PROCEEDINGS, Thirty-Ninth Workshop on Geothermal Reservoir Engineering

Stanford University, Stanford, California, February 24-26, 2014

SGP-TR-202

1

An Integrated Discrete Fracture Model for Description of Dynamic Behavior in Fractured

Reservoirs

Jack Norbeck1, Hai Huang

2, Robert Podgorney

2, and Roland Horne

1

1Department of Energy Resources Engineering, Stanford University, Stanford, CA, 94305, USA

2Idaho National Laboratory, Idaho Falls, ID, 83415, USA

e-mail: jnorbeck@stanford.edu

Keywords: dynamic fracture model, fracture mechanics, fracture propagation, leakoff

ABSTRACT

We present the framework for a numerical model that is capable of calculating the coupled interaction of mass transfer between

fractures and surrounding matrix rock, fracture deformation, and fracture propagation. We call the framework an integrated discrete

fracture model (iDFM) approach, because it is necessary to combine several numerical modeling strategies that were each developed

originally to solve particular types of problems in order to capture the dynamic behavior of fractured systems appropriately. In this

work, we extended the coupled fluid flow, geomechanics, and fracture propagation model introduced by McClure (2012) to incorporate

mass exchange between fractures and surrounding matrix rock. We adopted a technique called hierarchical fracture modeling for the

matrix-fracture mass transfer component of the model to ensure that the fracture propagation problem remained tractable in terms of

numerical efficiency.

In this paper, we first present the formulation for the iDFM approach. We verified the accuracy of the model against an analytical

solution to a fractured reservoir problem, and subsequently characterized several of the model’s numerical properties. We found that

the matrix-fracture mass transfer model was able to yield reasonably accurate solutions with only a modest increase in the total number

of degrees of freedom beyond what is required to solve the fracture flow and deformation problem. The convergence rate of the matrix-

fracture mass transfer model improved for low matrix permeability settings. Finally, we applied the model to a synthetic example of a

minifrac analysis in a geothermal well in order to demonstrate the model’s ability to calculate complex behavior in dynamic fracture

systems.

1. INTRODUCTION

The development of geothermal and unconventional hydrocarbon resources requires reservoir characterization and reservoir

management strategies that are tailored to systems in which the flow behavior is dominated by the engineering properties of connected

fracture networks. It is important to recognize that the fluid flow characteristics of fractured reservoirs will remain dynamic throughout

the entire lifecycle of a resource. In these settings, it is crucial that a strong understanding of the fundamentals of fractured reservoir

geomechanics be leveraged in order to optimize stimulation treatments and long-term production strategies.

The purpose of this research was to develop the framework for a numerical model that is able to capture the dynamic behavior of

fractured reservoir systems in which both the properties of individual fractures and the connectivity of fracture networks are expected to

evolve over time. We refer to this modeling framework as an integrated discrete fracture model (iDFM) approach because several

different numerical modeling strategies that are each customized for particular types of problems must be combined effectively in order

to capture the broad range of physics that is expected to have first-order impacts on reservoir behavior.

The local state of stress controls the permeability of individual fractures and governs fracture propagation behavior, both of which

contribute to reservoir-scale flow patterns. Poroelastic effects due to fluid transfer between fractures and surrounding matrix rock as

well as stress redistribution resulting from fracture deformation can have significant impacts on the local state of stress throughout the

reservoir. In the present work, we focused on a full-physics coupling of three distinct processes: mass exchange between fractures and

surrounding matrix rock, fracture deformation and mechanical interaction between fractures, and fracture propagation. The model

described in this paper merges two approaches that were developed previously into a unified framework. The framework is flexible and

could be extended to incorporate additional physics, such as poroelastic and thermoelastic effects that arise from pore pressure and

temperature gradients in the matrix rock, while maintaining the ability to treat fracture networks as dynamic systems. The model has

direct applications for reservoir stimulation modeling, well test interpretation, and production optimization in fractured reservoir

settings.

The fracture mechanics and fracture propagation calculations are performed using the strategy introduced by McClure (2012). The two-

dimensional displacement discontinuity method (DDM) and a discrete fracture network finite volume method are used to couple fluid

flow in fractures to mechanical deformation of fractures. The approach described in McClure (2012) assumed that matrix permeability

was negligible and required that only the fracture domain be discretized. In order to incorporate matrix-fracture mass exchange into the

geomechanical model, we adopted the hierarchical fracture modeling (HFM) approach proposed originally by Lee et al. (2000, 2001)

and Li and Lee (2008). Moinfar et al. (2013) implemented the HFM approach with a relatively simple treatment of geomechanics that

related fracture permeability to effective stress using a functional relationship and did not incorporate fracture propagation. Hajibeygi et

Norbeck et al.

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al. (2011) recognized that the HFM method is particularly well-suited for problems in which the fracture system grows over time, but

did not incorporate a geomechanical component in their model. In our model, we calculate fracture deformation with a rigorous

treatment of fracture mechanics, allow new tensile fractures to nucleate and propagate, and apply the HFM method to capture fluid

exchange behavior between fractures and surrounding rock.

