SPE 166222

Pressure and Temperature Transient Analysis: Hydraulic Fractured Well Application Priscila M. Ribeiro, Stanford University/ Petrobras, and Roland N. Horne, Stanford University

Copyright 2013, Society of Petroleum Engineers This paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in New Orleans, Louisiana, USA, 30 September–2 October 2013. 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 necessar ily 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 prohi bited. 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 Recent developments in bottomhole data acquisition techniques, such as distributed temperature sensing systems (DTS), have

brought attention to the potential increase of information that can be obtained from temperature data. Studies have shown the

application of temperature surveys to estimate flowrate profiles, resolve the kind of damage around the well, improve the

robustness of the history matching, among others. Nonetheless, Temperature Transient Analysis (TTA) is not a mature

technique and its capabilities have not been explored fully yet.

In order to investigate the application of temperature analysis to the hydraulic fracturing problem, in addition to pressure

analysis, a numerical model was developed to calculate pressure and temperature responses. Regarding the fracture and

reservoir fluid flow, a general approach can be adopted, where the formation permeability and fracture characteristics dictate

how the fluids flow during and after fracture growth. We developed a comprehensive model, which accounts for the pressure

effect on the temperature response, as well as a dynamic fracture that grows and eventually is allowed to close during falloff.

In this research we analyzed the temperature and pressure responses during and immediately after hydraulic fracturing in

order to improve our knowledge of this complicated physical problem. Based on this study, we can better understand not only

the fracture properties, but also the reservoir itself. In addition, sensitivity analysis shows how reservoir permeability can

impact final fracturing performance, as well as pressure and temperature responses. The developed model is also applied to

simulate minifrac analysis, and a field example is presented that shows a good agreement with the simulated behavior during

fracture closure.

This paper highlights the potential of Temperature Transient Analysis, expanding the application of temperature in

addition to pressure transient analysis to improve the characterization of fractured wells:

a) TTA has potential to reduce uncertainty related to fracture length and reservoir permeability.

b) TTA also adds value to DTS information, which is commonly measured but often not fully used.

c) Temperature analysis has the potential to give reliable information about the flow dynamics of the reservoir and

especially about near well zone.

Introduction Hydraulic fracturing is a widely applied well stimulation technique. Oil and gas operators around the world have used

hydraulic fracturing successfully to increase production and reserves. Recently the application of multiple hydraulic fractures

along horizontal wells has made possible the exploitation of unconventional reservoirs, such as shale gas and shale oil.

Hydraulic fracturing operations allow us to bypass near-wellbore damage and return a well to its “natural” productivity, and

extend a conductive path deep into a formation and thus increase productivity beyond the natural level (Economides and

Nolte, 2000). Although operational practices have shown considerable evolution over a few decades, hydraulic fracturing

characterization still has a lot of room for improvement.

The modeling of the hydraulic fracturing process involves the coupling of at least three processes: the mechanical

deformation induced by the fluid pressure on the fracture surfaces; the flow of fluid within the fracture; and the fracture

propagation. Usually, the solid (rock) deformation is modeled using the theory of linear elasticity, which is represented by an

integral equation that determines the nonlocal relationship between the fracture width and the fluid pressure (Economides and

Nolte, 2000).

Nolte and Smith (1981) presented the basis for interpretation of pressure behavior during hydraulic fracturing based on

Carter’s leakoff model (Carter, 1957). They demonstrated that a log-log plot of fracturing pressure above the closure stress

2 SPE 166222

versus treatment time can be used to identify periods of unrestricted extension, confined height, excessive height growth and

restricted penetration. In the same year, Nolte (1979) presented a pressure decline analysis theory (pressure after injection and

before fracture closure). He suggested procedures to quantify the fluid loss coefficient, fracture length and width, fluid

efficiency and time of fracture closure. The practice of performing minifracs was introduced by his development.

A frequent assumption among the hydraulic fracturing models is that fluid loss from fracture to reservoir rock is described

by Carter’s leakoff model (Carter, 1957), where leak-off velocity is a function of time and the leak-off coefficient. Different

from the previous assumptions, Plahn et al. (1997) presented a numerical model that uses the reservoir pressure as a control

to the flux from fracture to the reservoir. The pressure response during fracture growth is modeled in order to study pump-

in/flowback tests (short time injection followed by production). Plahn et al. (1997) used the two well-known fracture

geometry models, PKN and KGD (Figure 1), to represent a dynamic fracture during the pressure change due to fluid

injection.

Pressure is by far the most commonly used data for fracture analysis, but there are others tools to investigate fracture

geometry and effectiveness, such as near-wellbore radioactive tracing or microseismic fracture imaging (Barree, 2002).

Recent developments in bottom-hole data acquisition techniques have allowed real-time monitoring of hydraulic fracturing

using fiber-optic distributed temperature sensing systems (DTS) to estimate the fracture initiation depth, vertical coverage,

number of generated fractures, effects of diverting agents and undesired flow behind casing (Sierra et al., 2008).

Originally, the temperature profiles, usually obtained by production logging tools (PLT), have been a tool to estimate

qualitatively where the flow took place. However temperature data are a rich source of information that has been collected for

years, together with the pressure, but not fully used. Recently, Duru and Horne (2008) built a comprehensive reservoir

thermal model considering conductive and convective mechanisms and also other thermal phenomena, like viscous

dissipation and the adiabatic expansion heating/cooling effect. Following their previous development, Duru and Horne (2010)

introduced the potential of using temperature data during history matching. The use of temperature information improved the

accuracy of estimation of the porosity field.

Sui et al. (2008) applied temperature transient analysis to commingled reservoirs. They presented a method to determine

multilayer formation properties from pressure and distributed temperature data, and concluded that layer permeability,

damage permeability and damage radius can be determined uniquely using single-point transient pressure and transient

distributed temperature data.

When the focus turns to temperature modeling of the hydraulic fracturing process, many temperature models presented in

the literature are focused on better prediction of fracturing fluid behavior to optimize proppant placement and/or acid

reactions in acid treatments (Kamphuis et al., 1990; Settari, 1980; Craig et al., 1996). However, the use of temperature

analysis applied to quantitative fracture and reservoir characterization is not a common practice.

