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SPE-187265-MS
Drilling Mud Loss in Naturally Fractured Reservoirs: Theoretical Modelling
and Field Data Analysis
Omid Razavi, Hunjoo P. Lee, Jon E. Olson, and Richard A. Schultz, Center for Petroleum and Geosystems
Engineering, The University of Texas at Austin
Copyright 2017, Society of Petroleum Engineers
This paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in San Antonio, Texas, USA, 9-11 October 2017.
This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents
of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect
any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written
consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may
not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.
Abstract
Real-time analysis of drilling mud loss data is an effective tool to characterize the size, distribution and
permeability of natural fractures. The estimated properties of natural fractures can then be used to modify the
drilling fluid properties and design remedial mud loss treatment to stop further drilling fluid loss. Moreover,
characterization of natural fracture provides crucial information for the design and application of reservoir
stimulation operations.
In this paper, we present a theoretical model to simulate the invasion of the drilling fluid into natural
fractures. Mathematical modelling was performed for two types of rheology constitutive models which are
commonly used for drilling fluids, namely Bingham Plastic and Herschel-Bulkley. The fluid leak-off into
the formation is modelled using Carter's model. Moreover, a simplified local elasticity equation is used for
the fracture width variation due to fluid pressure. A forward finite-difference numerical scheme is used
to solve the pressure drop equation. The developed model is then validated against well-known analytical
solutions and numerical simulations. Parametric studies were conducted to evaluate the impact of the
fracture deformability, formation leak-off properties, drilling fluid rheology, and the pore fluid contribution.
Finally, the output of the developed model is compared with the field data of a welldocumented mud loss
incident.
The obtained results indicate that the incorporating the effect of the fluid leak-off may significantly
increase the rate of mud loss volume. Therefore, for highly permeable formations (such as carbonates),
considering the impact of the fluid leak-off effect seems necessary for a realistic characterization of the
mud loss problem. Fracture deformability affects the mud loss volume, with higher deformability leading
to a larger mud loss volume. Also, the numerical results confirmed that the rate and magnitude of mud
loss volume is significantly affected by the rheological properties of the drilling fluid, in particular the
yield stress. In contrast, the effect of pore fluid seems relatively negligible for typical formation fluid
types. Comparing the results of our simulation with mud loss field data shows that the developed model
can accurately predict the rate and total volume of mud loss into the fractures. Moreover, it shows that
incorporating the impact of the fluid leak-off and the fracture stiffness may significantly affect the fracture
width prediction. Therefore, these factors should be considered to obtain an accurate estimate of the natural
fracture aperture.
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SPE-187265-MS
Introduction and Background
Accurate characterization of natural fractures is of crucial importance for well construction and reservoir
stimulation. A reliable estimate of the natural fracture size is essential for the design of remedial mud loss
treatments (Razavi et al., 2015, 2016, and 2017a). Also, the knowledge of pre-existing natural fracture
properties can significantly improve the application of hydraulic fracturing treatments and enhance the
overall productivity of the reservoir (Olson et al., 2009, Bahorich et al., 2012, Lee et al., 2015 and 2016).
Real-time monitoring of mud flow data is a common practice in the drilling industry to detect kick or
loss circulation incidents. A systematic analysis of the mud loss data may also be utilized to characterize
naturally fractured reservoirs. Dyke et al. (1995) pioneered the analysis of mud-loss data to determine the
permeability and aperture of natural fractures which intercept the borehole. They presented a qualitative
description of the variation of the mud tank volume based on the aperture of natural fracture. Sanfillippo et
al. (1997) proposed an analytical solution to predict the invasion of drilling fluid within a single fracture.
The proposed solution is then utilized to find the aperture and permeability of the natural fracture from the
mud loss data.
Lietard et al. (1999) developed a model for the radial flow of a Bingham-Plastic fluid into a single natural
fracture assuming no fluid leak-off into the formation. The authors proposed a curve matching technique
to predict the average natural fracture width for arbitrary drilling fluid rheology and wellbore overpressure.
They also showed that for drilling fluids with non-zero yield stress, the final radius of mud invasion within
the natural fracture is finite and proportional to the yield stress of the drilling fluid. In addition, Civan and
Rasmussen (as presented in the discussion section of Lietard et al. 1999) derived a closed-form analytical
solution for the radial flow of drilling fluids into natural fractures, assuming no fluid leak-off and fracture
deformability. Considering the effectiveness and simplicity of the model developed by Lietard et al. (1999),
it may be considered as a major step in the quantitative analysis of mud loss data. However, the application of
this model is relatively limited due to several simplifying assumptions. As stated previously, Lietard's model
assumes no fluid leak-off to the formation matrix. Although this assumption is plausible in tight formations
like shales, its application to mud loss incidents in more permeable formations, such as carbonates, seems
problematic. Also, this model is derived assuming constant fracture width and does not incorporate the
impact of the fracture deformability on the mud loss problem. This assumption may result in a significant
underestimation of the mud loss volume in the natural fracture. Majidi et al. (2010b) extended the model
developed by Lietard et al. (1999) to simulate the radial flow of a yield power-law (Herschel Bulkley) fluid
into a single natural fracture. Also, their model incorporates the effect of formation fluid on the mud invasion
problem. However, similar to Lietard's work (Lietard et al., 1999), they neglected the effect of drilling fluid
leak-off and the fracture deformability.
Lavrov and Tronvoll (2003, 2004, 2005, and 2006) developed a theoretical framework for mud loss
problem in naturally fractured formation and borehole ballooning phenomenon for power-law fluids. They
also studied the effects of formation fluid pressure and borehole diameter on the mud loss volume. In order
to incorporate the impact of the fracture deformability, they have proposed a simplified elasticity equation,
which relates the fracture width variation to the local pressure through a fracture stiffness parameter. Using a
relatively similar approach, Shahri and Mehrabi (2012) and Mehrabi et al. (2012) presented a mathematical
modelling of the fracture ballooning phenomenon in naturally fractured reservoirs.