This paper is organized as follows. In Section 2, we present the numerical formulation for the iDFM framework. The matrix-fracture

mass transfer approach used in the iDFM approach was verified against an analytical solution to a fractured reservoir problem as shown

in Section 3. Several of the method’s numerical properties were investigated and characterized in Section 4. As an example of the

utility of the model, a practical application of the model for interpretation of minifrac tests in geothermal wells is presented in Section 5.

Finally, several concluding remarks are discussed in Section 6.

2. FORMULATION OF THE INTEGRATED DISCRETE FRACTURE MODEL

In this section, the formulation for the integrated discrete fracture model (iDFM) is presented. In Section 2.1, we describe the

hierarchical fracture modeling approach for matrix-fracture mass transfer and emphasize its utility for fracture propagation problems. In

Section 2.2, the fracture mechanics and fracture propagation model is described. In Section 2.3, we illustrate the overall iterative

coupling strategy of the algorithm.

2.1 Matrix-Fracture Mass Transfer

In this work, we made use of the hierarchical fracture modeling (HFM) approach introduced originally by Lee et al. (2000). In the HFM

approach, it is recognized that small scale fractures affect flow locally and can be homogenized (i.e., upscaled) to form an equivalent

matrix permeability. On the other hand, large scale fractures affect flow on a reservoir-scale and must be treated explicitly through the

use of a discrete fracture network approach.

The HFM approach is unique from other discrete fracture modeling (DFM) strategies (e.g., see Karimi-Fard and Firoozabadi, 2003;

Karimi-Fard et al., 2004) in that two completely independent computational domains are used for the matrix and large-scale fracture

systems (see Fig. 1). Mass conservation is strictly enforced through a coupling term that is similar in concept to well source terms in

conventional reservoir simulation. The mass conservation equations are written separately for the fracture and matrix domains

(modified from Aziz and Settari, 1979) and are discretized separately so that an unstructured, conforming mesh is no longer required.

The HFM approach provides an elegant framework for fracture propagation problems, because new fracture elements appear simply as

source terms in the matrix system of equations.

For a single-phase fluid, the mass conservation equations can be written, for flow in the fracture domain, as:

,f f wf mfek p q Et

(1)

where e is fracture hydraulic aperture, fk is fracture permeability, is fluid mobility, fp is fluid pressure in the fracture, is fluid

density, wfq is a normalized volumetric source term from a well, and E is fracture void aperture. For flow in the matrix domain, the

mass conservation equation is:

,m m wm fmk p qt

(2)

where mk is matrix permeability, mp is fluid pressure in the matrix, wmq is a normalized volumetric source term from a well, and is

matrix porosity. In Eqs. 1 and 2, mf and fm represent the mass transfer terms between the fracture and matrix domains. These mass

transfer terms are necessary to guarantee mass conservation between the fracture network and surrounding matrix rock and take the

following form:

,mf m fp p A (3)

and

.fm f mp p V (4)

In Eqs. 3 and 4, the parameter is called the fracture index and is analogous to the Peaceman well index (Peaceman, 1978). The terms

are normalized by the fracture control volume surface area, A , and the matrix control volume, V , to ensure continuity upon integration

over the respective control volumes (Hajibeygi et al., 2011). Similar to the treatment of wells in traditional reservoir simulators, the

fracture index serves to capture subgrid behavior of the pressure gradient near fractures. In this work, we followed the derivation of Li

and Lee (2008) to calculate the fracture index. The assumptions in the derivation are: a) flow in the vicinity of the fracture is linear, b)

the fracture fully penetrates the matrix control volume in the vertical direction, and c) the matrix pressure represents the average

pressure over the control volume (Li and Lee, 2008).

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The mass flux from a fracture control volume into a matrix control volume is defined as:

.fm f mp p (5)

This term has units of mass per time. Using Darcy’s law and assuming that flow is linear in the local region near the fracture, we can

alternatively describe the mass flux term as:

fm f mA k p , n (6)

where fA is the fracture surface area, n is the unit normal vector to the fracture face, and the pressure gradient term is:

.

f mp pp

d

n (7)

Here, d represents the average normal distance from the fracture surface in the matrix control volume. Equating the right hand side

expressions in Eqs. 5 and 6 allows for the determination of the fracture index:

,mIk (8)

where I is a grid dependent property with units of length that can be calculated as:

.fA

Id

(9)

The quantity d can be calculated numerically for complex fracture and matrix control volume geometries (Hajibeygi et al., 2011).

It is important to recognize the utility of the fracture index for problems that involve complex networks of preexisting fractures.

Traditional DFM approaches that use unstructured, conforming grids can quickly become computationally expensive due to the detailed

grid refinement that is necessary near fracture intersections. Furthermore, problems that involve fracture propagation are intractable

with traditional DFM because they would require the domain be continually rediscretized as each new fracture propagates, resulting in a

huge amount of computational overhead. Treating the fractures as source terms through the use of fracture indices completely

eliminates this issue. The HFM approach becomes more attractive as the level of fracture network complexity increases.

