3) discrete fracture network, and 4) single equivalent continuum. In addition, multiple-interacting
continua and multi-porosity/multi-permeability conceptual models (Sahimi, 1995) have recently been introduced in
Bear and Berkowitz (1987) describe four scales of concern in fracture flow: 1) the very near field, where flow
occurs in a single fracture and porous medium exchange is possible; 2) the near field, where flow occurs in a
fractured porous medium and each fracture is described in detail; 3) the far field, where flow occurs in two
overlapping continua with mass exchanged through coupling parameters; and 4) the very far field, where fracture
flow occurs, on average, in an equivalent porous medium.
Developments in laboratory imaging of core samples (David, 1993; Fredrich, 1993; Fredrich et al., 1995; Lindquist
et al., 1996; Doughty and Tomutsa, 1997) provides us with useful means of obtaining sufficient details of the flow
paths of reservoir rock samples in order to model the flow in a single fracture at a pore (micro) scale.
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Explicit Discrete Fracture Formulation
Several investigators have published numerical models incorporating explicit discrete representations of fractures.
Like Travis (1984), most of the models incorporate fractures explicitly, but these models are restricted to fractures
with vertical or horizontal orientation. The advantage of explicit discrete-fracture models is that they allow for
explicit representation of fluid potential gradients and fluxes between fractures and porous media with minimal non-
physical parameterization. But the fact that fractures may be very tortuous and may have significant impacts on the
characterization of flow is often unaccounted for in these models.
Dual-continuum approaches were introduced by Barenblatt et al. (1960) and later extended by Warren and Root
(1963). Dual-continuum models are based on an idealized flow medium consisting of a primary porosity created by
deposition and lithification and a secondary porosity created by fracturing, jointing, or dissolution. The basis of
these models is the observation that unfractured rock masses account for much of the porosity (storage) of the
medium, but little of the permeability (flow). Conversely, fractures may have negligible storage, but high
permeability. The porous medium and the fractures are envisioned as two separate but overlapping continua. Fluid
mass transfer between porous media and fractures occur at the fracture-porous medium interface.
Discrete Fracture Network
Discrete fracture network (DFN) models describe a class of dual-continuum models in which the porous medium is
not represented. Instead, all flow is restricted to the fractures. This idealization reduces computational resource
requirements. Fractures ―legs‖ are often represented as lines or planes in two or three dimensions.
Statistical Models for Fractures
In a series of papers, M. Oda and coworkers presented a statistical approach to describing and modeling the elastic
deformation and fluid flow properties of fractured rocks (Oda, 1982; Oda, 1985; Oda et al., 1987). The tensors for
each physical property are derived by taking a volume average of the expected effect of each fracture in the
population. The volume average contains functions of the fracture orientation, length, and aperture in such a way
that long fracture or wide fractures contribute relatively more than their smaller cousins.
Oda (1982) envisioned that a general geometric property of cracked rock, termed the fabric, determines many
mechanical properties of geological materials. He developed a mathematical description, a fabric tensor, which
considers the following elements of crack geometry: 1) position and density of cracks; 2) shape and dimension of
cracks, and 3) orientation of rocks. One of the primary simplifying assumptions in the derivation of the fabric tensor
concerns the relative position of fractures. Fractures are assumed to have a random position in the network, and all
fractures are distributed uniformly throughout the network, as is often described by a Poisson point process.
Oda (1985) developed a tensor model for the fluid permeability of fractured rock. An assumption required for fluid
flow in his derivation requires that the fractures be comprised of smooth parallel plates with a constant separation, or
aperture h. thus fluid flow is described by the parallel plate model, where the volumetric flow rate is proportional to
the aperture raised to the third power (h ).
The parallel plate model for fluid flow can only be considered a qualitative description of flow through real
fractures. Real fracture surfaces are not smooth parallel plates but are rough and contact each other at discrete
points. Fluid will take a tortuous path when moving through a real fracture.
It is worth noting that further empirical studies showed a relationship between aspects of the crack tensor and the
anisotropy of acoustic wave velocities in laboratory specimens (Oda et al., 1986). This result suggests the potential
to develop an inverse method to derive important aspects of fracture geometry from geophysical methods. Hence the
importance and future prospect of a simulation tool that would handle the details and complexities of a real fracture
and model the flow of reservoir fluids through it is not trivial.
Conventional reservoir flow simulation has its roots in the macroscopic description of fluid flow through Darcy’s
equation. The mathematical fundamentals of reservoir simulators are based on the substitution of Darcy’s equation
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into the mass balance equation for a system and obtaining expressions for pressure in discrete grid-blocks within that
system. The pressure equations contain some terms that are related to the rock and fluid properties. One of the most
important and probably the most uncertain among all is called the transmissibility – a lumped parameter that retains
information about the property of the porous medium, the fluid flowing through the medium, the direction of flow,
and the position in space.
Disadvantages of conventional methods
Because of the tortuousity of the flow paths, the actual fluid velocity will vary from point to point within the rock.
But the velocity used in Darcy’s equation (often referred to as Darcy velocity) is actually an average velocity over
the core, and hence the conventional simulators are limited in scope to analyze the flow at the microscopic (pore)
scale. Because actual velocities are difficult to measure, they are rarely used in reservoir-engineering calculations.
