Below is a list of publications, with accompanying abstracts, authored
or co-authored by members of Meyer & Associates, Inc. from 1980 to
March 2008.
2005
A new solution methodology is available for the pseudosteady-state
behavior of a well with a finite-conductivity vertical fracture based on
a reservoir/fracture domain resistivity concept (SPE Paper 95941).
It is easily implemented for fracture optimization by maximizing the
productivity index for a given proppant mass (or volume).
This article presents a summary of the pseudosteady-state equations
for finite-conductivity fractures in square formations, and focuses on
fracture optimization based on the unified fracture design methodology of
Prats and Economides. That is, for a given proppant volume and
conductivity, there exists an optimum fracture conductivity and
penetration that will maximize productivity.
Method
Optimizes Frac Performance (PDF; 315 KB)
A novel solution methodology for pseudosteady-state behavior of a well
with a finite-conductivity vertical fracture is formulated using a
reservoir/fracture domain resistivity concept. The formulation
encompasses a transformed resistivity domain based on an equivalent or
effective wellbore radius. The resulting pseudosteady solution is
presented in the form of the dimensionless productivity index,
JD.
Some of the major advantages of this pseudosteady solution for
finite-conductivity vertical fractures are:
- The methodology is based on fundamental principles,
- The solution is analytical and easily implemented,
- The equations are formulated for rectangular reservoirs,
- The model includes wellbore flow (i.e., wellbore radius effect)
which is important for low-conductivity fractures, and
- The formulation accounts for a piecewise continuous linearly
varying fracture conductivity including: proppant tail-ins,
over-flushing, pinch zones, choked flow (internal), and external skin
mechanisms.
The stimulation ratios for finite-conductivity fractures with an
undamaged well are compared with those of McGuire and Sikora (1960), and
Holditch (1974). The accuracy and validity of the pseudosteady model is
also illustrated by comparison with the works of Prats (1961, 1962),
Gringarten et al. (1974), Cinco-Ley et al. (1978, 1981), Barker and
Ramey (1978), and Valko and Economides (1998).
A summary of the fundamental building blocks, effective wellbore
radius concept, pseudo-skin functions and fracture skin are discussed. An
improvement to Gringarten’s dimensionless productivity solution for
infinite-conductivity vertical fractures in rectangular closed reservoirs
is also presented.
2002
Limited entry design techniques have proven successful for the
simultaneous fracture stimulation of the Fruitland Coal and the Pictured
Cliffs sandstone in the T28N-R7W Federal Unit of the San Juan basin, Rio
Arriba County, New Mexico. Optimization of this completion technique is
dependent upon determining and placing the required effective propped
fracture length in the coal and sandstone formations. This manuscript
addresses utilizing limited entry techniques and a 3D fracture
model. The model is then used to design N2 foam proppant
fracture treatments in the coal and sandstone formations. This completion
methodology allows reserves to be recovered from the Fruitland Coal at a
significant cost reduction. With the different mechanical and reservoir
properties of the two formation types, created fracture geometry will
vary in each formation. Methods used to model these varied fracture
geometries are discussed. Net pressure and production data analysis
provide estimates for the effective propped fracture lengths in each
formation. Radioactive tracer and production logs are presented as
supporting evidence to validate the well completion design, the fracture
modeling inputs and the stimulation of both formations simultaneously.
Using this fracture modeling technology leads to increased reserve
recovery and cost effective proppant fracture treatments.
2000
Hydraulic fracturing simulators currently assume that rocks are linear
elastic materials with a linear stress-strain relationship. Although many
formations do behave in a linear elastic manner when fractured, other
soft rocks have been studied in triaxial and plane-strain compression
tests that show a highly stress-dependent Young’s Modulus even at low
strain levels. The fundamentals of the stress-strain relationship and the
importance of distinguishing between the tangential and secant Young’s
Modulus for hydraulic fracture designs are discussed. Various
stress-strain models are presented for nonlinear elastic and
elastic-plastic behaviors of materials. The first order impact of a
stress-dependent Young’s modulus on the fracture geometry and pressure
behavior is discussed. Various examples of stress-dependent Young’s
Modulus are presented that illustrate the importance of including
non-linear stress-strain behavior in hydraulic fracture modeling.