With the development of new measurement technologies, such as DTS, the characterization practices of temperature

analysis are likely to be improved. Recently, Seth et al. (2010) presented a numerical solution for temperature response

during a hydraulic fracture job using a simplified fracture growth model by approximating the interaction between fracture

and reservoir by an overall leak-off coefficient, without any pressure consideration. They showed that the temperature data

can give useful information about the rock thermal properties, such as conductivity, and also about the leak-off coefficient.

In this work, we analyzed the temperature and pressure responses during and immediately after hydraulic fracturing in

order to improve our knowledge of this complicated physical problem. We developed a more comprehensive model, which

accounts for the pressure effect on the temperature response, as well as a dynamic fracture that grows during injection and

eventually is allowed to close during falloff. Based on this study, we are able to better understand not only the fracture

properties, but also the reservoir itself. In addition, sensitivity analysis shows how reservoir permeability can impact final

fracturing performance, as well as pressure and temperature responses. Also, the developed model can be applied to

understand minifrac pressure response. Minifracs appear as a reliable alternative for tight formation characterization. We

explored the fracture creation during injection and also the type curves and data transformation technique for pressure decline

analysis during the falloff. A field data example from a nanodarcy reservoir was compared with the simulated results and

showed the same data characteristics.

Model Description Pressure and temperature responses during hydraulic fracturing and after the injection stops (falloff/flowback) are the focus

of this work. The hydraulic fracturing problem requires a coupled model that encompasses well, fracture and reservoir

behavior. The fracture is able to grow and eventually to decrease in volume (by closure) due to fluid loss to the reservoir. The

model described in this paper couples reservoir and fracture flow to the energy balance. No prior assumption about fluid loss

from fracture to reservoir is required, and so Carter’s model is not assumed. The coupling between mechanics and mass

balance follows an approach similar to the one described by Plahn et al. (1997). The approach generates the pressure

response and predicts the flow velocities throughout the reservoir during fracture creation by coupling the mass balance

inside the reservoir and within the created fracture with well-known simplified two-dimensional fracture geometries, named

PKN (Nordgren, 1972) and KGD (Geertsma and Klerk, 1969). The reservoir flow and mechanical properties and fluid

injection rates are the parameters that determine final fracture length.

Temperature responses during fracture growth and closure are governed by the energy balance coupled with mass balance

and simplified geomechanical relations. The problem described here accounts for variable fracture length and width, and also

SPE 166222 3

for pressure effects on the energy balance. The solution is obtained by numerical methods, where the mass balance coupled

with geomechanics is solved first and then applied to the energy balance to obtain the equivalent temperature response, not

only during the injection period, but also in the subsequent falloff or flowback. The detailed pressure and temperature

modelings are described in the following sections.

Mass balance and fracture growth: Fracture growth and closure modeling requires coupling two-dimensional fracture

geometries, such as in the PKN and KGD models (Figure 1), to the reservoir model. Both reservoir and fracture are

discretized and solved numerically by the finite difference technique. Finite difference reservoir simulators are commonly

used to model fluid flow in reservoirs.

Figure 1 – PKN (Nordgren, 1972) and KGD (Geertsma and Klerk, 1969) fracture geometries.

Considering a representative elemental volume (REV) of reservoir, the mass balance is represented for a block (i, j) as:

jijijijijiji Vt

Sqqqq ,,2/1,2/1,,2/1,2/1 )(

( 1)

(q) denotes the mass flow rate normal to a particular face of the grid-block, S is the rate at which mass is removed from the

grid-block via sinks (or added via sources), is the porosity of the grid-block, and V is the bulk volume of (i, j) grid block.

The analysis assumes a two-dimensional system with uniform thickness, h, and the flow velocities inside the reservoir are

given by Darcy’s law.

The initial and boundary conditions describe a closed reservoir in equilibrium prior to the injection:

iptyxp 0,, ( 2)

injww qtyyxxS )(,, ( 3)

0,,)(,,)( tyyxqtyxxq ee ( 4)

After the discretization of the reservoir, and stating the conservation laws, a system of equations can be written honoring

the boundary and initial conditions.

The coupling between fracture and reservoir starts by assuming a direction of growth in advance, defining the potential

fracture as special grid-blocks. The fracture grids have variable properties, which relate pressure, stress, fracture width and

length. For a homogeneous reservoir, lines of symmetry can be traced from the wellbore, so the reservoir can be represented

by only one fourth of the total, as is shown in Figure 2. Dynamic fracture behavior (growth and shrinkage) is incorporated

into the numerical model by changing fracture grid-block properties such as porosity and permeability. As pressure increases

due to the injection of fluid, the fracture grows and the grid-block properties are not constant anymore. As the fracture grows,

the fracture grid block has to advance and the permeability and porosity have to change due to the increase in fracture width

and length. When the injection stops, the fluid pressure decreases and so does fracture aperture. To honor fracture volume

and conductivity, grid-block permeability and porosity also decrease.

Fracture growth is controlled by predefined propagation criteria. In this work, we applied the two-dimensional geometry

models: PKN and KGD. The propagation criterion for a PKN fracture is based on fluid velocity at the tip. While the fracture

is growing, pressure at the tip is equal to the closure pressure pc. The choice of simplified geometries is related to the fact that

the full mechanical description is not the goal of this first analysis, but to understand pressure and temperature behavior when

4 SPE 166222

fracture properties are not constant. For KGD fractures the propagation criterion is based on the magnitude of the mode I

stress intensity factor, KI:

( 5)

When the KI overcomes the rock critical value (KIC) the fracture is allowed to grow, otherwise the fracture stays at the

same position.

Figure 2 – Reservoir and fracture volume discretization: the model couples reservoir, well and fracture.

The modeled fracture is vertical with fixed height equal to the reservoir thickness. Fracture deformation (i.e., width) is

modeled using England and Green’s (1963) well-known solution for a pressurized plane-strain crack, in an infinite, linearly

elastic medium:

)(

0 2222

2

)](,0[ , ),()1(8

),(td u cf

tddsdusu

pts

u

u

Etw

( 6)

where d is the characteristic length of the fracture, is the position along d, w(x) is the fracture width distribution, σf(x) is

total internal fracture stress, and pc is the pressure of fracture closure. E and are the mechanical properties of the rock,

Young’s modulus and Poisson’s ratio. During the injection period σf(x) can be considered equal to pf(x), where pf(x) is the

fluid pressure distribution in the fracture. During fracture closure, the existence of asperities is considered (Figure 3), which

introduces an additional contact stress (σm) that acts against complete fracture closure. In this case the total fracture stress is

written as:

mff p ( 7)

Figure 3 – Asperities on fracture walls (modified from Danko, 2013).