Figure 1: Illustration of the hierarchical fracture modeling conceptual approach for using separate computational domains for

the fracture and matrix systems. Only the matrix control volumes that contain at least one fracture element (shaded in

gray) pick up a matrix-fracture coupling term. Subgrid scale fractures can be homogenized to form an equivalent

matrix permeability as discussed in Li and Lee (2008) and Haijibeygi et al. (2011).

2.2 Fracture Mechanics and Fracture Propagation

In this work, we adopted the approach introduced by McClure (2012) to couple fluid flow in fractures and fracture mechanics. Friction

evolution, fracture deformation, and the mechanical interaction between fractures were modeled directly. In addition, tensile fractures

were allowed to propagate along prespecified planes according to fracture propagation criteria based on stress intensity factors at

fracture tips. A detailed description of the assumptions and equations used in the geomechanical model are given in McClure (2012)

and McClure and Horne (2013). In this section, we will present the major components of the model.

The model assumes a two-dimensional, linear elastic fractured medium with homogeneous mechanical properties in the matrix rock.

The domain is initially saturated with a single-phase fluid and is in mechanical equilibrium. Fractures are able to deform as fluid

pressure changes, giving rise to a discontinuous displacement field. A boundary element method called the displacement discontinuity

method (DDM) was used to calculate the opening and sliding displacements along the fractures and the displacements in the matrix rock

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that result from changes in traction boundary conditions along the fracture surfaces. Fractures were discretized into discrete elements,

and the approach described by Crouch and Starfield (1983) was used to arrive at a system of equations that relate opening and shear

displacements to changes in normal and shear traction boundary conditions:

,t = Au (10)

where t is a vector of changes in normal and shear traction boundary conditions along the fracture surfaces, A is a matrix of DDM

interaction coefficients, and u is a vector of unknown displacement discontinuities (i.e., opening and shear displacements). The

interaction coefficients were calculated using the higher order DDM approach of Shou and Crouch (1995). As with all boundary

element methods, the normal and shear displacements of each element affect every other element, leading to a fully dense system of

equations. An algorithm called HMMVP is used to solve the system of equations efficiently (Bradley, 2012).

A sequential approach is used to solve for the mechanical displacements and fluid pressure of the fracture elements. The basic strategy

is to first solve for shear displacement while holding opening displacement and fluid pressure constant. Upon convergence, the normal

stress and fluid pressure equations are solved while holding shear displacement constant. This process is repeated iteratively until the

changes in all the primary variables (i.e., shear displacement, opening displacement, and fracture fluid pressure) fall below a prescribed

tolerance. This approach is not guaranteed to converge, but has been applied successfully by other researchers for coupled flow and

geomechanics problems and has proven to work well in practice (e.g., see McClure and Horne, 2010; Kim et al., 2011; McClure, 2012).

During the iterative process, the state of stress is continually updated and evaluated for all fracture elements. The effective normal

stress acting on a fracture element is:

' ,f

n n p (11)

wheren is the resolved normal stress on the fracture plane, and compressive stresses are taken to be positive. If the effective normal

stress is positive, the fracture is bearing compression and is considered closed. The opening displacements for closed fractures are zero

and can be removed from the set of primary variables in Eq. 10. The shear stress acting on closed fractures is compared against the

frictional resistance to slip based on the Coulomb failure criterion:

' ,crit s n S (12)

wherecrit is the frictional resistance to slip, s is the static coefficient of friction, and S is the cohesion of the fracture surface. If the

resolved shear stress acting on a fracture is less than the frictional resistance to slip, the fracture is considered stuck. The shear

displacements are zero for stuck fractures can be removed from the set of primary variables in Eq. 10. The aperture of closed fractures

is a function of the effective normal stress acting on the fracture and the amount of dilation due to shearing (Willis-Richards et al.,

1996):

0

' '

,

tan ,1 9 1 9n e ref n ref

ee D

(13)

where0e is aperture at zero effective stress, ,e ref is a laboratory derived constant, D is cumulative shear slip, and is shear dilation

angle.

If the effective normal stress acting on a fracture element is negative, the walls of the fracture element are in tension and the element is

considered open. For open fractures, the opening and shear displacements are determined from the solution of Eq. 10, and the apertures

are calculated as:

0 tan ,e e e D (14)

where e is the opening displacement. The transmissivity of a fracture is calculated according to the cubic law (Snow, 1965):

3

.12

f f eT ek (15)

It should be noted that in Eq. 1, a distinction was made between the hydraulic aperture, e , that is related to the flux term and the void

aperture, E , that is related to the storage term. For fractures that could be considered parallel plates, the hydraulic aperture is equal to

the void aperture. For fractures or faults with a damage zone, these parameters could differ significantly. Eqs. 13 and 14 are used for

void aperture as well, and the constants are allowed to be different as necessary.