Parametric studies of limiting non-linear behaviors using
analytical 2D fracture propagation solutions and a numerical
simulation results are also presented. Numerical hydraulic fracturing
solutions for a stress-dependent Young’s modulus in soft rock fracturing
are also discussed. General conditions identifying when non-linear
elastic behavior should be included are also addressed.
Fracture calibration tests (minifracs) are very successful methods for
providing estimates of the fluid efficiency, closure pressure, fracture
geometry and leakoff coefficient prior to the main treatment. The
pressure decline data is normally analyzed using a Nolte type method for
calibration and redesign of the main treatment.
Many times it has been observed that the main treatment has a higher
efficiency (less fluid loss) than the minifrac which can adversely impact
the fracture treatment. This paper addresses the effect of fluid loss
during fracture calibration tests on the main treatment.
An analytical method for leakoff controlled by the filter cake and/or
filtrate fluid has been developed for analyzing the effect of fluid loss
in the formation prior to the main treatment. The main treatment leakoff
velocity and volume loss equations account for the effects of the
minifrac fluid loss behavior, including filter cake, spurt loss,
mobility, time of fracture creation and relative fracture planes.
This paper presents the foundation for a generalized set of equations
quantifying the effects of the minifrac fluid loss on the main treatment
by conservation of mass and Darcy’s law. Equations are formulated for the
main treatment leakoff velocity and volume loss. Numerous figures are
provided that illustrate the parametric effects of the minifrac fluid
loss on the main treatment efficiency and fraction of pad volume.
The relative popularity and success of the frac & pack technique
in hydraulic fracturing has resulted in some misconceptions regarding the
objective, procedure and pressure analysis after a screen-out. This paper
addresses frac & pack procedures and the pressure response after a
tip screen-out (TSO).
An analytical method has been developed for analyzing pressure slope
behavior after a TSO in high permeability formations. These equations
incorporate the first order parameters affecting the fracture pressure,
rate of change of pressure (derivative) and pressure slope behaviors
after a screen-out.
The fundamental equations for pressure slope analysis are similar to
those originally developed by Nolte for pressure decline analysis. The
major difference is that after the fracture stops propagating (i.e.,
after a TSO) the injection rate is not zero. Consequently, if the
injection rate is greater than the leakoff rate, the fracture volume and
net pressure (constant compliance) must increase. If the injection rate
falls below the leakoff rate, the fracture net pressure must
decrease.
Although analytical equations will not replace three dimensional
fracturing simulators normally used for design and real-time history
matching, they do provide insight into the major parameters affecting
pressure behavior after a TSO without running a numerical simulator. The
analytical equations presented in this paper demonstrate why pressure
slopes after a screen-out are typically much greater than unity for low
efficiency fractures.
A generalized set of equations is presented for analyzing the pressure
slope behavior after a screen-out. Numerous graphs are provided which
illustrate the parametric effects of fracture efficiency, spurt loss and
fracture net pressure at the time of a screen-out on the pressure,
derivative and slop behaviors after a TSO. Comparisons of the analytical
pressure slope equations with a three dimensional fracturing simulator
are presented to show the application of the analysis.
A new methodology of frac & pack post analysis is presented using
the pressure slope technique. This methodology utilizes the pressure
slope during a screen-out as a check on the minifrac and fracture
efficiency. Two frac & pack cases with bottomhole data are analyzed
using a three dimensional hydraulic fracturing simulator to illustrate
the pressure slope analysis for low efficiency fractures.
Nolte originally developed a pressure decline analysis method to
provide an estimate of the fluid efficiency, closure pressure, fracture
geometry and leakoff coefficient. The Nolte method of pressure decline
analysis has become a standard for fracture calibration in the industry.
Implementation of a proper calibration test is critical for the
successful design and evaluation of hydraulic fracturing treatments.
Although Nolte’s original work has many simplifying assumptions
regarding fracture closure (e.g., constant leakoff coefficient, no spurt,
no fracture growth after shut-in, constant compliance, etc.) it still
provides the foundation for the basic first order parameters affecting
pressure decline behavior.