Recently, Xia et al. (2015a and 2015b) developed a methodology to detect fracture aperture in naturally
fractured formations using the mud loss data. Their methodology focuses on characterizing a network of
natural fractures (as opposed to a single natural fracture). A relatively complex methodology is developed
by coupling the impacts of formation matrix and the fracture. Also, a fracture generation module and a
global optimization module is employed for natural fracture characterization. Given the complexity of the
developed model, it requires significant computational time and most likely high performance computing
SPE-187265-MS
3
systems to conduct the simulations. This, however, limits the application of the developed model for most
practical cases of mud loss data analysis in real-time.
In this paper, we present a new model for the invasion of a drilling fluid into natural fractures. Our goal is
to develop a simple model which provides a realistic characterization of the mud loss problem and delineate
its dependence on main factors which affect the rate and volume of mud loss. In addition, we would like to
develop a model which can be used for real-time analysis of mud loss data using regular computing systems.
Our work in this paper is a continuation of our previous study (Razavi et al., 2017b), in which we investigated
the effect of the fluid rheology on the mud loss prediction for both radial and linear flow types in natural
fractures assuming constant fracture width. In that study, we have shown how incorporating the effect of
leak-off and flow types can affect our prediction of the mud loss volume. In this study, we aim to build on the
results of our previous work by incorporating a more general fracture geometry and fluid rheology model,
which is applicable to a range of different formation types and drilling fluids. In the Approach section, we
briefly discuss the mathematical formulation of our model for both Bingham Plastic and Herschel-Bulkley
fluids. Next, we validate the developed model by comparing it with the analytical solution and numerical
simulation for the mud loss invasion problem. In the Results and Discussion section, we present the results
of several studies to illustrate the effect of leak-off coefficient, the fracture deformability, and the drilling
fluid rheology. Moreover, we have compared the prediction of our model with the field data of a mud loss
incident. In the Summary and Conclusion section, we present a summary of major findings of this work
and suggest some ideas for future work in this field.
Approach
We have derived the transient solution of the radial flow of drilling fluid from the borehole into a natural
fracture with an infinitely large radius (Figure 1). In order to characterize the viscosity of various drilling
fluids, mathematical formulations were derived for the two commonly used rheology models: Bingham
Plastic and Herschel-Bulkley models. The rheology constitutive equation which relates the shear rate to the
shear stress for Bingham Plastic is
Eq. 1
in which, τy is the yield stress, μp is the viscosity, and
is the first derivation of the fluid velocity over the
distance perpendicular to the fracture surface. Also, the constitutive equation for Herschel-Bulkley model
is presented in Eq. 2; where k is the consistency factor, and m is the flow-behavior index.
Eq. 2
Fluid leak-off into formation matrix is simulated using Carter's model (Carter, 1957) as presented in Eq.
3. In this equation, CL is the fluid leak-off coefficient, ta(r) is the arrival time of fluid at location r along the
fracture, and it is given by the inverse function of fluid penetration radius (rf(t)) as shown in Eq. 4.
Eq. 3
Eq. 4
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SPE-187265-MS
Figure 1—The schematics of the radial flow of drilling fluid into a
penny-shaped natural fracture: (a) plan view; (b) elevation view.
Using the model for the width of natural fracture proposed by Lavrov and Tronvoll (2004), the fracture
width at every location is assumed to be a linear function of the local pressure, through fracture stiffness
parameter (KFrac) (Eq. 5). The fracture stiffness parameter determines variation of the fracture deformation
as a function of applied normal stress (i.e., fluid pressure in this case). The assumption of local elasticity
for fracture width calculation is reasonable when the overpressure is relatively small (Lavrov and Tronvoll,
2003). Several experimental studies have attempted to measure the fracture stiffness values of various
naturally fractured samples at various applied normal stresses using uniaxial compressive loading/unloading
cycles (e.g., Raven and Gale, 1985, and Pyrak-Nolte et al., 1987). Moreover, Cook (1992) presented a
theoretical analysis of natural joint closure in terms of the deformation of two rough surfaces. In this study,
we have selected the value of the fracture stiffness based on the range of applied stresses (i.e., fluid pressure
for mud loss problem) and the formation types using the abovementioned experimental and theoretical
studies.
In Eq. 5, the natural fracture assumed to have a non-zero fracture width (w0) at zero overpressure, which
implies that the natural fracture remains open at the reservoir pore pressure. This may be caused by various
geological reasons such as the cement precipitation in the rock matrix around the natural fracture, during
and after the opening process, which makes the rock stiffer and preserves the opening against the ambient
compressive stress state in subsurface (Laubach et al., 2004; Gale et al., 2007). Interface asperities and
surface roughness are two other reasons which can create open natural fractures (Roko and Daemen, 1988).
Eq. 5
In order to simplify the numerical scheme, a pressure distribution along the natural fracture is employed
to characterize the local pressure (p(r)) in Eq. 5.
Eq. 6
This equation satisfies the boundary conditions of the pressure distribution, by providing borehole
overpressure (Δp) at the borehole face (r = rw) and zero pressure at the fluid front (r = rf). It should be
noted that the proposed pressure distribution in Eq. 6 is developed assuming that the amount of the pressure
SPE-187265-MS
5
drop required for displacing the formation pore fluid in the natural fracture is negligible. In this study, we
have studied the effect of the formation fluid on the total pressure drop and showed that typical formation
fluids have a negligible effect on the mud loss problem. Therefore, the proposed pressure distribution in
Eq. 6 is valid for most practical applications. For further discussion on this topic look at the section entitled
"Contribution of the Formation Fluid" in this paper. The exponent value (c) determines the shape of the
pressure profile along the natural fracture zone invaded by the drilling fluid. At each time step, this exponent
value may be altered to match the pressure profile along the fracture.