Hydraulic fracture propagation is governed by the state of stress near fracture tips. In the current version of the model, potentially

forming fracture planes are prespecified as a numerical convenience for discretization purposes. Initially, the potentially forming

fractures are considered inactive and are not included in any of the systems of equations. During the simulation, the state of stress is

continually evaluated at the potentially forming fracture elements, and if the effective normal stress acting on an element goes into

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tension that element is activated. Upon activation, the mode one stress intensity factor for fracture tip elements is calculated based on

the opening displacement discontinuity (Shultz, 1988):

1/22

,4 1

I

GK e

a

(16)

where G is shear modulus, is Poisson’s ratio, and a is the fracture tip element half-length. If the stress intensity factor reaches a

critical threshold, then the fracture will propagate.

2.3 Overall Coupling Strategy

In this work, we have extended the McClure (2012) model to allow for fluid leakoff into the surrounding matrix rock. Mass transfer

between the fracture and matrix domains is computed using the HFM approach described in Section 2.1 through an iterative strategy,

and the mechanics equations are solved using a sequential iterative strategy as described in Section 2.2. Here, we illustrate the overall

coupling strategy.

In the outer loop, matrix pressures, fracture pressures, and opening displacements are held constant while the shear stress equations are

solved for shear displacements (Eq. 10). In the inner loop, the algorithm first calculates matrix pressures while holding fracture

pressures and opening displacements constant (Eq. 1). Upon solving for the matrix pressure distribution, the matrix pressures are held

constant while solving for fracture pressures and opening displacements (Eqs. 2 and 10). The matrix pressure residual equations are

then evaluated for convergence. This inner loop is repeated until matrix pressures, fracture pressures, and opening displacements have

converged. At this point, the shear stress residual equations are evaluated for convergence, and the outer loop is repeated as necessary.

A flow chart of the algorithm is shown in Fig. 2.

Figure 2: Overall coupling strategy for iDFM framework.

3. VERIFICATION OF NUMERICAL MODEL

In this section, we show the verification that the HFM matrix-fracture mass transfer term (see Eq. 5) is capable of capturing leakoff

behavior of fractured systems accurately. The numerical model was compared against the analytical solution presented by Ghassemi et

al. (2008) for a reservoir that contains one production well and one injection well connected by a single vertical fracture. The problem

configuration is illustrated in Fig. 3. Fluid is injected at a constant volumetric rate and the production well is maintained at a constant

pressure equal to the initial reservoir pressure. Poroelastic effects were neglected so that the fracture aperture remains constant. The

fluid in the fracture is incompressible.

In order to obtain an analytical expression for pressure distribution in the fracture, the fluid leakoff rate was assumed to be constant

along the fracture and also in time. The resulting fracture pressure distribution was then used as a boundary condition to solve the

slightly compressible diffusivity equation for the transient pressure distribution in the surrounding matrix rock. Fluid leakoff was

assumed to be one-dimensional flow in the direction perpendicular to the fracture. Fluid pressure in the matrix is then given by the

following expression (Ghassemi et al., 2008):

1 2 0, , erfc ,2

m yp x y t x L C x L C p

t

(17)

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where L is the length of the fracture, is hydraulic diffusivity, and0p is the initial reservoir pressure. The two constants are:

1 3

12,Lq

Ce

(18)

and

02 3

12,

qC

e

(19)

whereLq is the constant leakoff rate and

0q is the constant injection rate.

All relevant model parameters are listed in Table 1. Several levels of matrix discretization refinement were tested. In Fig. 4, the

analytical solution is compared to the numerical solution for the times of 100 days and 1000 days after initiation of injection and

production. It is clear that the numerical solution with the highest level of grid refinement was able to capture the leakoff behavior very

well at both early times and late times. The late time solution appears to have some minor differences near the boundaries of the

domain. This difference is most likely due to the fact that the analytical solution assumes one-dimensional flow while the numerical

solution is calculating the more realistic case of two-dimensional flow. Boundary effects may also be introducing additional error

because the analytical solution assumes an infinite domain. Visual inspection of Figs. 4e and 4f might seem to indicate that the pressure

is underestimated significantly for the lowest level of grid refinement, especially in the region near the injection well. Of course, the

reservoir pressures yielded from finite volume schemes represent an average pressure over the control volume and so the accuracy must

be quantified with respect to volume averages.

To quantify the error introduced by the numerical scheme, the root mean square error relative to the analytical solution was calculated

across the domain and normalized by the pressure drop between the injection and production well. The error is presented in Fig. 5 for

five different levels of grid refinement at different solution times. The error for the lowest level of grid refinement ranged between 1%

and 6% over the duration of the simulation. The error for the highest level of grid refinement ranged between roughly 1% and 2%.

The goal of this numerical experiment was to verify that the HFM matrix-fracture mass transfer approach is capable of accurately

calculating leakoff behavior by treating fractures essentially like wells, in contrast to more conventional DFM approaches. These results

indicate that the assumptions involved in deriving the HFM fracture index are well founded, at least for this relatively simple model of a

single vertical fracture. This experiment was performed from the point of view of the matrix rock, because the fracture pressure

distribution was held constant throughout time. This was done purely for the sake of having the ability to compare against an analytical

solution. In Section 4, we will show more realistic simulations in which both fracture pressure and matrix pressure had transient effects.