The assumption of pressure independent fluid loss may have an adverse
affect on the pressure decline analysis if leakoff is controlled by the
filtrate viscosity and/or reservoir compressibility and mobility effects
or fissure opening effects. An improved procedure is presented for
identifying and implementing pressure dependent leakoff. This methodology
is formulated from the generalized Nolte G function approach utilizing a
dimensionless pressure function based on the original work of Castillo
for a pressure dependent leakoff coefficient and Barree for naturally
fractured reservoirs. The procedure first makes use of the standard Nolte
analysis to identify the presence of pressure dependent leakoff, and then
establishes the appropriate leakoff pressure dependent parameters by
matching the pressure decline behavior. Deviation of the measured data
from the dimensionless pressure function versus Nolte G time indicates
closure. The methodology also utilizes the first derivative and pressure
superposition derivative to help identify and quantify closure.
Guidelines are presented to identify cases where pressure dependent
leakoff should be considered. Implications of neglecting pressure
dependent leakoff or using a pressure dependent leakoff coefficient
improperly are addressed. The governing equations and field examples are
presented for clarity.
1999
Identification of fracture geometry is necessary for the proper design
and execution of a hydraulic fracture treatment. The Appalachian Basin
consists of many different producing formations. These formations can
vary in depth from 600 ft. to 7700 ft. Because of the
low reservoir pressure (many of these formations are also underpressured)
and low permeability, most of these formations require a hydraulic
fracturing treatment to be economically successful.
1996
Tip screen-out (TSO) and frac pack designs have gained popularity in
the industry, especially in high-permeability wells in the Gulf of Mexico
where inadequate conductivity and formation damage have traditionally
been problems.
TSOs and frac packs are generally performed in moderate- to
high-permeability reservoirs that require greater conductivities than
conventional hydraulic fracturing. The implementation of TSOs and frac
packs over the past few years has resulted in substantially greater
fracture conductivities and improved proppant placement. Fundamentally,
these techniques are similar up to the step of fully packing a
fracture.
TSO methodology as presented in “Tip Screen-Out Fracturing: A
Technique for Soft Unstable Formations” by M.B. Smith, W.K. Miller and J.
Haga and applied to the Ravenspurn South natural gas field is used to
deliberately create a proppant screen-out or bridging condition around
the perimeter of the fracture to prevent further propagation and height
growth. Continued pumping results in “ballooning” or an increase in the
fracture aperture with continued increasing fracture pressure. The
increased aperture results in a greater propped width and increased
fracture conductivity. Typically, only the perimeter of the fracture is
packed.
Frac packs differ from TSOs by packing the entire fracture with
proppant from the tip to the well bore at the settled bank concentration,
which greatly increases fracture conductivity. This technique is
typically performed in higher-permeability formations that require large
average conductivities to increase productivity.
The relative popularity and success of the frac pack technique for
hydraulic fracturing has resulted in many misconceptions regarding the
objectives and procedures used for these non-conventional treatments. The
frac pack methodology presented here (and compared to the classical TSO
methodology) was originally developed and implemented into a
three-dimensional hydraulic fracturing simulator (MFrac-IITM
version 7.1). An analytical form of this methodology was presented
at the 1995 annual meeting of the Society of Petroleum
Engineers.
1993
Two-dimensional hydraulic fracture simulators are used to design the
majority of hydraulic fracture treatments in West Texas. These models
assume a fixed hydraulic fracture height. In many cases, hydraulic
fracture treatments often do not stay confined within a fixed height as
the two dimensional models assume. In the Spraberry/Dean siltstones of
West Texas there are no strong barriers to vertical fracture migration,
and two dimensional models are clearly inappropriate. Three dimensional
hydraulic fracturing simulators overcome this limitation of a fixed
height by calculating the fracture height. The inputs required are
considerably more involved, and the 3D models require data from both the
pay sand and boundary layers. If the raw input data are not available or
if sufficient engineering effort is not invested the model output may be
misleading. Wireline logs can provide valuable inputs to these three
dimensional simulators. These inputs will be discussed, along with case
studies where the inputs helped optimize the completion design.
1992
An extension to the current mini-frac procedure is presented to
improve the quality of the analysis. The methodology is based on history
matching the pressure response during pumping and closure for the
mini-frac treatment. This process couples the traditional mini-frac
analysis with a three-dimensional hydraulic fracturing simulator.
The fracture propagation solution for geometry and extension pressure
is coupled with the post injection pressure decline analysis. The
coupling process reduces the possible number of parameter
uncertainties.
The measured fracture extension and decline pressure data is used to
establish the appropriate fracture geometry model and to perform
parametric studies. Reservoir and fluid parameters are then varied to
perform further sensitivity analyses.