Implementing Eq. 6 in Eq. 5 yields w(r) as a function of radius (r), and the exponent (c). The obtained
equation for the fracture width (Eq. 7) enables simplifying the elastic equation and the local pressure drop
equation and thereby significantly reduces the computational cost of simulations. In Eq. 7, using a relatively
large value of the fracture stiffness (Kfrac) results in a small variation in the fracture width. In fact, it can be
shown that the fracture width will be equal to w0 when KFrac → ∞. This indicates that when using significantly
large values for Kfrac, the results of our simulation should be similar to the models which assume a constant
width along the entire length of the natural fracture.
Eq. 7
The flowchart shown in Figure 2 presents the numerical algorithm of our model to simulate the drilling
fluid invasion into a natural fracture. The algorithm employs the initial condition for the fluid front radius
as expressed in Eq. 8, which indicates no fluid invasion into the fracture at the beginning of the simulation.
The input parameters include the borehole overpressure, fluid rheology, formation leak-off, and geometrical
properties. Also, a pressure distribution function (Eq. 7) is assumed to determine the fracture width along
the invaded zone of the natural fracture. For the first step, the initial guess for the pressure distribution may
be assumed using an arbitrary exponent value. We have conducted several sensitivity analyses to investigate
the effect of the initial guess for the pressure distribution on the final results of the simulation. These
analyses indicated that when the employed time step is small, the overall effect of the assumed distribution
for pressure at the first time step is negligible. For all other time steps after the first time step, the initial
guess for the pressure distribution may be obtained from the calculated pressure distribution of the previous
step. Using the input parameters and the fracture width distribution, the rate of the fluid penetration into
the natural fracture (drf/dt) may be obtained by solving the local pressure drop equation. The details of
mathematical derivation for the drf/dt solution are explained in Appendix A (for Bingham Plastic fluids) and
Appendix B (for Herschel-Bulkley fluids). After the drf/dt is calculated, the pressure distribution function
may be determined from the pressure drop equation and compared with the assumed pressure distribution
at each time step. An iterative approach is employed to determine the appropriate pressure distribution
function (i.e., to find the most suitable exponent value). Once this appropriate pressure distribution function
is found, the value of the mud penetration radius (rf) and the mud loss volume (VML) are determined using
a forward finite difference scheme and the simulation proceeds to the next time step.
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SPE-187265-MS
Figure 2—Numerical algorithm employed to simulate the invasion of drilling fluid into a natural fracture.
Eq. 8
Mathematical modelling of Bingham Plastic Fluids Penetrations in Natural
Fractures
In Appendix A, we derived the mathematical equations for the one-dimensional radial flow of an
incompressible Bingham Plastic fluid into an infinitely long natural fracture. As explained in the Appendix
A, the derived equation may be solved using an explicit finite difference scheme. In order to validate the
developed mud loss model for Bingham Plastic fluids, we have compared the results of our model with
the closed-form analytical solution derived by Civan and Rasmussen (presented in the discussion section
of Lietard et al., 1999). This solution assumes constant fracture width and no fluid leak-off. Therefore,
we have used a very large fracture stiffness value (KFrac = 108 psi/inch) and zero leak-off coefficient (CL
= 0) to enable comparing our results with Civan and Rasmussen's solution. It should be noted that the
conducted experimental studies by Raven and Gale (1985), and Pyrak-Nolte et al. (1987) indicate that the
value of fracture stiffness at 1000 psi normal stress is ranging between 5 × 105psi/inch and 5 × 106 psi/inch
for different formation types. Thus, the used fracture stiffness values for validation case (108 psi/inch) is
significantly larger than the typical fracture stiffness value and provides a nearly constant fracture width
along the entire zone of natural fracture invaded by the drilling fluid. The other input parameters used for
validation case are shown in Table 1.
SPE-187265-MS
7
Table 1—Input parameters used for validation of Bingham Plastic Model
Geometry
Overpressure
rw(inch)
w0(μm)
Δp(psi)
4.32
620
1200
Fluid Rheology
μp(cP)
19.5
30.5
Figure 3 shows the results of the mud loss volume as a function time for the analytical solution and
the numerical solution from this study. Complete agreement is observed between the analytical and the
numerical solutions. Moreover, since we assume zero leak-off volume, the mud loss volume approaches a
finite asymptotic value, as t → ∞. This is in line with the findings of Lietard et al. (1999) and Majidi et al.
(2010b) that for drilling fluids with non-zero yield stress there exists a finite maximum mud loss volume
(assuming no leak-off to the formation). However, it can be shown that when we consider the effect of the
fluid leak-off, the volume of mud loss may continuously increase due to fluid filtration into the formation
matrix. This topic is further discussed in the Results and Discussion section.
Figure 3—Comparison between the analytical solution (derived by Civan and Rasmussen, as presented in the
discussion section of Lietard et al. 1999) and the numerical solution (developed in this study) for the mud loss
volume of a Bingham Plastic drilling fluid assuming no leak-off and constant fracture width. The obtained results
demonstrate perfect agreement between the Civan and Rasmussen analytical solution and the calculated numerical
solution for the mud loss volume of a Bingham Plastic fluid with constant fracture width and no fluid leak-off.