Nonetheless, we have demonstrated that the conceptual approach of using two separate computational domains for the fracture and

matrix system can provide reasonably accurate solutions without the need of an unstructured, conforming grid.

Figure 3: Schematic for the one-dimensional leakoff model (modified from Nygren and Ghassemi, 2006).

Table 1: Model parameters for model verification study.

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Figure 4: Comparison of the analytical solution to a one-dimensional leakoff model for a vertical fracture to the HFM numerical

solutions at very high refinement (111 x 111 grid blocks) and very low refinement (5 x 5 grid blocks). The left side (a, c,

e) shows the solutions after 100 days of injection and production. The right side (b, d, f) shows the solution after 1000

days of injection and production.

Figure 5: Normalized mean square error for different levels of grid refinement for the matrix domain. The different symbols

represent the error at various simulation times. Errors are calculated with respect to control volume averages of the

analytical solution and normalized by the maximum pressure drop over the system.

(a) (b)

(c) (d)

(e) (f)

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4. CHARACTERIZATION OF NUMERICAL PROPERTIES

In hydraulic fracture applications, the leakoff effect has two main physical implications. The first is that fluid pressure inside the

fracture is reduced, which acts to limit the growth and aperture of the hydraulic fracture. The second is that the surrounding matrix rock

expands, acting to reduce the hydraulic fracture aperture. The latter is termed the poroelastic effect, and is ignored in this paper. In this

section, we show the ability of the HFM matrix-fracture mass transfer approach to calculate the pressure distribution within the fracture

accurately, because the fracture pressure will be the key component governing the mechanical behavior of fracture deformation.

The McClure (2012) model assumed that matrix permeability was negligible so that no fluid was allowed to leak off from the fractures.

The motivation for this assumption was to reduce numerical complexity in order to focus on a rigorous treatment of fracture mechanics

and fracture propagation. One of the main goals of this research was to extend the capabilities of that model to incorporate leakoff

effects without increasing the computational burden significantly. Therefore, it was important to understand the numerical properties of

the matrix-fracture mass transfer model and leverage that understanding to achieve a balance between physical accuracy and

computational efficiency.

To test some numerical properties of the method, we used a model that is similar to the one described in Section 3. A single vertical

fracture connects two wells. Fluid is injected at a constant mass flow rate and the production well is held at constant pressure. The

fracture aperture is constant. Fluid in the fracture is now taken to be slightly compressible. The leakoff rate is no longer assumed to be

constant in space or time, but is now fully coupled to both fracture and matrix pressures and is determined iteratively using the HFM

fracture index approach.

In addition to testing the sensitivity to matrix discretization refinement, we also tested the sensitivity to contrast between fracture

permeability and matrix permeability. One of the main numerical advantages of the HFM approach is that two separate systems of

equations are solved for the fracture and matrix domains, whereas traditional DFM strategies solve a single system of equations. The

coefficient matrices for DFM models therefore involve terms related to both fracture and matrix transmissibilities that can differ by

many orders of magnitude, especially for settings with very tight matrix rock. These coefficient matrices can become ill-conditioned,

which can have negative consequences on convergence rates. Fluid exchange between the fracture and matrix system is expected to

decrease as matrix permeability becomes tighter, and so, conversely, the convergence rate of the HFM iterative strategy is expected to

improve in these settings.

The relevant model parameters for the base case simulations are given in Table 2. The matrix permeability was then modified as

necessary to observe the effect of contrast in fracture permeability to matrix permeability. For this model, an analytical solution is not

easily obtainable, and so errors are calculated with reference to a simulation with high level of grid refinement (101 × 101 grid blocks)

for each permeability contrast.

The fracture pressure distribution along the length of the fracture is illustrated in Fig. 6 for the base case permeability contrast. For

reference, the pressure distribution for the case of no leakoff is also shown (black line). It is clear that the solutions are convergent upon

grid refinement. A more interesting observation is that the solutions tend towards the zero leakoff case as the level of refinement

becomes coarser. This result indicates that the matrix-fracture mass transfer model tends to underestimates the amount of leakoff for

coarser matrix grids. Therefore, the solution can be considered to yield a conservative estimate of fracture pressure with respect to the

original McClure (2012) model assumption of negligible leakoff. The implication is that solutions with improved physical accuracy can

be obtained by paying only a small price in terms of additional computational burden.

The error in fracture pressure due to matrix discretization refinement is presented in Fig. 7 for an early time and late time solution. In

this problem, the maximum error occurs in the fracture control volume that contains the injection well. The reported errors are defined

as the difference in fracture pressure from the reference case normalized by the pressure drop between the injection and production well

for the impermeable matrix case. As expected, the error is observed to decrease as the level of grid refinement increases. In addition,

the solution generally becomes more accurate as matrix permeability decreases, especially for lower levels of discretization refinement.