This property ensures that an enhanced conductivity path along the fracture will be preserved, even though the fracture

conductivity decreases continuously until a stable fracture volume is reached during shut-in (or flowback). The minimum

fracture aperture at which the walls will first touch (asperities), wf,min, is an input in this approach, which considers that the

)(

0 22 )(

),()(122)(

tL

f

cff

I

f

dxxtL

ptxptLtK

SPE 166222 5

asperities contact one another along the centerline of the fracture. The contact stress equations are computed using the

Barton-Bandis discontinuity closure model (Brady and Brown, 2004):

),(

),(),(

txwBA

txwtx

f

f

m

( 8)

where A and B are constants, and

),(),( min, txwwtxw fff ( 9)

The vertical plane-strain assumption simplifies the elasticity problem because fracture width for each vertical cross

section depends only on the local pressure (i.e., there is no lateral coupling), which is the basis for the PKN fracture model.

The PKN model is indicated as the appropriate geometry model when fracture length, 2Lf, exceeds fracture height, h

(Economides and Nolte, 2000). The vertical plane-strain assumption requires that each vertical cross section deforms

independently of all others. This results from the implicit assumption that pressure gradients along the fracture are relatively

small. Since pressure is uniform over each vertical cross section, Equation (6) reduces to a simple form:

]2,2[ , 4

12),( 22

2

hhzzhE

ptzw f

cf

( 10)

Equation (10) gives an elliptical width profile with maximum width, wmax, at z = 0, the centerline of the fracture. The

average width that gives the same area as the elliptical cross section is:

cf

fp

E

hww

2

1

4

2

max ( 11)

For the KGD geometry Equation (6) has to be solved by numerical integration.

The fracture grid properties need to be modified at each time step to honor the fracture geometry. The fracture

permeability is determined by the fracture width, as stated by the following equation:

12

2w

k fr ( 12)

Porosity is also changed in order to honor variable fracture volume in a fixed-dimension grid-block representation. The

system of equations is solved by a fully implicit numerical scheme. As permeability and porosity along the fracture grid

blocks are not constant anymore, their changes and derivatives have to be incorporated into the Jacobian of the Newton-

Raphson scheme. In addition to that, contact stress terms also have to be considered during shut-in or flowback, when

asperities on the opposite fracture walls touch. In this case, the Jacobian increases in size (Figure 4), as for each fracture grid-

block there will be a contact stress σm as unknown in addition to pressure. The additional equations added to the system are

the contact stress equations for each fracture grid block (Equation 8).

Figure 4 – Jacobian and Newton Rapshon matrix equations when mechanical stress inside the fracture is applied.

Energy balance: During hydraulic fracturing, cold fluid is injected into a warm reservoir. Part of the injected fluid creates

the fracture and part of it is lost to the formation. The difference between injected fluid and reservoir temperatures creates an

altered temperature zone not only inside the well and fracture, but also inside the reservoir, in the fracture neighborhood. In

the energy balance equation we consider not only heat exchange by convection and conduction, but also heat changes due to

pressure effects. The pressure influence is named the Joule-Thomson effect and the adiabatic expansion. There is also the

work done by the fluid to break the rock during fracturing. The Joule-Thomson effect is a change in the temperature of a fluid

during expansion in a steady flow process involving no heat transfer or work at constant enthalpy. “The effect occurs in

6 SPE 166222

throttling-type processes, such as adiabatic flow through a porous plug or an expansion valve, and as fluids flow into and up

the wellbore, where a drop in pressure occurs” (Steffensen and Smith, 1973).

The temperature change per unit pressure change at constant enthalpy is defined by the Joule-Thomson coefficient,

C

T

p

T

H

JT

1

( 13)

where T is temperature, p is pressure, H the enthalpy, is the coefficient of thermal expansion, fluid density, and C the

specific heat capacity.

This effect means that as liquid or gas flows in the reservoir toward the well, depending on its Joule-Thomson coefficient,

it heats up or cools down because of the pressure drop at the wellbore. A liquid or gas also heats up or cools down as it flows

up the wellbore because of the friction pressure drop there. In general, a pressure drop causes slight heating of flowing oil and

water but a large Joule-Thomson cooling for flowing gas (Brown, 2003).

Adding those effects, the reservoir energy balance equation becomes:

t

TCSpvT

t

pTTvCT r

effeffrrrllreff

1 ( 14)

where:

sleff 1 ( 15)

sleff CCC 1 ( 16)

In addition to that, the growing fracture problem accounts for the work done by the fluid to increase the size of the

fracture by breaking the reservoir rock:

t

ApW

f

( 17)

where Af = wf h = f(x,t). For fracture the equation is:

t

TCWTThhpvT

t

pTTvC

frllfrrlfrfrfrfrfrll

1 ( 18)

The right hand side of Equation (14) represents the transient temperature variation. The left hand side terms are heat

conduction, heat convection, temperature change caused by temporal fluid expansion, temperature change caused by spatial

fluid expansion and viscous dissipation, and finally the heat source term. For the fracture equation, Equation (18), in addition

to the previously described terms there is the work done by the fluid to deform the rock and open the fracture (W) and heat

conduction through the fracture wall between fluid inside the fracture and the reservoir.

The injection temperature at the top of reservoir is given by Ramey’s classic analytical solution based on the heat transfer

between well and reservoir and the convection of fluid downhole. The analytical solution states that:

Azsurfinjsurfinj eAaTtzTAazaTtzT /,0, ( 19)

where:

2

)(tfqCA l ( 20)

z is the distance down from the well head, a is the geothermal gradient, Cl is fluid thermal heat capacity and f is a

dimensionless time function representing the transient of the heat transfer coefficient between the well and the formation.

More details can be found in Ramey (1962). This temperature will be part of the source term for the well grid-block energy

balance equation (Sinj).