Mathematical modelling of Herschel-Bulkley fluids Penetrations in Natural
Fractures
In Appendix B, we derive the mathematical equation for the one-dimensional radial flow of an
incompressible Herschel-Bulkley fluid into an infinitely large natural fracture. Unlike Bingham Plastic
fluids, there exists no closed-form analytical solution for the mud loss problem when using the
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SPE-187265-MS
HerschelBulkley rheology model. In order to validate the developed model for Herschel-Bulkley fluids,
we have compared our results with the numerical solution provided by Majidi et al. (2010b). Similar to
Civan and Rasmussen solution, Majid et al. (2010b) solution is derived assuming constant fracture width
and zero fluid leak-off. Therefore, in order to compare the results of our model with Majidi et al. (2010b)
work, we have used zero leak-off coefficient and high fracture stiffness value (KFrac = 108 psi/inch). Other
input parameters for validation of the Herschel-Bulkley model are shown in Table 2. Figure 4 shows the
results of the mud loss volume as a function of time for Majidi et al. (2010b) and the numerical simulation
performed in this study. Similar to the Bingham Plastic fluid, perfect agreement is observed between the
results of our model and Majidi et al. (2010b) work.
Table 2—Input parameters used for validation of Herschel-Bulkley Model
Geometry
Overpressure
Fluid Rheology
m
rw(inch)
w0(μm)
Δp(psi)
4.125
500
1000
10
0.5
1.2
Figure 4—Comparison between the numerical solution (derived by Majidi et al. (2010b)) and the numerical solution (developed
in this study) for the mud loss volume of a Herschel-Bulkley drilling fluid assuming no leak-off and constant fracture
width. Perfect agreement is observed between the results of this study and the solution calculated by Majidi et al. (2010b).
Contribution of the Formation Fluid
In this section, we have conducted a sensitivity analysis to evaluate the contribution of the formation
fluid to the mud loss problem. In Appendix C, we present the transient solution for the radial flow of
HerschelBulkley drilling fluids into an infinitely large natural fracture which is filled with a pore fluid. For
the sake of simplicity, we have neglected the impact of the fracture deformability in this section. Similar
to the previous derivations, the effect of fluid leak-off is simulated using Carter's model (Carter, 1957). We
have used the solution derived by Majidi et al. (2010b) for the contribution of formation fluid on the total
pressure drop. This solution is obtained by solving the diffusivity equation for the flow of a compressible
Newtonian fluid in natural fractures. Therefore, the formation fluid is characterized using two properties:
compressibility (cpore) and viscosity (μpore). The contribution of formation fluid may then be combined with
SPE-187265-MS
9
local pressure drop of the drilling fluid (due to fluid rheology) to obtain a general equation for the total
pressure drop. Next, this total pressure drop equation is solved to find the rate of fluid penetration into the
natural fracture and thereby determine the volume of mud loss.
We have conducted sensitivity analysis to investigate the impact of formation fluid on the invasion of a
Bingham Plastic drilling fluid. The input parameters used for the analyses are shown in Table 3. Numerical
simulations were conducted for three different formation fluid types: typical reservoir water (with [μpore
= 1cP & cpore = 10-6psi-1) and two types of mineral oils with similar compressibility (cpore = 10-5psi-1) and
different viscosities (μpore = 10cP & 100cP). Figure 5 shows the volume of mud loss against time for the
above mentioned formation fluids. Also, these results are compared with the solution obtained assuming
no formation fluid effect. The results clearly indicate that for typical drilling engineering conditions, the
formation fluid contribution to the mud invasion problem is relatively negligible. In fact, it can be shown
when natural fractures are filled with water or moderately viscous mineral oil (μpore ≈ 10 cP), incorporating
the pore fluid effect has no discernible effect. For natural fractures filled with a highly viscous mineral oil
(μpore ≈ 100 cP), incorporating the effect of the formation fluid results in varying the mud loss prediction by
a maximum of 5 percent, compared to the case assuming no formation fluid. Also, the effect of formation
fluid becomes less important as the mud loss time increases. Therefore, in order to reduce the computational
costs of simulations, we neglect the effect of the formation fluid in the remainder of this study.
Table 3—Input parameters used to study the contribution of the formation fluid
Geometry
Overpressure
Fluid Rheology
μp (cP)
rw(inch)
w0(μm)
Δp(psi)
4.32
620
1200
19.5
30.5
3 · 10-6
Figure 5—The volume of mud loss for different types of formation fluids. Three types of formation fluids were
-6
-1
-5
-1
tested: water with μpore = lcp & cpore = 10 psi , and two mineral oils with similar compressibility (cpore = 10 psi )
and with two distinct viscosities (μpore= 10cp & 100cp). The results were compared with the case assuming no
formation fluid in the natural fracture. The obtained results indicate that the contribution of the formation fluid
on the mud loss problem is relatively negligible for typical formation water or low viscosity mineral oil. However,
the effect of formation fluid may become significant when the pore fluid has a very high viscosity (μpore ≥ 100cP).
10
SPE-187265-MS
Results and Discussion
Base Case Results
In this section, we present the results of our simulation for a typical mud loss incident which we used as
the base case. Table 4 presents the input parameters used for the base case simulation. The fluid rheology
is characterized using Herschel-Bulkley model. Moreover, the values of the leak-off coefficient (CL) was
selected using the approach explained in Carter (1957) based on typical formation properties, the borehole
overpressure and the drilling fluid properties. It has been observed that the leak-off coefficient is primarily
affected by the capability of the drilling fluids to control fluid loss into the formation. The values of the
fracture stiffness (KFrac) was selected based on the experimental studies to determine the normal fracture
stiffness (e.g., Raven and Gale, 1985, and Pyrak-Nolte et al., 1987) for the assumed borehole overpressure
(≈ 1000 psi).