For the relatively high matrix permeability cases, the error never exceeded 6.5%. The error tended to drop off rapidly as the

permeability contrast increased, which can be attributed mainly to the fact that leakoff becomes negligible for low matrix permeability.

In Fig. 7, the positive errors indicate that fracture pressure was overestimated for all levels of grid refinement and permeability contrast,

and that the magnitude of the overestimate was larger for coarser grids. We reiterate that this result indicates that the matrix-fracture

mass transfer model tends to underestimate the amount of fluid leakoff, yielding a conservative estimate of fracture pressure. From a

modeling perspective, this result is encouraging because the modeler is able to safely ramp up the level of grid refinement as necessary

to achieve the desired level of accuracy.

The average number of matrix-fracture mass transfer coupling iterations over the duration of the simulation is shown in Fig. 8. It is

observed that the coupling process typically converged after 0 – 2 iterations. If the amount of mass transfer over a timestep is

insignificant, then the algorithm does not require any iteration. In timesteps that did require iteration, it was observed that convergence

usually occurred after one or two iterations. Occasionally, convergence was not observed after five coupling iterations at which point

the timestep was discarded for a smaller timestep length. These wasted iterations are not reflected in Fig. 8. As expected, the number of

coupling iterations decreased for the cases with relatively low matrix permeability. It is also interesting to note that the coarse grids

tended to require fewer coupling iterations.

Norbeck et al.

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Table 2: Model parameters for study to evaluate numerical properties of the matrix-fracture mass transfer model.

Figure 6: Fracture pressure distribution after a) 7 days and b) 1000 days for the base case permeability contrast. As the level of

grid refinement increases, the solutions appear convergent. More interestingly, the solution is tending towards the zero

leakoff case (black line) for coarser grids. This result is encouraging, because it shows that the amount of leakoff is

underestimated for coarser grids. This can be considered a conservative estimate with respect to the McClure (2012)

assumption of zero leakoff, and implies that improved solutions can be obtained with only a relatively small number of

additional degrees of freedom.

Figure 7: Error in the fracture pressure of the control volume that contains the injection well with respect to the high grid

refinement case (101 x 101 grid blocks) for different fracture-matrix permeability contrasts. Error is normalized by the

pressure drop between the injection and production well for the zero leakoff model. a) error after 7 days of and b) error

after 1000 days of injection and production.

(a) (b)

(a) (b)

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Figure 8: Average number of matrix-fracture mass transfer coupling iterations over 1000 days of simulation time. Reducing

matrix permeability improves the convergence rate of the algorithm.

5. PRACTICAL APPLICATION OF MODEL TO MINIFRAC ANALYSIS

In this work, we extended a tool used to model hydraulic stimulation to incorporate fluid leakoff effects. Here, we present an

application of the newly developed model to a practical geothermal reservoir engineering problem. Minifrac tests or extended leakoff

tests are commonly performed in the geothermal and oil and gas industries to measure reservoir parameters that are required to design

hydraulic stimulation treatments. In a minifrac test, a relatively small volume of fluid is injected, typically at constant rate, for a short

duration of several minutes in order to propagate a small hydraulic fracture. In some cases, the well is then shut in for a period of time

and fluid is allowed to bleed off into the matrix rock. The pressure response at the well can be used to infer information about hydraulic

fracture initiation, propagation, and closure (Nolte, 1979; Zoback, 2007). One of the most important measurements gained from these

tests is an estimate of the in-situ minimum principal stress. There are five distinct measurements that can be used to infer minimum

principal stress: leakoff pressure (LOP), formation breakdown pressure (FBP), fracture propagation pressure (FPP), initial shut-in

pressure (ISIP), and fracture closure pressure (FCP). Whether the target reservoir stimulation mechanism is hydraulic fracturing or

shear stimulation, accurate knowledge of the minimum principal stress is key to optimizing the treatment design because the magnitude

of this stress controls the fluid pressure necessary to drive hydraulic fracture propagation.

In oil and gas settings, minifrac tests are typically performed under relatively controlled conditions. The wells are usually cased, and a

target location (free of preexisting fractures or faults) is identified, isolated with packers, and perforated. In geothermal settings, the

tests may be performed in less ideal conditions. For example, wells are usually open-hole and intersect several conductive natural

fractures. Regardless, the results from geothermal field tests are often analyzed using principals that were developed based on the more

ideal assumptions.

In this section, we illustrate the extent to which natural fractures interfere with the interpretation of minifrac field tests. The problem

setup was motivated by a recent field test at a geothermal injection well targeted for stimulation. It was clear from image log and

temperature log data that this well was accepting fluid at a single natural fracture. Borehole breakouts and drilling-induced tensile

fractures identified in the image log allowed for the determination of the orientation of the horizontal principal stresses, and the

conductive natural fracture was oriented roughly 45 degrees from the principal stress directions. One of the unknowns at this site was

the length that the natural fracture extended away from the wellbore.