The well will provide fluid to fracture and reservoir, in case of open completion, and it also will exchange heat by

conduction with the formation around it. The overall heat transfer coefficient will depend on the kind of completion, for a

cased and cemented well, for example, both materials have to be considered while calculating U (Willhite, 1967). The contact

area is given by 2πrwh. So, the well grid-block equation can be written as:

t

TrCSTvCrTTUrpqT

t

pT w

wllinjwfrllwwrwww

2221

( 21)

A fine discretization is necessary for the finite difference scheme in order to increase the accuracy and capture the thermal

transients.

SPE 166222 7

Results and Discussions The finite difference model described in the previous section was used to study pressure and temperature transients during

hydraulic fracturing. The base case consists of a vertical fracture propagating in a homogeneous reservoir. Reservoir

dimensions were set in a way that boundary effects are not felt during the total simulation time. The injected and reservoir

fluids are assumed to have similar properties, and the single-phase solution represents the scenario adequately. The base case

reservoir parameters are shown in Table 1. The case considered 30 minutes injection of 12ft3/min of fluid at 20

oC at the

surface. The injected fluid exchanges heat with the wellbore wall and the surrounding formation during its way down to the

reservoir, which makes the injected temperature at the sand face different from the specified Tinj at the surface. After 30

minutes the well is shut in and the falloff period starts. Figure 5 presents the reservoir pressure maps at the end of injection

(Figure 5 - top) and during falloff (Figure 5 - bottom). During the injection period the fracture propagates away from the

wellbore and the injected fluid creates a pressure disturbance around the fracture. During shut-in, the pressure alteration

diffuses further into the reservoir, reducing its magnitude.

Figure 5- Pressure maps (psi) at the end of injection (top) and falloff (bottom) periods for PKN fracture geometry.

Table 1 – Base case fluid and reservoir properties

Property Value

Porosity (ø) 0.15

Permeability (k) 0.5md

Reservoir thickness (h) 50ft

Well radius (rw) 0.3ft

Reservoir depth 2200ft

Initial pressure (pi) 2500psi

Fluid compressibility (cl) 5.0x10-6 psi-1

Fluid density at standard conditions 49 lb/ft3

Fluid density 125 lb/ft3

Fluid viscosity (µ) 1cp

Minimal horizontal stress (σmin or pc) 3500psi

Poisson ration (υ) 0.2

Young’s Modulus (E) 2.0x106psi

Asperities size (wf,min) 7x10-4ft

Fluid heat capacity (Cl) 4186.8 J/kg.K

Rock heat capacity (Cr) 921.1 J/kg.K

Rock thermal conductivity (λr) 1.44 W/m.K

Fluid thermal conductivity (λl) 0.52 W/m.K

Injection temperature at surface (z=0ft) 20oC

Initial Temperature (Ti) 80oC

The net pressure and fracture half-length histories for PKN fracture geometry are shown in Figure 6. Net pressure consists

of wellbore pressure minus the closure pressure, pc. The well pressure increases moderately during injection, while the

fracture is growing. The injection causes not only fracture length growth but also width growth related with the increase in

pressure along the fracture, as shown in Figure 7 (top). During shut-in the pressure declines and the fracture volume

decreases. The fracture closure happens through a decrease in fracture aperture, while the fracture length generated during

injection stays constant (Figure 6 - right). Figure 7 (bottom) presents width profiles along the fracture during shut-in. As

pressure decreases inside the fracture, it is not able to stay open, so it gradually gets narrower. When the width reaches the

size of the asperities, the opposite walls start to touch and contact stress is generated against the complete closure. In part, this

8 SPE 166222

is responsible for the behavior of pressure change behavior during falloff presented in Figure 6. The pressure reflects the

behavior of dynamic geometry, where the fracture grows and closes, changing volume and conductivity.

Figure 6 - Wellbore pressure (left) and fracture length (right) histories for PKN geometry.

Figure 7 –Average width profiles during fracturing (top) and falloff (bottom) for PKN geometry.

For the KGD model the pressure behavior during injection and fracture propagation differs, it decreases slightly as the

fracture propagates (Figure 8). The width profiles during injection show how the fracture grows and during shut-in they

indicate how fracture volume decreases and the effect of asperities on the residual fracture aperture (Figure 9).

Figure 8 - Wellbore pressure and fracture length histories for KGD geometry

SPE 166222 9

Figure 9 – Fracture width profiles during fracture growth (top) and closure (bottom) for KGD geometry.

Figure 10 – Pressure difference (blue) and its logarithmic derivatives with respect to ∆t (red) and equivalent time (green) during falloff for PKN geometry.

Figure 11 – Pressure difference (blue) and its logarithmic derivatives with respect to ∆t (red) and equivalent time (green) during falloff for KGD geometry.

The log-log plot of falloff pressure and its derivative with respect to the shut-in time (∆t=t-tinj) for PKN fracture is

presented in Figure 10. The linear flow is observed in early times, followed by a steep bump in the derivative, which

decreases towards a slope of -1 after that. The derivative spike is coincident with the moment at which the fracture width

10 SPE 166222

around the well becomes as small as the specified minimum width of the asperities. The derivative tends to zero because

during the shut-in the pressure tends to return to the initial pressure of the reservoir, remembering that the injection period is

small compared with the total falloff time.

The equivalent log-log plot for KGD fracture geometry is presented on Figure 11. It is important to notice that the same

spike in the pressure derivative is present, regardless the assumption of fracture geometry. The spike is related to the fracture

“closure” at the wellbore, when the wall asperities start to touch.

When we performed a sensitivity analysis to reservoir permeabilities it was clear that fracture length and the pressure

response are directly affected. Table 2 shows the final fracture length for different permeabilities in a massive hydraulic

fracturing scenario, which are performed at high rates in a 60ft thick reservoir. For all the sensitivity analysis performed in

this section the injection rate was 21.5bbl/min. All cases have the same input properties, and PKN geometry, except for

reservoir permeability. The lower the permeability, the longer the generated fracture will be. This is related to the fact that for

constant injection rate, the lower the permeability, the lower the quantity of fluid lost to the formation and the more effective

the fracturing treatment.

Table 2 – Fracture length sensitivity to reservoir permeability.