Table 4—Input parameters used for the base case
Geometry
Overpressure
rw(inch)
w0(μm)
Δp(psi)
4.32
590
1120
Fluid Rheology
Leak-Off
Coefficient
Fracture Stiffness
3 · 10-6
106
m
19.5
0.9
30.5
The results of the base case simulation are presented in Figure 6. Figure 6a shows the variation of the
fluid penetration radius (ry) against time. The results indicate that the penetration radius reaches a finite
asymptotic value as t→ ∞. The value of mud loss volume is presented in Figure 6b. The results indicate
that the rate of mud loss decreases significantly after around 10 hours. This is the time when the penetration
radius reaches the maximum penetration radius. However, there exists a secondary increasing trend in the
rate of mud loss volume. Our analysis indicates that this secondary increasing trend is associated with the
fluid leak-off to the formation. Figure 6c presents the value of the pressure distribution exponent (c) against
time. The obtained exponent values indicate that the exponent value decreases from 1.5 at the beginning of
the simulation to 1 as the penetration radius reaches its maximum value. This can be explained by the fact
that when rf reaches a constant value (i.e., when vm = 0), the fluid pressure drop becomes a linear function of
radial distance from the borehole (r). In order to evaluate the validity of the proposed pressure distribution,
we plot the coefficient of determination (R2) for the proposed pressure distribution in Figure 6d. The results
indicates that the R2 coefficient is above 0.8 throughout the entire simulation time. Moreoever, for simuation
time after 12 minutes, the R2 value remains above 0.9. Similar trend is observed in the rest of the parametric
studies conducted in this study. Therefore, it seems that the proposed pressure distribution function provides
an acceptable representation of the actual pressure values along the fracture.
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11
Figure 6—Base case analysis: (a) radius of mud invasion into natural fracture; (b) mud loss volume; (c) pressure
2
distribution exponent value (c); (d) coefficient of determination (R ) for the proposed pressure distribution function.
The Effect of Fluid Leak-Off
In order to study the effect of fluid leak-off, we have conducted sensitivity analyses using four leakoff coefficients
. Other input parameters are similar
to the base case as reported in Table 4. We plot the values of the mud invasion radius (rf) against time
in Figure 7a. The results indicate that the mud penetration radius is relatively similar for the cases with
no leak-off and
. However, higher leak-off coefficient (as shown by
, results in a slower rate of fluid penetration into the natural fracture. However,
as the time goes to infinity, all penetration radius curves converge to the same maximum invasion radius
irrespective of the leak-off coefficient. Figure 7b shows the total mud loss volume against time. The results
indicate that greater leak-off coefficient results in greater mud loss volume. Moreover, the effect of leakoff
coefficient on the mud loss volume becomes more dominant at later simulation time, when the fluid invasion
radius approaches its asymptotic maximum value. It should be noted that even for the leak-off coefficient
which yield a similar rate of fluid penetration to the no leak-off case
, there exist
a significant increase in the mud loss volume compared to the no leak-off case.
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SPE-187265-MS
Figure 7—The results of numerical simulation using different leak-off coefficients:
(a) radius of mud invasion into the natural fracture; (b) mud loss volume.
We plot the value of drf/dt against time for the tested leak-off coefficients in Figure 8, to study the effect of
the leak-off on the rate of fluid penetration. The results clearly show that fluid leak-off results in reducing the
indicates a significant reduction in
drf/dt. Furthermore, the results obtained using
rate of fluid penetration compared to the zero leak-off case. The results obtained here may explain why solid
plugging of natural fracture - as performed using lost circulation materials - is typically more challenging in
formation with low permeability (Razavi et al., 2015). Due to the lack of fluid leak-off in these formations,
the fluid penetration rate is faster. A faster penetration rate then complicates the solid plugging of fractures
by preventing the deposition of lost circulation materials along the fracture.
Figure 8—The effect of the fluid leak-off on the fluid penetration rate
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13
The Effect of the Fracture Stiffness
The fracture stiffness (KFrac) affects the fracture width, as explained in Eq. 7. Variation of fracture width will
change the rate of fluid invasion and the volume of mud loss. In order to study the effect of the fracture
stiffness, we have tested two KFrac values (106 and 3 × 105 psi/in) for two distinct w0 values (590, 390 μm).
Moreover, results were compared with the solution obtained assuming constant fracture width (when KFrac
→; ∞). Other input parameters are similar to the base case as listed in Table 4.
Figure 9a shows the variation of rf against time for all tested KFrac and w0 values. The results indicate that
smaller w0 results in a smaller invaded radius. This is due to a sharper pressure drop in narrower fractures
and consistent with the pressure drop equations, shown in Eq. B 2. Relatively similar rf values were observed
for the tested fracture stiffness values. This can be due to the fact that the ratio of Δp/KFrac is relatively small,
compared to the w0.
Figure 9—The results of numerical simulations using different fracture
stiffness and width values: (a) fluid invasion radius; (b) mud loss volume.
Figure 9b shows the mud loss volume for different tested cases. For both tested natural fracture widths
(590 & 390 μm), the total mud loss volume significantly increases as the fracture stiffness decreases. This
is because a smaller fracture stiffness results in a larger fracture width under the same local pressure, and
therefore larger mud loss volume. Moreover, wider fracture results in a faster penetration rate (drf/dt),
which further increases the mud loss volume. For both tested fracture widths, the results obtained assuming
constant fracture width (KFrac → ∞) are respectively 10 and 30 percent smaller than those obtained assuming
106 and 3 × 105 psi/in fracture stiffness values. For typical carbonate and shale formation types, the value
of the fracture stiffness under the tested overpressure range (≈ 1000 psi) ranges between 105 - 106 psi/inch
(Raven and Gale, 1985). Therefore, it is clear that neglecting the effect of the fracture stiffness may result
in a significant underestimation of the mud loss.
The Effect of the Fluid Rheology
In this section, we have investigated the effect of drilling fluid rheological properties. Parametric studies
were performed to investigate the effect of the fluid yield stress (τy), the flow-behavior index (m), and the
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SPE-187265-MS
consistency factor (k). For all conducted parametric studies, the input parameters are similar to those of the
base case, as presented in Table 4.