In this synthetic example, we modeled a minifrac test performed in a strike-slip environment. In the model, fluid is injected at a rate of

2.0 kg/s for a period of four minutes, after which the well is shut-in for 48 hours. The matrix rock permeability is 0.1 md. As a base

case, we simulated a conventional minifrac test for a well that did not intersect a natural fracture. Six additional cases were designed to

simulate minfrac tests for wells that intersected a single natural fracture oriented at 45 degrees from the minimum horizontal stress

direction with lengths ranging from 1.4 m to 70.7 m. We hypothesized that the stress perturbation due to slip on the natural fracture

would influence the estimate of minimum principal stress inferred from the wellbore pressure measurements (see Fig. 9). To test for

this effect, two simulations were performed for each fracture length. In one simulation, the fracture was given an artificially high

coefficient of friction so that slip was prevented from occurring. In the other simulation, a normal coefficient of friction was assigned so

that slip could occur. The model parameters are listed in Table 3.

The wellbore bottomhole pressure observations for the full duration of the test are shown in Fig. 10. The solid lines represent the

simulations where no shear slip was allowed and the dashed lines represent the simulations where slip occurred. A magnified

illustration of the pumping phase is shown in Fig. 11. In all simulations, hydraulic fractures did initiate and propagate from the tips of

the natural fracture with the exception of the two cases with the longest preexisting fracture (70.7 m). Upon shut-in, fluid leaked off

into the surrounding formation and the wellbore pressure eventually decayed to the initial reservoir pressure. Interpretations of the

LOP, FBP, FPP, ISIP, and FCP for all cases are summarized in Fig. 12.

As a reference, the case where no preexisting fracture intersected the wellbore behaved as expected. The LOP was exactly equal to the

minimum principal stress ( 3 ), a clear FBP was observed, and the FPP reached a steady value slightly larger than 3 . The cases with

preexisting natural fractures that were not allowed to slip (solid lines) each displayed behavior that is qualitatively similar to the

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reference case. The LOP, FBP, and FPP are higher than the reference case due to the injectivity of the well, frictional losses along the

length of the fracture, storage effects related to the length of the fracture, and storage effects related to changes in fracture aperture

resulting from changes in effective stress.

The cases in which the preexisting fractures were allowed to slip (dashed lines) display markedly different behavior. The LOP

consistently occurred significantly below3 (see Fig. 12a). This was the expected result, because shear slip induces local anti-

symmetric zones of tension and compression near the fracture tip (see Fig. 9). Hydraulic fractures are then able to propagate more

easily within the zones of induced tension. Therefore, using the LOP as a proxy for3 resulted in an underestimate of the minimum

principal stress of up to nearly 8% in this example. For the fractures longer than 5.7 m, a clear FBP was not observed. Our

interpretation is that the unsteady propagation behavior was masked by storage effects.

Once the fracture begins to propagate stably, it is typically assumed that the FPP will reach a steady value slightly above3 because it

does not take much additional fluid pressure to extend a fracture once it has reached a length greater than about 1.0 m. In these

examples, we see a steady increase in the FPP as fracture length increases (see Fig. 12c). This is most likely due to frictional effects.

Additionally, an interesting behavior is observed for the smaller length fractures that display a clear FBP. After the FBP is reached, the

pressure drops quickly, but then begins to rise again, ultimately reaching a FPP value that is higher than the FBP. Our interpretation is

that the fracture initiates and propagates relatively easily in the zone of induced tension near the fracture tip, but once the fracture has

extended beyond the perturbed stress zone it takes a relatively higher pressure to continue to extend the fracture. This interpretation is

also guided by the observation that the FPP values for the fractures that were allowed to slip appear to be converging towards the FPP

values for the counterparts that were not allowed to slip.

Upon shut-in, the pressure decline behavior can also provide information about the in-situ stress state. In fact, the ISIP and FCP can be

more accurate measures of3 because they should not include the effects of friction that are present when the well is flowing (Zoback,

2007). Because we have extended our model to incorporate leakoff effects, we are able to interpret the falloff data. The results of this

numerical experiment indicate that the measurements of ISIP and FCP are not greatly affected by the shear slip-induced stress

perturbation. Of the five different minifrac pressure measurements, ISIP and FCP are able to predict3 most accurately over the range

of cases studied (see Figs. 12d and 12e). Note that clear ISIP and FCP could not be observed for the cases with the 70.7 m natural

fracture because a hydraulic fracture did not propagate a significant distance in either of these two simulations.

Figure 9: Schematic of conceptual model for minifrac analysis. The injection well (black dot) intersects a natural fracture that

is well-oriented for shear slip in the current stress field. As the fracture slips, the near-tip stress field is perturbed. The

cool colors indicate areas where tensile stresses are induced. The red dashed lines indicate potentially forming hydraulic

fractures.

Table 3: Model parameters for minifrac numerical experiment.

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Figure 10: Wellbore bottomhole pressure measurements over the entire duration of the minifrac test. The different colors

represent simulations with different preexisting natural fracture length. Solid lines are for simulations where shear slip

was not allowed to occur. Dashed lines are for simulations where shear slip did occur.