Repeating the procedure for a multi-cycle mini-frac operation provides
the analyst with an opportunity to refine the parametric analyses and
select the most probably set of parameters to optimize the treatment.
This results in an improved understanding of hydraulic fracture
propagation and formation response, which results in a higher probability
for further improving production.
The fracture-pressure analysis and procedures provided in this paper
generally follow the theory of mini-frac analysis originally formulated
by Nolte. An important difference, however, is that the pressure-analysis
used here couples a mini-frac simulator with a three-dimensional
hydraulic fracture simulator to predict fracture propagation and pressure
response. The mini- frac simulator is first used for preliminary analysis
and quick estimates of parameters prior to performing more rigorous
parametric studies with the 3D fracturing model.
Parametric studies and field case examples are presented to illustrate
the improved mini-frac/ treatment techniques. Simulated values for
permeability and fracture height are then compared to more traditional
sources of information from lab analysis and logs.
1990
This paper presents a foundation and methodology for real time
three-dimensional hydraulic fracturing simulation and analysis. The
equations governing fracture propagation are summarized for both rate and
net pressure boundary conditions. A new dimensionless pressure slope
parameter is introduced which prevents using chaotic measured or
calculated pressures. This parameter also helps identify near wellbore
restrictions.
The real time fracturing simulator utilizes the same numerical modules
and routines as the design program. This will insure design, real time
and post-design simulation compatibility. Simulated results for the rate
and net pressure driven models can be displayed concurrently during the
hydraulic stimulation.
Comparative studies of the rate and net pressure driven numerical
results are included. A post-treatment analysis of a real time field case
study is presented to illustrate the application of this real time
fracturing system. The importance of using a dimensionless pressure slope
is also discussed.
1989
The solution methodology of a three-dimensional hydraulic fracturing
simulator (MFRAC-II) for use on personal computers is described. The
Simulator is design oriented and user-friendly with menu-driven pre- and
post-processors. It has been formulated and structured to be used as an
everyday design tool.
The coupled rock and fluid mechanics equations governing fracture
propagation are presented. These non-linear partial differential
equations are then transformed and solved using integral methods. The
criteria for fracture propagation and the effect fracture toughness and
confining stress contrast has on fracture propagation are also
discussed.
Comparative studies with full 3D hydraulic fracturing simulators are
included to illustrate the diversity of MFRAC-II. These comparison
studies range from fractures dominated by pure rock mechanics (fracture
toughness) to fractures dominated by viscous driven fluids. The fractures
range from highly uncontained to very well contained fractures.
1988
A new mini-frac methodology is presented for the automated simulation
of mini-frac analysis. The solution is based on solving the governing
momentum, mass conservation and constitutive relationships for
Perkins-Kern/Nordgren (PKN), Geertsma-Deklerk (GDK) and penny shape type
fracture geometries.
The analytical technique is unique because the fracture propagation
characteristics of length, width, pressure and efficiency are numerically
calculated from conservation of momentum and mass principles for
power-law type fluids.
Major enhancements to current mini-frac analysis include:
- momentum and mass conservation
- spurt loss
- fluid flow-back after pumping
- time-dependent fluid loss coefficients
- interference closure
- determination of formation permeability.
The analysis has the additional advantage of using the measured
pressure decline data as a history matching parameter to determine the
appropriate fracture model and sensitivity of input data.
A number of mini-frac designs, parametric studies and filed case
examples are presented to illustrate the applicability of the numerical
technique.
1987
An analytical heat transfer model is presented for the combined
convection along a vertical fracture with conduction and convection in
the reservoir. The model couples the energy (heat transfer) and fracture
propagation equations resulting in a closed form integral dimensionless
temperature solution.
Major Model enhancements include incorporation of the following
mechanisms:
- a finite film coefficient
- time-dependent fracture temperature for calculating the
instantaneous heat flux
- energy storage in the fracture
- coupled energy and fracture propagation equations.
The introduction of a power-law Nusselt number to determine the
convection film coefficient is also new and unique.
The influence of fluid leakoff and a finite convection film
coefficient as heat reduction mechanisms is discussed. The effect of
energy storage and a time-dependent fracture temperature on the heat
transfer rate is also illustrated. Finally, simulations studies
illustrating the effect of heat transfer on fracture propagation
characteristics are presented.