Reservoir permeability [md] 0.05 0.5 1 5 10

Fracture half length (Lf) [ft] 2297 1301 980 507 450

The pressure transient analysis for the different reservoir permeabilities is expressed in terms of falloff pressure and the

logarithmic derivative with respect to the shut-in time in Figure 12. The commonly observed signature contains the linear

flow at early times, when the fracture is still open, followed by a steep jump in the logarithmic derivative. For all the cases

analyzed, this jump occurs at the moment when the fracture closes at the wellbore wall. The lower the permeability, the later

the spike in the logarithmic derivative happens. Koning and Niko (1985) and van den Hoek (2002) presented semianalytical

solutions for fracture closure during falloff applied to water flooding scenario (long time of injection when compared with the

falloff), where a similar spike is present when fracture closes. Those models described fracture closure by a change in

wellbore storage, keeping the fracture conductivity constant and infinite.

Figure 12 – Permeability sensitivity analysis: pressure and its logarithmic derivative in a log-log plot.

Solving the energy balance equation yields the temperature during the hydraulic fracturing process. Figure 13 (left) shows

the average temperature inside the wellbore at reservoir depth for the different permeability cases. As can be seen, the

average temperature inside the wellbore shows a very small sensitivity to the different reservoir permeability cases. On the

other hand, analyzing the temperature inside the fracture (right plot in Figure 13), in the well neighborhood, we can observe a

significant difference in the way the temperature recovers, in other words, in the way it warms back. It can be observed that

the lower the permeability, the faster will be the temperature recovery over time. Paying close attention to the early times, the

lower permeability cases, which have larger fractures, warm up slower than higher permeability cases. At early times the

fracture is still open and more fluid will be inside the fracture in lower permeability cases, so the heat that comes from the

reservoir has to warm up a bigger mass and larger fluid film; then, explaining why the temperature recovery is slower. After

the fracture shrinks, the reservoir rock will easily warm the fluid inside the fracture. Lower permeability will have smaller

alterations in the temperature inside the reservoir in the neighborhood around the fracture. So for those cases the fracture

temperature will warm up faster towards the initial reservoir temperature (Tres). After the fracture closure, the heat transfer is

basically governed by conduction (diffusive mechanism). This process characterizes a slower temperature increase during

later times of shut-in.

SPE 166222 11

Figure 13 shows that during shut-in the highest distinction between different reservoir properties in terms of temperature

occurs inside the fracture and reservoir. So, if instead of only shutting the well in it is flowed back, the temperature

disturbances that have happened inside the fracture and in its neighborhood would be recovered at the well. Figure 14

presents the temperature response measured inside the well during injection and flowback. As can be seen, the difference

between permeabilities is also reflected in the temperature measurement, and this difference is able to be observed inside the

well.

Figure 13 – Permeability sensitivity analysis: average temperature inside the wellbore (left), and temperature history inside the fracture adjacent to the wellbore (right).

Figure 14 – Average temperature inside the wellbore at the reservoir depth considering flowback rate history.

Temperature derivatives: Pressure transient analysis has the pressure derivatives as main tool for flow regime diagnosis,

especially after the logarithm derivative type curves introduced by Bourdet et al. (1983). Inspired by such a useful tool we

analyzed the derivative characteristics of temperature transients for hydraulic fracturing scenario.

The first important observation is about the location of temperature sensor. For cases where DTS is deployed in a well

there are two main possibilities: the optical fiber can be installed inside the tubing, in direct contact with the fluid inside the

well; or it can be installed behind the casing and cemented, which will lead to higher sensitivity to the reservoir temperature

(Sierra et al. 2008). If installed behind the casing, it has sometimes been the practice that several fibers are placed around the

well to ensure that after the perforating job at least one of the fibers will not be damaged by the guns.

When the temperature sensor is placed inside the well it measures the average fluid temperature at the reservoir depth. At

the end of injection the temperature inside the well is close or equal to the injected temperature, and the surrounding reservoir

is at higher temperature. Figure 15 shows the difference between the temperatures recorded for different sensor locations: at

the reservoir wall (green line, Tr) and at the wellbore (blue line, Tw), in addition to the fluid temperature inside the fracture

(red, Tfr). When the well is flowed back the temperature warms faster and the temperature tends to be the same for the

different locations after a couple of hours.

When the temperature is recorded behind the casing (Tr), close to the reservoir rock, the measurement might behave as

represented by the green line in Figure 15. When the injection stops, the temperature still decreases due to the heat transfer

through the wall of the well. This period will be shorter for smaller well radius and also when the flowback is performed.

Inside the fracture the temperature (Tfr) changes very quickly when the fracture is closing. The temperature transits from the

well temperature to the formation temperature as the fluid film gets thinner. Inside the well the temperature (Tw) will reflect

the injected temperature during injection and after it be influenced by the volume of fluid at lower temperature and the heat

12 SPE 166222

transfer through the casing. In case of flowback the temperature increases faster mainly due to the entrance of warmer fluid

that comes from the fracture and the reservoir.

Figure 15 – Temperature response measurement for different sensor locations: injection followed by shut-in (left), and injection followed by flowback (right).

To understand the effect of injection and falloff/flowback on the temperature derivatives, analyses of sensitivity to

reservoir permeability and to injection rate were performed. The higher the injection rate the longer will be the fracture

length. The same is true for lower permeabilities, less fluid is lost and a longer fracture is created. In order to avoid

differences in injected temperature at the reservoir depth due to different injection rate, the injection temperature for all the

cases analyzed in this subsection was specified as constant at the reservoir depth.

Figure 16– Temperature logarithmic derivative for flowback case: sensitivity to reservoir permeability (left) and injection flow rate (right).

Figure 17 Temperature logarithmic derivative for warmback case: sensitivity to reservoir permeability (left) and injection flow rate (right).

SPE 166222 13

The semilog plot of logarithmic derivative of temperature during both flowback (Figure 16) and falloff (Figure 17)

presents a maximum point for all the analyzed cases. For the flowback cases the maximum is located at the same time for

most of the different values of reservoir permeability or injection rate, as long as the fluid properties, well characteristics and

flowback rate remain the same.

Figure 18 – Simplified representation of wellbore heat transfer during falloff (left) and flowback (right).