Figure 11: Wellbore bottomhole pressure measurements during the pumping phase of the minifrac test.

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Figure 12: Summary of the results of the estimate of minimum principal stress inferred from the minifrac test. Minimum

principal stress was estimated using five different parameters available from the minifrac pressure data: a) leakoff

pressure (LOP), b) formation breakdown pressure (FPP), c) fracture propagation pressure (FPP), d) initial shut-in

pressure (ISIP), and e) fracture closure pressure (FCP). The LOP was observed to significantly underestimate the

minimum principal stress when shear slip of preexisting fractures occurred. In contrast, both ISIP and FCP were

relatively unaffected by the shear slip effect.

6. CONCLUDING REMARKS

In this paper, we have presented the framework for an integrated discrete fracture modeling (iDFM) approach. The concept of iDFM is

to combine various numerical modeling strategies in a unified framework in order to capture the multitude of coupled physical

processes that are expected to have first-order effects on the fluid flow behavior in fractured reservoirs. These physical processes

include: mass exchange between fractures and surrounding matrix rock, fracture deformation and friction evolution, mechanical

interaction between fractures, poroelasticity, and fracture propagation. In the current version of the model, we have included each of

these effects with the exception of poroelasticity in the matrix rock.

We have extended the coupled fluid flow and geomechanical model introduced originally by McClure (2012) in order to incorporate

fluid exchange between fracture and matrix rock domains though the use of a hierarchical fracture modeling (HFM) technique. We

demonstrated that the HFM conceptual model is particularly well-suited for problems that involve fracture propagation due to the

unique implementation of matrix-fracture mass transfer coupling terms that involve a parameter called the fracture index. We evaluated

(e)

(c) (d)

(a) (b)

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several numerical properties of the matrix-fracture mass transfer model, and found that reasonably accurate solutions can be obtained

with relatively few additional degrees of freedom beyond what is needed to capture the fracture flow and deformation problem. In

addition, we found that the numerical efficiency of the method improved for increasing contrast in fracture to matrix permeability (i.e.,

tight formations), an advantage over traditional DFM strategies.

A practical application to the analysis of minifrac tests in geothermal wells showcased the utility of the model. A numerical experiment

was performed in order to investigate the influence of shear slip of preexisting natural fractures that intersect the well on the

interpretation of the minimum principal stress inferred from minifrac data. It was observed that using the leakoff pressure as a proxy for

the minimum principal stress could result in significant underestimates of the stress magnitude. The result is a direct effect of the stress

perturbation near the fracture tip due to fracture deformation. However, initial shut-in pressure and fracture closure pressure were not

affected significantly by shear slip, and were found to provide reliable estimates of the minimum principal stress. These results are not

meant to be used to make definitive arguments about well test interpretation strategies, but rather to indicate that incorporating a

rigorous treatment of the mechanical deformation of fractures and associated stress perturbations into a numerical model is necessary to

accurately model behavior in fractured reservoirs.

In order to optimize the exploitation of geothermal and unconventional hydrocarbon resources, engineers must leverage an

understanding of the dynamic behavior of fractured reservoir systems. Fluid flow behavior in these systems can be highly nonlinear,

largely due to complex geomechanical processes that occur during all phases of a reservoir lifetime, including both stimulation and

production. The model presented in this work can be used as an effective engineering tool in both research and practical applications in

order to enhance understanding of dynamic behavior in fractured reservoirs.

NOTATION

a fracture element half-length

A surface area of fracture control volume fA fracture surface area

A matrix of DDM interaction coefficients

1C first constant in analytical pressure solution

2C second constant in analytical pressure solution

d average normal distance from fracture surface

D shear displacement discontinuity

e fracture hydraulic aperture

0e fracture aperture at zero effective stress

e normal displacement discontinuity

E fracture void aperture

G shear modulus

I connectivity index fk fracture permeability mk matrix permeability

IK mode one stress intensity factor

L fracture length n unit normal vector to fracture surface

fp fracture pressure mp matrix pressure

Lq fluid leakoff rate

0q fluid injection rate

wfq normalized well-fracture volumetric source wmq normalized well-matrix volumetric source

S fracture cohesion

t vector of shear and normal tractions fT fracture transmissivity

u vector of shear and normal displacements

V volume of matrix control volume

Greek Symbols

hydraulic diffusivity

porosity

fracture index

fluid mobility

fluid viscosity

s static coefficient of friction

Poisson’s ratio

shear dilation angle

fluid density

,e ref reference effective stress

n normal stress

'

n effective normal stress

crit resistance to shear slip

fm fracture-matrix mass flux fm normalized fracture-matrix mass flux mf normalized matrix-fracture mass flux

ACKNOWLEDGEMENTS

This work was supported by the Stanford Center for Induced and Triggered Seismicity. The authors would like to thank Dr. Mark

McClure for the use of the code, CFRAC, and for the many helpful conversations.

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