1986
A generalized drag coefficient correlation for particulate settling in
power-law-type fluids is applicable for all flow regimes from Stokes’
(laminar) to Newton’s flow (turbulence).
The major advantage of this correlation for hydrocarbon reservoirs is
that it correctly asymptotes to a modified Stokes’ law (viscous effects)
at low Reynolds vale (Re) numbers and to the turbulent drag coefficient
CD ≈ 0.44 at large values of Re (inertia
dominance).
The use of Stokes’ law can greatly underestimate the drag coefficient
and grossly overestimate the terminal settling velocity at large Re.
In many respects, this article is a review and extension of drag
coefficient correlations used in hydraulic fracturing on particulate
settling in Newtonian and non-Newtonian fluids.
Since most non-Newtonian correlations are for creeping flow (Re′
< 0.1), the major emphasis will be to extend these correlations
to Newton’s flow.
A comparison with experimental data and empirical correlations of
other investigators is included to verify the validity and applicability
of the extended correlation.
This paper presents asymptotic analytical and numerical solutions for
two-dimensional (2D) and three-dimensional (3D) type hydraulic fracture
geometries. The fracture propagation models investigated include: a
Geertsma-Deklerk (GDK), Perkins-Kern/Nordgren (PKN) and a 3D type
model.
Comprehensive 2D and 3D design formulae for power-law
fracturing fluids are given for various cases of fluid loss and
containment. These formulae can be used to benchmark 2D and 3D hydraulic
fracturing simulators. Characteristics of equilibrium 3D height growth
are also discussed. Parametric studies based on the design formulae are
performed to show the effect of any one variable.
Typical fracture designs are simulated for various containment and
leakoff examples using each of the fracture propagation models, along
with a comparison of proppant scheduling and design. The simulation
studies identify and illustrate the basic characteristics and design
differences of each model.
1985
A new three-dimensional, hydraulic fracturing simulator (MFRAC-II) is
introduced in this first of a four-part series of articles.
Design-oriented and user-friendly, it has been structured and formulated
to be used as an everyday design tool.
Subsequent installments will include proppant transport, confining
stress, variable injection rates, fracture closure, and post-fracture
production simulation.
Menu-driven, pre and post-processor modules are included to facilitate
ease of data input and graphical display of numerical results. The
simulator is written in standard Fortran 77 for use on a personal
computer as well as mini/main frames. Typical execution time on a PC is
on the order of minutes.
1981
Analytical models have been developed to calculate steady and unsteady
one-dimensional nonhomogeneous multiphase dissolution flows in variable
area inclined channels. The models are based on conservation of mass and
momentum of each of the phases for a liquid containing dissolved gas
which tends to come out of solution in a supersaturated pressure
field.
The steady state system of nonlinear initial value differential
equations is solved numerically by an iterative variable step-size
predictor-corrector algorithm for the pressure, void fraction and
velocity profiles. This method incorporates the capability of predicting
choked flow. The method of calculating transient multiphase flow is
presented based on a modified implicit multifield (IMF) method for the
field variables using a drift-flux model. The unsteady numerical finite
difference method incorporates the flexibility for variable advanced
timing of convective terms from a fully implicit to purely explicit form.
The method has the capability of treating the mass phases fully
implicitly. This numerical method incorporates spatial coupling of
pressure and incremental pressure terms which results in
order-of-magnitude gains in computation time.
Steady and unsteady numerical calculations are presented to
demonstrate the numerical validity and accuracy of each code for a wide
range of multiphase steady and transient dissolution flows. These
problems include the numerical calculation of steady full dissolution and
single-phase flows and unsteady calculations of pressure pulse
propagation, blowout phenomena, flow oscillation and asymptotic
behavior.
The immediate application of the present work was to analyze
multiphase dissolution flows for compressed air energy storage (CAES)
systems. The steady-state numerical results are summarized as a
parametric study of the effect mass transfer and system parameters have
on pressure, void fraction and velocity profiles for the RPI high
solubility carbon dioxide-water system. A comparison of numerically
predicted and RPI experimental results is provided. The unsteady analysis
is performed for a typical CAES start-up transient to demonstrate the
dynamic system response of multiphase flows. The resulting time-dependent
pressure, void fraction and velocity profiles are reported at various
nodal positions.
1980