This observation can be explained by solving a simplified equation for each of the cases, falloff and flowback. For the

flowback case, the temperature inside the well can be simplified by assuming that heat convection is the dominant effect in

the heat transfer, and also that reservoir fluid that enter the wellbore has constant temperature and equal to Tres (Figure 18 -

right). The governing equation is:

TTCqdt

dTCV resllfbllw ( 22)

where qfb is the flowback rate. The solution of Equation (22) is given by Equation (23):

w

fb

V

tq

D eT

( 23)

where:

ires

resD

TT

tTTtT

( 24)

The logarithmic derivative of wellbore temperature has a maximum, which is given by the ratio of well volume by the

flowback rate (Equation 25). For the simulated flowback cases the ∆tmax is equal to 10.6min. Multiplying ∆tmax by qfb

(1.6ft3/min) we recover the input volume of the well in front of the porous medium (16.95ft

3). This effect can be seen as an

analog of wellbore storage in pressure transient analysis (PTA). The temperature wellbore storage (TWBS) was also

acknowledged by Ramazanov et al. (2010). This relationship between the temperature maximum and the wellbore volume

and rate was true for many different well radia and flowback rates tested, but for low permeability where the fracturing was

performed at high rates the maximum position is shifted to a little late in time (see 50µd case in Figure 16-left). A possible

explanation is that for lager fractures, and consequently large volume of liquid inside it at shut-in, the fluid that flows first to

the well is at low temperature and it decreases the accuracy of this simplification.

fb

w

q

Vt max ( 25)

For the falloff scenario, a possible simplification is to describe the heat transfer between the well and reservoir as a fluid-

filled cylinder at low initial temperature (Ti) surrounded by a medium at high temperature (Tres), where the heat is transferred

by conduction through the cylinder wall (Figure 18 - left). The higher the temperature variation in the reservoir in the vicinity

of the well the worse will be this approximation compared with the full physics solution. The governing equation for

simplified warmback is presented in Equation (26).

TTUhrdt

dTCV reswllw 2 ( 26)

The solution of this first-order ordinary equation subjected to the initial condition (T=Ti) is given by Equation (27).

T

t

D eT

( 27)

where

⁄ and TD is given by Equation (24).

The logarithmic derivative of the analytical solution for well temperature during falloff has a maximum which happens at

a shut-in time given by Equation (28). From Equation (28) we can obtain the effective heat transfer coefficient between well

and reservoir (U) by taking the time of maximum dT/dlnt semilog plot, when the fluid density and heat capacity are known.

14 SPE 166222

The slope of straight line formed by the semilog plot ln(dT/dt) vs. t can also be used to calculate U, because it is given by -

1/ηT. But this simplification is not fully accurate, if we calculate U using the ∆tmax from the simulated data the value of heat

transfer estimate (6.15W/m2K) is smaller than the actual one (15.5W/m

2K).

U

rCt wll

T2

max

( 28)

For the falloff case this approximation is very rough and becomes less accurate as the wellbore radius increases, because

the decrease in the wall temperature and heat conduction inside the reservoir are not accounted for, which reflects a smaller

heat transfer coefficient estimate than the actual one. High injection rates and permeabilities also contribute to decrease the

accuracy of this representation. So the calculated value from Equation (28) can be seen as the lower bound for the effective

heat transfer coefficient.

Considering the conduction inside the reservoir rock around the well, but assuming the temperature inside the reservoir is

constant and equal to Tres at the shut-in time, the solution is given in the Laplace space by:

11

01

0

ssKssKsUr

k

sK

sT

w

effwD

( 29)

where ⁄ and:

effT

weff

effll

weff rC

C

rUC

22

( 30)

The comparison between the temperature logarithmic derivatives obtained from Equations (27) and (29) with the numeric

solution is shown in Figure 19. It is observed that even considering the conduction inside the reservoir is not enough to

reproduce the numeric behavior. The numeric solution considers the cooling effects during injection, which has the

conduction with the wellbore and also the convective effect of cold fluid percolating the reservoir by the leakoff. As the

fracture is created the shape of cooling zone inside the reservoir is elliptic. This first cooling effect influences the temperature

derivative in a way that its maximum happens later in time and has smaller absolute value.

In spite of the approximations, in a practical application the calculated values from either Equation (28) or Equation (29)

can be seen as lower bounds for the effective heat transfer coefficient. This would be useful information in that the heat

transfer coefficient is often unknown.

Figure 19 – Comparison between the temperature logarithmic derivatives during shut-in: analytical solution considering constant rock temperature (green); analytical solution considering conduction inside the rock (red); and full physics numerical solution (blue).

When we analyze the wellbore temperature first derivative with respect to time it is observed that there is a considerable

difference when sensitivity analysis is performed (Figure 20). For different injection rates, and consequently different fracture

length, the flowback derivative shows some difference at the early times, and after a maximum all the curves collapse in a

descendent mode (Figure 20 - right). For different permeabilities the distinction in the temperature derivative is seen for

longer time (Figure 20- left), which shows the permeability indeed affects the thermal response in fracturing scenarios.

Comparing Figure 20 left and right we have observed that for the same permeability the curves have the same derivative

behavior for most of the time and different behavior for all different permeabilities studied here. This characteristic can be

explored for inverse problem solution.

SPE 166222 15

Figure 20 – Temperature derivative during flowback: permeability (left) and injection flow rate (right) sensitivities.

Minifrac A straightforward application of the presented model is the study of minifrac tests. Minifrac is characterized by a very short

injection time, usually few minutes, at high rate capable of breaking the rock and creating a small fracture. The test is

followed by a long falloff, where the pressure decline is recorded during the shut-in period. This test is normally performed in

order to estimate the fracture closure pressure, and when the falloff is long enough to reach the radial flow it also can provide

a valuable estimate of permeability, which will be used for the main fracturing job. Minifracs are not conventional well tests,

because a fracture is created during injection and closes during falloff.