In order to study the effect of the yields stress, we have conducted simulation using three different yield
stresses: 15, 19.5, 25 lb/100ft2. Figure 10a plots the values of the invasion radius against time. Overall,
similar trend is observed in all three curves during the first hour of the simulation. Beyond this time
threshold, however, there exists a significant discrepancy between the three fluids. The fluid with the lower
yield stress penetrates deeper and faster into the natural fracture network. This can be explained by analyzing
the fluid pressure drop equation (as presented in Eq. B 2), in which the effect of the yield stress becomes
dominant when the fluid velocity is small. As shown by this equation, the effect of the yield stress is more
dominant when the velocity of mud front drops (i.e., smaller q). This means that the yield stress effect
becomes more pronounced as the rf reaches its maximum threshold, which is consistent with the obtained
results in our simulations. Figure 10b illustrates the mud loss volume against time for different yield stresses.
Similar to the observation made for the fluid invasion radius, lower yield stress results in larger total mud
loss volume. These results confirmed the importance of engineering the yield stress of drilling fluids to
control the mud loss problems while drilling naturally fractured formations.
Figure 10—The results of simulation using various yield stress values: (a) fluid invasion radius; (b) mud loss volume.
Numerical simulation were performed using four different flow-behavior index (m = 1, 0.9, 0.8,0.7);
where m = 1 represents the Bingham Plastic fluid and other m values characterize Herschel-Bulkley fluids.
Figure 11a shows the mud invasion radius (rf) variation with time. The obtained results show that the drilling
fluid with a lower m value penetrates faster into the natural fracture. However, the total penetration radius
is approximately similar for the tested m values, with the fluid with the lower m penetrates slightly deeper
into the formation. Unlike the yield stress, the flow-behavior index primarily affects the mud loss invasion
during the beginning of the mud loss incident. This is because the effect of the flow-behavior index is more
dominant when the fluid flow rate (i.e., rate of fluid invasion) is larger. This happens at the initial stage of
the drilling fluid invasion into the fracture. Figure 11b shows the total mud loss volume increases for the
tested m values, which indicates a similar trend to what is observed in the invasion radius. For the drilling
fluids with m = 0.7, the majority of the mud loss occurs during the first 2 hours and, the rate of the loss
volume slow down significantly after 4 hours. In contrast, the mud loss volume with m = 1 grows slowly in
the early stage of the mud loss but continues with a larger rate of mud loss after the first 2 hours. Overall, the
SPE-187265-MS
15
mud loss with a lower m value results in a larger mud volume in the beginning of the mud loss incident. This
is because a lower m value results in a lower apparent viscosity at higher shear rates, which therefore leads
to a larger lost volume during the primary phase of the mud loss incident (when the shear rate is larger).
Figure 11—The effect of the flow-behavior index: (a) fluid invasion radius; (b) mud loss volume.
Numerical simulation were conducted to investigate the effect of the consistency factor (k) on the mud
loss problem. Three different consistency factors were tested, namely 20, 30.5, and 40 cP/sec1-m. The results
of the fluid invasion radius and mud loss volume are shown in Figure 12. For the tested consistency factors,
the impact of the consistency factor on the mud loss problem seems less important, compared to other
rheology parameters. Similar fluid invasion radius is observed for all tested consistency factors. A lower
consistency factor results in a faster mud loss volume during the initial phase of mud loss incident. However,
the long-term mud loss volume is not affected by the consistency factor. This is because when the fluid
invasion rate decreases (as rf approaches its finite asymptotic value) the effect of the consistency factor
becomes negligible.
Figure 12—The effect of the consistency factor: (a) fluid invasion radius; (b) mud loss volume.
Comparison with Field Data
In order to better evaluate the capability of our model to predict the rate and volume of mud loss into natural
fractures, we have compared the output of our model with the field data of an actual mud loss incident.
16
SPE-187265-MS
Figure 13 shows the results of a severe mud loss incident in which a significant volume of drilling fluid (≈
400 bbls) was lost in two hours. The details of this mud loss incident was presented in Majidi et al. (2010b).
Table 5 shows the drilling parameters of the field data of this mud loss incident, as reported by Majidi et
al. (2010b). In their study, Majidi et al. (2010b) analyzed this field data using a mud loss model which
neglects the impact of the fracture deformability and leak-off. Their analysis estimated a natural fracture
width of 880 im (assuming that the pore fluid is a low viscosity mineral oil). Moreover, when the pore fluid
is assumed to be a highly viscous mineral oil, they estimated a natural fracture width of 970 μm.
Figure 13—The predictions of our model is compared with the field data of a real-life mud loss incident.
The mud loss field data is adopted from Majidi et al., 2010b. Fracture width predictions were obtained
using two different sets of assumptions for fluid leak-off and fracture stiffness: (a) zero fluid leak6
6
off and Kfrac = 10 psi/inch; (b)
and Kfrac = 10 psi/inch. The values of the natural
fracture width were determined for each set of assumptions, using conventional curve fitting techniques.
Table 5—Drilling Parameters of the Mud Loss Incident Field Data (adopted from Majidi et al., 2010b)
Borehole Radius
Borehole Overpressure
Drilling Fluid Rheology
rw(inch)
Δp(psi)
m
4.125
700 - 800
8.4
0.94
0.08
In order to evaluate the impact of the fluid leak-off and the fracture deformability on the fracture width
prediction for a real-life mud loss incident, we have revisited the above mentioned mud loss dataset using the
theoretical model developed in this study. It should be noted that our analysis in this section is solely aimed
at demonstrating the capability of the developed model to predict the rate and volume and mud loss and to
emphasize the importance of considering the effect of the fluid leak-off and the fracture deformability. An
exact prediction of the size of natural fractures, however, is not the focus of our study here. This is because
SPE-187265-MS
17
such an exact prediction requires information regarding the type and lithology of the drilled formation, and
the fluid loss properties of the drilling fluid.
