B037 DEVELOPMENT AND APPLICATION OF A HYDRAULIC
FRACTURE RESERVOIR SIMULATION TOOL
Torsten Friedel and Frieder Häfner
Institute of Drilling Engineering and Fluid Mining,
Freiberg University of Mining and Technology,
Agricolastraße 22, D-09596 Freiberg (Germany)
Abstract
A single component, three-phase reservoir simulator is presented for detailed investigations of
hydraulically fractured wells. The tool features a gas and loadwater phase, capturing inertial
non-Darcy flow, and a special gel phase to represent the highly viscous fracturing fluid. Its nonNewtonian fluid characteristics is modelled by means of the Herschel-Bulkley fluid model with
yield stress. Using state of the art numerical techniques, the tool is built around a fully unstructured discretisation framework, providing an accurate capture of the flow around fractures and
wells and the opportunity for sophisticated wellbore flow modelling. The fracture Voronoi grids
allow for simulation with well-test accuracy. The simulator is validated against analytical and
numerical solutions.
Introduction
The benefits of a precise analysis of the transient flow in the fracture vicinity, with the aim of
improving well performance, have been well recognized. Furthermore, the enormous development
costs in, e.g., tight-gas reservoirs using multiple fractured horizontal wells, make accurate and
reliable production forecasts a necessity.
Fig.1 presents specific processes at fractured wells. A realistic cleanup scenario may imply: (i) the
simultaneous flow of three phases, (ii) the formation of a load water invasion zone accompanied
by hydraulic and mechanical damage in the fracture vicinity, (iii) filtercake buildup and erosion,
(iv) gel residue damaging the proppant pack, incorporating complex non-Newtonian rheology,
and (v) viscous fingering through the proppant pack. During the subsequent production, inertial
non-Darcy flow and geomechanical effects, e.g., stress dependency of reservoir permeability and
fracture closure, impact the behaviour of the fractured well. Furthermore, the fracture-reservoir
system is characterised by highly discriminative magnitudes in space, with fracture width in the
range of millimeters and by distinct time scales of cleanup and post-fracture production periods.
A hydraulic fracture reservoir simulation tool has been developed to allow for detailed cleanup
investigations as well as post-fracture evaluation within one unified workflow model. The tool
aims to consider physical and technological aspects as accurately as possible. Primary objectives
of the tool development and its application are: (i) to obtain a considerable increase in the quality
of predictions from tight-gas reservoirs with fractured wells, by taking into account the specific
conditions with the corresponding petro-physical functions, (ii) improving the insight into complex and highly nonlinear multi-phase flow phenomena, and (iii) contribute to a more realistic
simulation technology for such reservoirs.
9th European Conference on the Mathematics of Oil Recovery - Cannes, France, 30 August - 2 September 2004
Gel residues
unbroken polymers with
non-Newtonian rheology
Undamaged
reservoir
(< 1 km)
Stress dependent
reservoir permeability
Cleanup processes
Multi-phase flow
effects
Inertial (non-Darcy) flow
=f(p)
Leakoff-zone
(invaded loadwater)
(< 1m)
Capillary endeffects
Fractureclosure
Fracture
(width < 10 mm)
Well
(diameter < 0.10 m)
Proppant
Mechanical
damage zone
Filter cake
Figure 1: Specific processes at hydraulically fractured wells
Physical and Mathematical Model
The starting point for the mathematical formulation is the integration of the mass balance equation
for Np phases, over a finite control volume, ∆V , in its simplified form:
−
Np Np
∂
(φρp Sp ) dV −
(ρpup ) n dA =
qp ,
∂t
p=1
p=1
p=1
Np
∆A
(1)
∆V
where the volume integral of the advection term is already revised in a surface integral ∆A of
the total surface ∆A by means of the Gaussian Theorem. For practical purposes, the gas, water
and gel phases can be considered immiscible. The fluid system present is characterised by a highly
discriminative rheology with different flow regimes. The classical method of using a single motion
equation is hence insufficient. Rather, different kinds of motion equations needs to be implemented
and switched in the simulator dependent on the fluid type.
Traditionally, Darcy’s law has been used in petroleum reservoir engineering to account for viscous
forces while neglecting inertial forces. These, however, are crucial for gas reservoirs, even in
a low rate tight-gas environment, and are captured with the well known Forchheimer-equation.