As the time of injection is short compared with the falloff, the minifrac could be approximated as an instantaneous

injection. As described by Gu et al. (1993) and Abousleiman et al. (1994), the instantaneous injection pressure solution for

the falloff period is given by the derivative of corresponding constant rate solution multiplied by the volume injected:

dt

CtpdVtp sTw

injwv

, ( 31)

Noticing this fact, Craig et al. 2006 realized the slug-test analysis methods (Peres et al. 1993) can be applied to the falloff

data as though the created fracture was preexisting. From Equation (31) the equivalent constant and unitary rate injection

pressure solution can be obtained by integrating the recorded falloff pressure. Equation (32) describes the data

transformation, and using the same principle the logarithmic derivative is calculated directly (Equation 33).

t

wvinj

sTw dpV

Ctp0

1, ( 32)

inj

wvsTw

V

tpt

td

Ctpd

)ln(

, ( 33)

We used our numerical model to simulate a minifrac performed in a 25ft thick reservoir with a permeability of 1md,

where the injection time was 5 minutes at 12ft3/min. The result is shown in Figure 21. At the early times the log-log plot

presents the effect of changing size fracture, with the equivalent time derivative assuming higher values than the pressure

difference for intermediate and later times.

Subsequently, we applied the integral transformation suggested by Craig et al. (2006) to our simulated falloff data when

the fracture is closing (Equation 32). Figure 22 shows the effect of such transformation. The Bourdet derivative presents a

more familiar characteristic: wellbore storage, fracture influence, followed by radial flow.

Using the integral transformation of closing fracture pressure during falloff period as input data into a commercial well

test analysis software allows the data to be interpreted as a regular constant-injection test, with unit rate (-1). The model is

able to recover the reservoir permeability and also the fracture dimensions (Table 3). Figure 22 shows the model (black lines)

fitting the transformed data with parameters given by nonlinear regression. The initial closing fracture behavior is

transformed into unit slope characteristic of wellbore storage.

Table 3 – Comparison between input and interpreted parameters.

Parameter Simulation Input and Final

Fracture Geometry

Data Interpretation

k (md) 1 1.03 Lf (ft) 117. 118

C (bbl/psi) - 0.009

16 SPE 166222

To understand the signatures displayed by the falloff pressure in a closing fracture scenario we compared the dynamic

fracture simulation with the respective response for a fixed fracture that has the same characteristics as the stable final stage

of the closing fracture (same reservoir properties, fracture length and conductivity). We also compared with the response of a

simple vertical well. The falloff response comparisons are shown in Figure 23 (left). As can be seen, after the closing fracture

reaches a more stable geometry its derivative overlays the derivative for the traditional fixed fracture scenario, showing not

only the fracture effect but also transitioning together to the radial flow. The fact that for later time the falloff response for

fixed fracture overlays the derivative for growing/closing fracture case implies that after sufficient time the reservoir behavior

is not influenced by the fracture propagation, but by the final fracture shape. After stable fracture geometry is reached, the

influence of bilinear flow is observed. For even later time the radial flow is obtained, which is in agreement with Gu et al.

(1993) and Abousleiman et al. (1994). When the integral transformation is applied the log-log plot shows a more familiar

characteristic, as shown in the right plot in Figure 23.

Figure 21- Bourdet derivative of minifrac pressure falloff for 1md reservoir.

Figure 22 – Bourdet derivative and finite conductivity model math for integral transformed minifrac pressure falloff (psi.h) for 1md reservoir.

Figure 23- Comparison between closing fracture, fixed fracture and vertical well falloff type curves (left), and equivalent integral transformed analysis (right).

1E-3 0.01 0.1 1 10 100Time [hr]

100

1000

Pre

ssure

[psi]

Log-Log plot: p-p@dt=0 and derivative [psi] vs dt [hr]

1E-3 0.01 0.1 1 10 100Time [hr]

100

1000

Pre

ssure

[psi]

Log-Log plot: p-p@dt=0 and derivative [psi] vs dt [hr]

SPE 166222 17

The integral operation transforms the early behavior of falloff data for the closing fracture into pure wellbore storage

(Figures 22 and 23-right). For the growing fracture problem the total storage coefficient can be expressed by:

dp

tdVtVcVctC

fr

frlwlsT ( 34)

A comparison between the terms on the right hand side of Equation 34 (Figure 24) shows that the magnitude of derivative

of fracture volume with respect to pressure is higher than the storativities of the fracture (clVfr) and wellbore. More than that,

the derivative has a high value for the majority of time when I(p) exhibits the unit slope. Then dVfr/dp drops and the

transformed pressure behavior starts to transit to linear flow. The value of the wellbore storage coefficient interpreted from

the transformed data is 0.009bbl/psi.

For the PKN fracture the derivative of fracture volume with respect to pressure during the falloff for ideal cases depends

on fracture stiffness. Substituting the input values used for our simulation into the fracture volume derivative expression, the

derivative of fracture volume with respect to pressure can be calculated as follows:

psibbl

S

hL

dp

pdwhL

dp

pdV

fr

ffrf

fr0095.0

22 ( 35)

where Sfr is the fracture stiffness and the expressions for PKN and KGD fractures are presented in Table 4.

Table 4 – Fracture Stiffness for PKN and KGD Geometries.

Fracture Model PKN KGD

Stiffness (Sfr) hE

212

fL

E21

For the case of constant wellbore storage in static reservoir geometry, the wellbore pressure is given in the Laplace space

by:

spsC

spsVsp

wcsT

wcinjwv

21 ( 36)

and the integral of falloff pressure will be equivalent to the constant rate at the wellhead with storage coefficient CsT:

spV

spsC

spVsp

sdttp winj

wcsT

wcinjwvwv

21

1 ( 37)

The fact that storage is high when the fracture is closing can explain why the unit slope is seen at the early times of the

transformed data, and also is covering the change in fracture conductivity with decrease in aperture. The data integration

brings the wellbore and fracture storages to influence the data in the same way the regular wellbore storage does in a

traditional drill stem test (DST).

A useful conclusion that was possible by analyzing the fracture behavior during closure and the data transformation by

integral is the fact that the moment in time when the pressure and its derivative start to first separate at the end of pure storage

is equivalent to the decrease in dVfr/dp. This statement can be visualized by comparing Figures 23 (right) and 24 (left). The

same was observed in all the simulated cases so far, with permeability ranging from 10md to 100nd.

Figure 24 – Fracture volume derivative with respect to pressure (left) and fracture storativity (right) during the falloff for 1md reservoir.

18 SPE 166222

One problem of the integral transformation approach is that it requires knowledge of the correct initial pressure (pi). Even

a small mistake can cause a large change in the ∆p and its logarithmic derivative such that they do not show the radial flow

clearly, as exemplified in Figure 25. 5 psi deviation from the correct initial pressure makes the integral data deform and the

interpretation is compromised severely. The plots in Figure 25 are based on simulated results, but the same behavior is seen

in real data.