In Figure 13, we present a comparison between the predictions of our model with the field data of the mud
loss volume. Numerical simulations were performed using two distinct sets of assumptions for the fracture
stiffness and leak-off coefficients. First, we perform the simulation assuming zero leak-off and a fracture
stiffness of (KFrac = 106 psi/in). Using these assumptions, we have obtained a natural fracture width of 820
[im. Second, we perform the simulation assuming a similar fracture stiffness with a leak-off coefficient of
. Using the input parameters of the second case, we have obtained a natural fracture
width of 740 [im. For both analysis cases, acceptable agreement may be observed between the predictions
of our model with the field data. Comparing these results with the predictions obtained by Majidi et al.
(2010b) indicates that neglecting the impact of the fracture deformability and leak-off coefficient results
in a significant overestimation of the natural fracture width. In fact, it seems that neglecting the fracture
deformability and leak-off coefficient may result to about 20-30 percent error in fracture width prediction.
The effect of fluid leak-off seems particularly important, as this parameter may be significantly reduced
using suitable fluid loss control agents in the drilling fluid.
Summary and Conclusion
•
•
•
•
•
•
In summary, we have developed a novel theoretical model which can be used for real-time analysis
of the mud loss data and characterize the size of natural fractures for arbitrary drilling fluid
rheology, fluid leak-off properties, and natural fracture deformability. The findings of this study
are relevant to reduce non-productive time in well construction by optimizing the selection of lost
circulation materials and fluid loss control agents. Also, the theoretical approach may be utilized
to simulate various problems related to the invasion of drilling fluids in the fractures, including
borehole ballooning/breathing phenomenon, and initiation and propagation of drilling induced
fracture.
The total mud loss volume significantly increases in more compliant fractures (i.e., fractures with
lower normal stiffness values). This is because a more deformable fracture results in a larger
fracture width expansion due to the fluid pressure, and therefore larger mud loss volume. Our
parametric studies indicate that for typical drilling engineering scenarios, considering the impact
of the fracture deformability may change the loss volume prediction between 10 to 20 percent.
Fluid leak-off can significantly affect the volume of mud loss into the natural fractures. Overall,
higher leak-off coefficient results in a faster rate and larger volume of mud loss. This finding is
also important from a practical standpoint as it emphasizes the importance of fluid loss control to
mitigate the rate and volume of mud loss in natural fractures.
Numerical simulations were performed to investigate the impact of the fluid rheology on the mud
invasion radius and volume into the natural fracture. Overall, our study confirms the findings of
Lietard et al. (1999) and Majidi et al. (2010b) that the rate and volume of mud loss is significantly
affected by the rheological properties of the drilling fluid, in particular the yield stress. Overall,
higher yield stress results in slower rate and smaller volume of mud loss. Compared to the yield
stress, the impact of the consistency factor and the fluid-behavior index are less significant.
The sensitivity analysis conducted on the effect of the contribution of the formation fluid indicates
that for typical pore fluids, the effect of the pore fluid on the mud loss volume is negligible. For
formations filled with a highly viscous pore fluid, the contribution of formation fluid may affect
the total mud loss volume by a maximum of 5 percent.
Comparing the results of our model with the field data of a mud loss incident indicates that the
developed theoretical model is capable of accurately predicting the rate and volume of mud loss
18
SPE-187265-MS
•
for a real-life drilling engineering scenario. Moreover, we have shown that neglecting the impact
of the fracture deformability and leak-off coefficient can results in a significant overestimation of
the natural fracture width. An excessively high estimate of the natural fracture width results in
using unrealistically coarse lost circulation materials for remedial loss control treatments. This can
complicate the mud loss control operation and increase non-productive time associated with lost
circulation incidents.
It should be noted that in this study we did not incorporate the impact of solid content of the drilling
fluid on the mud loss problem. However, this impact can significantly affect the rate and volume of
mud loss, as the deposition of solids along the natural fractures creates low-permeability pressure
barriers which can then drastically reduce the fluid pressure and prevent further losses. Therefore,
to reach a more accurate characterization of the mud loss problem, it seems necessary to incorporate
the impact of the solid content of drilling fluids. Further research effort is required in this field.
Acknowledgement
The authors thank the Fracture Research and Application Consortium (FRAC) at The University of Texas
at Austin for supporting this work.
Nomenclature
c Exponent value used in the local elasticity equation
cPore Pore fluid compressibility
CL Leak-off coefficient
k Consistency factor
KFrac Fracture stiffness
m Flow-behavior index
p Fluid pressure within fracture
q Flow rate
r Radial distance
rf Radius of fluid invasion
rw Wellbore radius
R2 Coefficient of Determination
t Time
ta Arrival time
u Leak-off velocity
vm Drilling fluid radial velocity within the fracture
VLO Leak-off volume
VML Total mud loss volume
W Fracture width
Wo Fracture width at zero overpressure
Z Distance perpendicular to the fracture surface
Δp Borehole overpressure
Δpmud Mud pressure drop
Δpore Pore fluid pressure drop
τ Shear stress
τy Yield stress
μp Plastic viscosity
μpore Pore fluid viscosity
V Fluid velocity along the fracture width
SPE-187265-MS
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Appendix A
Mathematical Modelling of Bingham Plastic
Drilling Fluids Penetration in Natural Fractures
The constitutive rheology equation for Bingham Plastic fluid relates the shear rate to the shear stress as:
Eq. A 1
where τy is the yield stress, is the viscosity, and
is the first derivation of the fluid velocity over the distance
perpendicular to the fracture surface. For the one-dimensional radial flow of an incompressible Bingham
Plastic fluid, the local pressure drop is given by (Lietard et al., 1999):
Eq. A 2
where vm is the velocity of the drilling fluid, and w is the width of the natural fracture at the radius r.