The inertial pressure drops are primarily caused by the continuous de- and acceleration of fluid
molecules travelling along a tortuous flow path through the interconnected pores and also in the
proppant pack. In vector form, it can be written as:
µ
(2)
+ βt ρ|u| u .
−grad Φ =
k
Since a value βt = 0 marks the transition to Darcy’s law, Eq.(2) is the default motion equation for
the gas and the water phase. The velocity of both phases is then:
up,ND = −δp
kkr,p
grad Φp ,
µp
δp =
1
,
βt ρp kkr,p
1+
|up,ND |
µp
(3)
already inhering the common multi-phase flow concepts. Permeability is assumed horizontally
isotropic. Permeability and non-Darcy flow coefficients can realistically be defined as a function
of the effective stress according, e.g., to the proppant type. Setting the fracture conductivity dependent on the effective stress also allows for consideration of fracture closure.
A variety of important fluids exhibit non-Newtonian behaviour, i.e., the shear rate depends nonlinearly on the shear stress. Such fluids are primarily based on polymers and are often utilised for
drilling or fracturing. The present approach facilitates a special gel phase to represent the fracturing fluid. This differs from the common concept to model polymers as a component of the
water phase, affecting its viscosity. The equation used to calculate the viscosity of the gel phase
is derived from the Herschel-Bulkley fluid model: τ = τ0 + K γ̇ n . Its derivation is based on one
dimensional flow through a single capillary, using the Blake-Kozeny equation for laminar flow
of Newtonian fluids in packed beds.1 In the rheology model, τ0 is the yield stress, K is the fluid
consistency index and n constitutes the fluid behaviour index. The velocity of a Herschel-Bulkley
fluid can be written analogously to the non-Darcy velocity in a generalised form:
⎧ 1/n
⎪
kk
φC
1
⎪
r,gel
⎨ −
, |grad Φ| ≥ φC
τ ;
τ0
grad Φ
1−
2k 0
µeff
2k |grad Φ|
, (4)
up,HB =
⎪
⎪
⎩ 0,
φC
|grad Φ| <
τ .
2k 0
where C is a tortuosity constant. Gravity effects are neglected. An effective viscosity coefficient
µeff for multi-phase flow of such fluids is defined as follows1–3:
n
1−n
K
3
(150Cφ(Sgel − Sgel,irr )kkr,gel ) 2 .
(5)
9+
µeff =
12
n
In order to adapt the numerical framework to accommodate the non-Newtonian fluid behaviour,
Eq.(4), a gel viscosity µgel is introduced. This viscosity reflects the equivalent viscosity of a nonNewtonian fluid moving at the same velocity as its Newtonian fluid counterpart1 and is derived by
equating the Darcian velocity (δp = 1), Eq.(3), with the Herschel-Bulkley velocity, Eq.(4), i.e.,
|up,HB | = |up,ND |. Thus, the viscosity becomes a function of the pressure gradient in the gel phase,
with a certain threshold pressure.
Numerical Model and Software Design
The physical model forms a set of residual equations which are solved fully implicitly using Newton’s method. Derivatives in the Jacobian matrix are derived numerically by finite-difference approximations. Derivation of underlying dependencies in the model are considered by either conventional table lookups or, preferably, in parametric form, such as the common Brooks-Corey
multi-phase flow correlations.
The framework facilitates the usage of Voronoi grids, generated either by an in-house pre-processor
or by third party gridding applications. A Voronoi block is defined as the region of space that is
closer to its grid point than to any other, and a Voronoi grid is made of such blocks.4 Mathematically, the major property of the Voronoi grid is its local orthogonality, i.e., the area open to flow
between adjacent grid blocks is always orthogonal to the stream line between the corresponding
grid points. Consequentially, the common two point flow stencil is applicable.
The treatment of the non-Darcy flow requires some special treatment since Eq.(2) is nonlinear.
9th European Conference on the Mathematics of Oil Recovery - Cannes, France, 30 August - 2 September 2004
Previously, fully implicit schemes have been introduced.5 The current code utilises an implicit
iterative scheme. In order to calculate the new solution, a control parameter δpk+1 is introduced to
begin with, where δpk from the preceding iteration is used to calculate the velocity u∗p .