Another problem with conventional reservoir simulation is how the rock permeability, to be used in the
transmissibility term, is assigned to each grid-block. Simulators often use a-priori knowledge of the permeability and
are rarely derived rarely from the transmission or conductive capacity of actual flow paths in a particular grid block.
The conventional methods sometimes cannot incorporate the proper anisotropy information in the flow model,
which results into inaccurate flow simulation. For more accurate reservoir simulations, and particularly for
identifying high potential production zones in fractured reservoirs, a a conventional approach of assigning the
permeability without having correlated with actual flow paths must be changed
The motion of a continuous medium is governed by the principle of classical mechanics and thermodynamics for the
conservation of mass, momentum, and energy. Fluids in a reservoir, as elsewhere, obey these principles; their flow
can be modeled with equations that balance these intrinsic physical properties within a region of investigation. A
brief discussion of the mathematical equations and the underlying assumptions that are generally used to describe
the basic physics of fluid flow at a microscopic level in a medium is presented in the following.
Assuming flow to be incompressible and Newtonian flow, among the petroleum reservoir fluids, gas-free oil and
water can be treated as incompressible fluids under typical reservoir conditions. The equation of motion for a
viscous, incompressible, Newtonian fluid is given by the Navier-Stoke’s equation:
where V and is the velocity vector and P is the pressure in the moving fluid at each point; fe is the external force per
unit volume acting on the flowing fluid; ρ and μ are density and dynamic (or absolute) viscosity of the fluid,
respectively; and ∇ and ∇ are the divergence and the Laplacian operator respectively.
Flow of oil or water in reservoirs, especially in narrow fractures, can be considered as laminar. Furthermore, it is
reasonable to assume that the flow is sufficiently ―slow‖ that inertial effects need not be considered in arriving at a
solution of the equations of motion. The relative importance of inertial and viscous effects is determined by a
dimensionless number, known as the Reynolds number, NRe, defined as:
In slow viscous flows, NRe is small because viscous forces arising from shearing motions of the fluid dominate over
inertial forces associated with acceleration or deceleration of fluid particles. Hence, in describing the motion of fluid
through fractures in reservoirs, the inertial terms, ρ(V•∇)V, can be omitted to give the governing equation of motion
for the so-called creeping flow, known as the Stoke’s equation. In the absence of any body forces, the time-
dependent Stoke’s equation has the following form:
where η = μ/ρ, is the kinematic viscosity of the fluid.
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In the absence of any source/sink inside the system, the equation of mass balance is known as the equation of
continuity, and for an incompressible fluid it is expressed as:
Equations (2) and (3) together provide a description of motion of a (single-phase) fluid through porous-permeable
media. Single-phase flow through fractures can also be modeled by this set of equations. For some simple cases,
such as the flow through a single fracture that can be approximated as flow through parallel plates, analytical
solutions to the Stoke’s equation exist.
A numerical model has been developed for fluid-driven opening mode fracture growth in a naturally fractured
formation. The rock formation contains discrete deformable fractures, which are initially closed but conductive
because of their preexisting apertures. Fluid flow that develops along fractures depends on fracture geometry
defined by preexisting aperture distribution, offsets along a fracture path, and intersections of two or more fractures.
The model couples fluid flow, elastic deformation, and frictional sliding to obtain the solution, which depends on the
competition between fractures for permeability enhancement. The fractures can be opened by fluid pressure that
exceeds the normal stress acting on them and by interactions with intersecting closed fractures experiencing
Coulomb-type frictional slip. The Newtonian fluid is assumed to flow through the conductive fractures according to
a lubrication equation that relates the cube of an equivalent hydraulic aperture to fracture conductivity. The rock
material is assumed to be impermeable and elastic. This paper provides the governing equations for the multiple
fracture systems and the solution methods used. Flow distribution and fracture growth in conductive fracture sets are
simulated for a range of geometric arrangements and hydraulic properties. Numerical results show that elastic
interaction between fracture branches plays a controlling role in fluid migration, although initial apertures can give
rise to a preferential fluid flow direction during the early stage. In the presence of offsets, fracture segments subject
to strong compression are difficult to open hydraulically, and their resulting smaller permeability can increase
overall upstream fracture pressure and opening. The patterns of fluid flow become more complicated if fractures
intersect each other. A portion of injected fluid is lost into closed empty fractures that cut across the main hydraulic
fracture, and this delays the pressure increases required for fracture growth past the crosscutting fracture. The
nonlinear fluid loss rate depends on the geometric complexities of the fracture sets and on the fluid viscosity.
Sometimes fracture growth can be accelerated by the fast fluid transport along an intersected, relatively conductive
With completion of technical paper prepared on topic “Flow through fracture‖, I would like to thank Dr. S.K.
Nanda sir for providing me an opportunity to write a technical paper on this topic.
I would like to thank my friends and colleagues for their kind cooperation.
1. Sudipta Sarkar, M. Nafi Toksöz, and Daniel R. Burns : Fluid Flow Simulation in Fractured Reservoirs,
2. Aziz, K., and Settari, A.: Petroleum Reservoir Simulation, Applied Science Publishers Ltd., London. (1979).
4. Fundamentals of Reservoir Fluid Flow- Tarek Ahmed
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Fig: Fracture development and quantitative Characterization within tightly stratifi ed low net-to-gross
Fig: highlighting the complexities of fracture characterization, development and predictability within a
folded, high net-to-gross clastic reservoir