Considering the effect of the fluid-off, the fluid velocity (vm) at radius, r, and time, t, may be obtained from:
Eq. A 3
In which VML, and VLO are the total mud loss volume and leak-off volume, respectively. VLO can be
expressed as:
Eq. A 4
Using Carter's leak-off model (Carter, 1957) for leak-off velocity u(r,t) yields:
Eq. A 5
CL is the fluid leak-off coefficient, ta(r) is the arrival time of fluid at location r along the fracture, and
it is given by the inverse function of rf(t) as:
Eq. A 6
Also, VML is given by:
Eq. A 7
The fracture width at each point is related to the local fluid pressure inside the fracture through the fracture
stiffness (Lavrov et al. 2003 and 2004):
Eq. A 8
where w0 is the fracture width at zero fluid overpressure (i.e., pore pressure), and KFrac is the fracture stiffness.
The pressure distribution along the natural fracture is assumed to be given by:
22
SPE-187265-MS
Eq. A 9
Implementing Eq. A 9 in Eq. A 8 yields w as a function of radius (r), and the exponent value (c). The
exponent at each time step may be calculated to capture the pressure distribution.
Eq. A 10
Implementing Eq. A 4, Eq. A 5, Eq. A 6, Eq. A 7, Eq. A 10 in Eq. A 3 yields:
Eq. A 11
In which, Q is given by:
Eq. A 12
Implementing the Eq. A 10 and Eq. A 11 in Eq. A 2 to calculate the fracture width and fluid velocity,
and integrating the pressure drop equation (Eq. A 2) from rw to rf, yields the following equation for drf/dt:
Eq. A 13
In which, F1, F2, and F3 are three integral functions given by:
Eq. A 14
Eq. A 15
Eq. A 16
To our knowledge, there exists no closed-form analytical solution for the above integral functions.
Therefore, these functions should be calculated numerically. Eq. A 13 is a first-order differential equation
and may be solved using an explicit finite difference scheme, with the following initial condition:
At each time step, once the drf/dt is calculated, the pressure distribution along the fracture, p(r), may be
calculated by integrating the pressure drop equation (Eq. A 2) from rw to r:
SPE-187265-MS
23
Eq. A 17
The obtained pressure distribution from Eq. A 17, then may be compared with the assumed pressure
distribution as presented in Eq. A 10. An iterative approach may be used to find the most appropriate
exponent value (c) in Eq. A 10 which provides the closest match between the assumed pressure distribution
in Eq. A 10 and obtained pressure in Eq. A 17. In fact, using a regression analysis the exponent value may
be obtained as:
Eq. A 18
In which, pi is the pressure values at various radius (ri) points as obtained from Eq. A 17. In order to
evaluate the validity of the proposed pressure distribution in Eq. A 9, the coefficient of determination (Æ2)
for the regression analysis may be calculated by:
Eq. A 19
24
SPE-187265-MS
Appendix B
Mathematical Modelling of Herschel-Bulkley
Drilling Fluids Penetration in Natural Fractures
The constitutive equation for Herschel-Bulkley fluid relates the shear rate to the shear stress as:
Eq. B 1
Where τy is the yield stress, k is the consistency factor, m is the flow-behavior index, and
is the first
derivation of the fluid velocity over the distance perpendicular to the fracture surface. Using an approximate
solution for the momentum balance of radial flow of Herschel-Bulkley fluids as derived by Majidi et al.
(2010a), the local pressure drop is given by:
Eq. B 2
In which, the flow rate (q) and the fracture width (w) may be calculated using the equations Eq. A 3 Eq. A 12, given in Appendix A. Implementing these equations in the pressure drop equation for HerschelBulkley fluids (Eq. B 2) and integrating from rw to rf gives:
Eq. B 3
In which, the leak-off velocity (u) is given by Eq. A 5, F2 function is given by Eq. A 15, and Q is given
by Eq. A 12. Eq. B 3 is a non-linear first order differential equation and may be solved using a forward
finite difference scheme. In this work, we have used the secant method to calculate the drf/dt at each time
step. Once the drf/dt is calculated, the pressure distribution may be obtained from:
Eq. B 4
An iterative approach, similar to what has explained in Appendix A, may be employed to determine the
pressure distribution exponent (c) and the coefficient of determination (R2) at each time step.
SPE-187265-MS
25
Appendix C
The Effect of the Formation Fluid
In this section, we derive the mathematical solution for the effect of the formation fluid on the drilling fluid
penetration in natural fracture with constant width. Using the solution derived by Majidi et al. (2010b),
the contribution of the formation fluid effect on the pressure drop for a Newtonian compressible fluid and
assuming infinitely large natural fracture is given by:
Eq. C 1
In which:
Eq. C 2
where μpore and cpore are the viscosity of compressibility of the formation fluid and Ei is the exponential
integral. For constant fracture width, the mud pressure drop may be calculated using:
Eq. C 3
Assuming constant fracture width and using Carter's model for fluid leak-off the fluid flow rate (q) is
given by:
Eq. C 4
Adding the mud pressure drop (Δpmud) and pore fluid pressure drop (Δppore) gives the borehole
overpressure (Δp):
Eq. C 5
Which yields:
Eq. C 6
Eq. C 6 is a non-linear first order differential equation and may be solved using a forward finite difference
scheme. Also, for the Bingham Plastic fluids, the Eq. C 6 may be simplified as:
26
SPE-187265-MS
Eq. C 7
Similar to Eq. C 6, Eq. C 7 is a non-linear first order differential equation and may be solved using a finite
difference scheme. In this work, we have used the secant method to calculate the (drf/dt) at each time step.
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