δpk+1
1
=
,
k k
k
βt ρp kkr,p
∗
1+
|up |
µkp
u∗p
=
−δpk
k
kkr,p
grad Φkp .
k
µp
(6)
Subsequently, the velocity for the new iteration k + 1 can be computed while simultaneously
damping the control parameter to avoid oscillations:
uk+1
p
=
−δpk+1
kkr,p
grad Φp
µp
k+1
,
δpk+1 = δpk + (δpk+1 − δpk ) ∗ RP ,
(7)
where RP denotes a relaxation parameter. Best results could be achieved with RP = 0.4...0.6.
The total number of Newton iterations required for non-Darcy flow are slightly increased when
compared with Darcy flow (δp = 1). For common three-phase problems, 2-4 iterations are taken
until the solution converges. Contrary to the fully implicit treatment, where non-Darcy effects
are not supposed to be large, the present approach facilitates even the consideration of extremely
"turbulent" flow with control parameters 0 < δp 1.
The code is realised by means of a MATLAB -FORTRAN coupling. Formulation of the set of
equations is accomplished within a highly efficient Fortran-environment and linked with MATLAB6,
where pre- and postprocessing as well as simulator control are performed. Both direct and iterative solvers are implemented for the solution of the system of linear algebraic equations. However,
most promising is the use of a new algebraic multigrid solver7, which is currently under evaluation.
Discretisation of Hydraulic Fracture
The fracture is an object with specific properties and particular pre-history that should be considered in the evaluation of hydraulic fracture stimulation. In terms of numerical simulation of
postfracture well performance, the problem can be addressed by (i) an adequate representation
of the fracture in a reservoir simulator and (ii) a reasonably accurate picture of the invaded zone
which formed by the leakoff process during fracturing. Both have been addressed in detail in one
of our previous papers.8
Besides, the unstructured discretisation framework facilitates a sophisticated geometrical representation of both single and multiple fractured wells depending on the specific fracture properties, the
well characteristics and the refinement parameters for the grids. Two examples of 2.5 dimensional
fracture grids are presented in Fig.2. Additionally, structured fracture grids can be used based on
the investigations of B ENNETT et al.9
Application Examples
The simulator and the gridding scheme has been verified and validated against a series of synthetic
examples. The checks include the comparison of results obtained, where available, from analytical
solutions and a commercial reservoir simulator. The input properties for the cases are summarised
in Tables 1 and 2.
(a) Voronoi-type grid for a vertical fractured well with
hexagonal background grid
(b) Voronoi-type grid for a horizontal fractured well
with 5 transverse fractures
Figure 2: Examples of unstructured fractured well discretisations
Case 1: Validation of Single-Phase Darcy Flow. Structured B ENNETT-type fracture grids are used
for the verification of the real gas single-phase option. Both finite and infinite conductivity
fractures are considered. As shown in the type-curves, Fig.3.a, the numerical solution is
de facto identical to the corresponding dimensionless analytical solution of G RINGARTEN et
al. for infinite conductivity fractures10 and C INCO -L EY et al. for finite conductivity fractures11.
The unstructured Voronoi fracture grids, Fig.2.a., are validated for a slightly compressible
fluid against the analytical solution for infinite conductivity fractures and the results of a
commercial simulator in Fig.3.b. The dimensionless fracture conductivity is 2400 with a
wellbore radius of 0.1 m and a reservoir permeability of 10 mD. The reservoir is rectangular
and bounded (unit slope in the derivative plot). In contrast, the analytical solution takes only
the infinite acting period into account.
Case 2: Verification of Single-Phase Non-Darcy Flow. Simulator results are compared against
the type-curves of G UPPY et al.12 for non-Darcy flow with real gas and constant pressure
boundary conditions in dimensionless form in Fig.3.c. Both solutions exhibit very close
agreement.
Case 3: Validation of Three-Phase Flow Model. The three-phase model is validated against results of a commercial simulator5. A fractured vertical well is initially filled with a gel phase
of constant viscosity. The fracture is surrounded by a water saturated zone (approx.0.5 m)
at critical gas saturation (0.1). The rest of the reservoir is gas filled with water at residual saturation (0.5). During cleanup, the well produces with constant bottom hole pressure.
Multi-phase flow functions in the formation are based on a Brookes-Corey model, with endpoint permeabilities 0.12 (gas) and 0.11 (water) and linear functions in the fracture. Capillary
pressure at residual water saturation is 330 bars. There is no capillary pressure in the fracture. Results are shown in Figs.3.d and 3.e. The first presents the temporal development
of phase saturations in the well block. The rates are shown in Fig.3.e. Both are in good
agreement with the results of the commercial simulator.
Case 4: Validation of Non-Newtonian Fluid Flow Model. The implementation of non-Newtonian
fluids is verified by means of the analytical solution of I KOKU and R AMEY for transient ra9th European Conference on the Mathematics of Oil Recovery - Cannes, France, 30 August - 2 September 2004
dial flow of non-Newtonian power law fluid in its corrected form13. The power law model
accounts for pseudo-plastic fluid behaviour without a yield stress. The effective viscosity is
calculated with Eq.5. Results, shown in Fig.3.f, exhibit very good agreement to the analytical results. Validation of non-Newtonian fluid with yield stress is impossible thus far, since
analytical solutions or appropriate fluid models in commercial simulators are not available.
The simulation tool has been successfully used for investigations on the performance of hydraulically fractured wells in tight-gas reservoirs with detailed analyses, e.g., of loadwater and gel
cleanup processes.14
Summary
This papers presents a novel reservoir simulation tool for fractured vertical and horizontal wells. It
features the specific physics such as non-Darcy flow and non-Newtonian fluids which are typical
for fractured wells. Unstructured fracture grids are presented, providing well testing accuracy and
very flexible handling of the fractures within a conventional full field model. The discretisation
scheme and the physics of the simulator have been validated against a variety of cases revealing
very good accuracy and reliability of the code.
Nomenclature
A
ct
C
k
K
n
n
Np
p
qp
Q
rw
RP
S
t
tDxf
u
V
z
area, m2
total compressibility, 1/Pa
tortuosity constant
permeability, m2
iteration counter
fluid consistency index, Pa.sn
normal vector
fluid behaviour index,
index of time
number of phases
pressure, Pa
mass source/sink, kg/(m3 .s)
(well) flow rate, m3 /s
well radius, m
relaxation parameter
saturation
time, s
dimensionless fracture time
velocity, m/s
volume, m3
depth, m
βt
γ̇
δ
µ
µeff
ρ
τ
τ0
φ
Φ
non-Darcy flow coefficient, 1/m
share rate, 1/s
control parameter
dynamic viscosity, Pa.s
effective viscosity, Pa.sn .m1−n
density, kg/m3
shear stress, Pa
yield stress, Pa
porosity
z
potential, Pa (with Φ = p + g 0 ρ(η) dη)
D
DNN
gel
HB
irr
ND
r
p
dimensionless
with non-Newtonian fluid
gel
Herschel-Bulkley
irreducible (saturation)
with non-Darcy flow
relative
phase
References
/1/ May E.A.; Britt L.K.; Nolte K.G. The Effect of Yield Stress on Fracture Fluid Cleanup. (SPE-Paper 38619),
1997. presented at the SPE Annual Technical Conference and Exhibition held in San Antonio, Texas.
/2/ Ikoku C.U.; Ramey H.J.Jr. Transient Flow of Non-Newtonian Power-Law Fluids in Porous Media. Society
of Petroleum Engineers Journal, (SPE-Paper 7139):164–174, June 1979. presented at 48th. Annual California
Regional Meeting held in San Francisco, California.
/3/ Al-Fariss T.; Pinder K.L. Flow of a Shear-Thinning Liquid With Yield Stress Through Porous Media. (SPE-Paper
13840), 1985.
/4/ Palagi C. Generation and Application of Voronoi Grid to Model Flow in Heterogeneous Reservoirs. PhD thesis,
Department of Petroleum Engineering, Stanford University, Stanford, California, 1992.
/5/ Schlumberger GeoQuest. Eclipse 100 Technical Description 2003a, 2003.
/6/ The MathWorks Inc. MATLAB 6.5: The Language of Technical Computing, 2002. Part: Using MATLAB,
Version 6.
/7/ Stüben K.; Clees T. SAMG User’s Manual Release 21c. Fraunhofer Institute SCAI, Schloss Birlinghoven D53754 St. Augustin, Germany, Aug. 2003.
/8/ Behr A.; Mtshedlishvili G.; Friedel T.; Häfner F. Initialization of Reservoir Model with Hydraulically Fractured Well for Simulation of Post-Fracture Performance. (SPE-Paper 82298), 2003. presented at the European
Formation Damage Conference held in The Hague, Netherlands.
/9/ Bennett C.O.; Reynolds A.C.; Raghavan R.; Elbel J.L. Performance of Finite-Conductivity, Vertically Fractured
Wells in Single-Layer Reservoirs. (SPE-Paper 11029), 1986.
/10/ Gringarten A.C.; Ramey H.G.; Raghavan R. Unsteady-State Pressure Distributions Created by a Well With
a Single Infinite-Conductivity Vertical Fracture. Society of Petroleum Engineers Journal, (SPE-Paper 4014):
347–360, Aug. 1974.
/11/ Cinco-Ley H.; Samaniego-V.F.; Dominguez N.A. Transient Pressure Behavior for a Well With a FiniteConductivity Vertical Fracture. Society of Petroleum Engineers Journal, (SPE-Paper 6014):253–264, Aug. 1978.
/12/ Guppy K.H.; Cinco-Ley H.; Ramey Jr. H.J. Effect of Non-Darcy Flow on the Constant Pressure Production of
Fractured Wells. Society of Petroleum Engineers Journal, (June 1981):390–400, 1981.
/13/ Ikoku C.U.; Ramey H.J.Jr. Numerical Solution of the Nonlinear Non-Newtonian Partial Differential Equation.
(SPE-Paper 7661), 1978.
/14/ Friedel T. Numerical Simulation of Production from Tight-Gas Reservoirs by Advanced Stimulation Technologies. PhD thesis, Fakultät für Geowissenschaften, Geotechnik und Bergbau der TU Bergakademie Freiberg,
Freiberg, Germany, 2004.
Appendix
Table 1: Input parameters Cases 1, 2 & 4
Well rate (m3 /s)
1.00E-03
10
Permeability (mD)
10
Thickness (m)
0.2
Porosity
1E-03
Viscosity (Pa.s)
5E-10
Total compressibility (1/Pa)
3
Density (kg/m )
1000
0.1
Well radius (m)
Non-Darcy flow coeff. βt (1/m) 5.00E+11
Table 2: Input parameters Case 3
Permeability (mD)
Thickness (m)
Porosity
Fracture half length (m)
Dimensionless fracture conductivity
Gel viscosity (mPa.s)
Gel compressibility (1/Pa)
Gel formation volume factor
Gel density (kg/m3 )
Water viscosity (Pa.s)
Water compressibility (1/Pa)
Water formation volume factor
Water density (kg/m3 )
Rock compressibility (1/Pa)
Initial field pressure (Pa)
Producer well pressure (Pa)
0.05
10
0.1
250
10
1.0
5.5E-10
1.05
800
0.25E-03
5.5E-10
1.05
1000
7.5E-10
600E+05
150E+05
9th European Conference on the Mathematics of Oil Recovery - Cannes, France, 30 August - 2 September 2004
(a) Case 1 - Real gas constant rate production of infinite
and finite conductivity fracture (structured grid)
(b) Case 1 - Water constant rate production with infinite
conductivity fracture (unstructured grid)
1.0
G as saturation:Sim ulator
W atersaturation:Sim ulator
G elsaturation:Sim ulator
5
G as saturation:R eference
5
W atersaturation:R eference
5
G elsaturation:R eference
0.9
W ellblock Saturation
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Tim e (days)
40
70000
35
60000
(d) Case 3: Three phase flow (well block saturation)
30
50000
G as rate:Sim ulator
5
G as rate:R eference
W aterrate:Sim ulator 40000
5
W aterrate:R eference
G elrate:Sim ulator
30000
5
G elrate:R eference
25
20
15
20000
G as rate (m ‡/day)
W aterand gelproduction rate (m ‡/day)
(c) Case 2: Single-phase non-Darcy flow in fractured
vertical wells
10
10000
5
0
0.01
0
0.1
1
Tim e (days)
(e) Case 3: Three phase flow (production rates)
(f) Case 4: Single-phase non-Newtonian (power law)
fluid flow
Figure 3: Application Examples