Figure 25 – Effect of initial pressure on integral transformed data. The actual pi used to create the simulated data is 2500psi. The figures from the left to the right were generated from ∆p calculated pi equals to 2500, 2505 and 2495psi, respectively.

Tight Reservoirs: Hydraulic fracturing is commonly applied to low permeability reservoirs, with permeability scaling from

micro- to nanodarcies. In order to analyze the reservoir and fracture behavior in those scenarios we simulated a minifrac

performed in a 100nd reservoir.

Figure 26 – Minifrac performed in 100nd reservoir: falloff Bourdet derivative with respect to Agarwal equivalent time (left) and of integral transformed data (right).

As we can see in Figure 26, the Bourdet derivative with respect to Agarwal equivalent time (teq) for the falloff period

represents very well the common characteristics of many real data from unconventional reservoirs published in the literature

(for example, Hawkes et al., 2013). The 3/2 slope appears in cases in which the permeability is very low and the fracture

remains open for a considerable amount of time. This is in agreement with Nolte’s solution (Nolte, 1979) for high efficiency

fracturing jobs (100% efficiency), because there is very low leakoff due to the formation being tight. This behavior has been

explored by recent papers in order to estimate the closure pressure, which would be the point where the pressure derivative

deviates from 3/2 slope (Marongiu-Porcu et al., 2011 and Bachman et al., 2012).

For this specific example the fracture does not stop growing after 5 minutes of injection. The fracture grows for 2 minutes

more at the beginning of falloff. In this case, also due to a very small leakoff, the fracture stays open for several hours, which

influences the pressure derivative directly. If we perform the integral transformation on the falloff part of the data we can see

that unit slope remains for 21 hours (Figure 26- right). There is a correspondence between the end of unit slope and a more

stable fracture volume. For this case, after the transformation we can see that radial flow is not seen during the 200 hours of

falloff. As the reservoir permeability is very low even the short injection was able to create a considerable fracture length

(400 ft), and the transition to fracture flow regime (linear or bilinear) is what we see after the unit slope.

Figure 27 (left) shows pressure derivative for the falloff period of a field test in which a minifrac was performed in a

high-pressure ultralow permeability reservoir. The injection time was 3 minutes, followed by 48 hours of falloff. As can be

seen, the pressure derivative shows a similar behavior to our simulated examples. The integral transformation led to the

permeability estimate of 130nd. In this case the estimated fracture length from interpretation of integral transformation in a

SPE 166222 19

commercial well testing software was small (6ft) and it is in agreement to the observation that unit slope starts to separate

after 1min of falloff.

Figure 27 – Field data of minifrac in an ultralow permeability gas reservoir: Falloff Bourdet pressure derivative with respect to Agarwal equivalent time (left) and integral transformation (right).

Conclusions In this work we developed a fully implicit numerical simulator that accounts for fracture creation and closure, providing both

pressure and temperature responses. This model was applied to a series of cases with different reservoir permeabilities and

injection rates. The fracture permeability appears as one of the key parameters to control final fracture length and the closure

behavior. The smaller the permeability the longer the fracture and later in time it will close.

The transient temperature data was also investigated and was shown to be impacted by the reservoir flow properties (such

as permeability and porosity); furthermore, temperature variation is affected by pressure drop. This fact provides us an

opportunity to apply temperature analysis together with pressure analysis to improve formation and fracture parameter

estimates. We also have demonstrated that measured temperature is very sensitive to the sensor location. If the sensor is

inside the well there will be an effect of the fluid volume in what has been referred to previously as temperature wellbore

storage (TWBS).

The preliminary temperature transient analysis shows that there is a potential application for these data and further studies

are needed in order to improve our understanding about the information that is carried by the temperature signal.

The simulator was also applied to minifrac analysis, where the simulated data were used to investigate the behavior of a

data transformation based on the integral operation technique. For all the examples analyzed so far, the end of unit slope of

the transformed data happens at the same time as when the value of dVfr/dp drops, and can be taken as fracture closure time.

It was possible to recover final fracture properties and reservoir permeability from traditional well testing technique, given

that the duration of falloff was long enough to develop the linear/bilinear and radial flow regimes after the fracture walls have

touched. The advantage of minifrac tests is the fact that short injection time allows those regimes to happen sooner than

would be seen in a traditional DST.

A field data example has shown that our model is able to reproduce the real behavior of falloff with closing fracture.

Based on the work presented here we can list the following conclusions:

1- It has been observed that reservoir permeability controls the fracture closure time and length;

2- Pressure derivative presents a spike when fracture closes at the well;

3- Real data from shale and tight reservoirs published in the literature show the same pressure derivative behavior

described here;

4- Temperature response after fracturing is influenced by reservoir permeability and final fracture length.

Acknowledgment The authors are grateful to Petrobras S.A. for allowing the publication of this work and the financial support of the current

PhD of the first author. We also appreciate the support from Stanford University Consortium on Innovation in Well Testing

(SUPRI-D).

Nomenclature Af Vertical cross sectional area of fracture

c Compressibility

Cl Heat capacity of the fluid

Clk Leak-off coefficient

Cs Heat capacity of the solid.

CsT Wellbore storage coefficient.

20 SPE 166222

d Characteristic length of fracture

E Young’s modulus

h Reservoir thickness / fracture height

k Permeability

kf Fracture permeability

KI Stress intensity

KIc Critical stress intensity factor

Lf Fracture half length

p Fluid pressure

pc Fracture closure pressure

pi Initial reservoir pressure

pf Pressure in the fracture

pnet Fracture net pressure

pw Pressure at the wellbore

q Flowrate

r Radius

s Laplace variable

S Source

Srf Fracture stiffness

t Time

T Temperature

U Equivalent heat transfer coefficient at well wall

vlk Leak-off velocity

V Volume

w Fracture width

Average fracture width for a cross section

wmax Maximum fracture width for a cross section

x Spatial coordinate in direction parallel to fracture

y Spatial coordinate in direction perpendicular to fracture

Coefficient of thermal expansion

Thermal conductivity

Fluid viscosity

Porosity

Poisson’s ratio

Density

Subscripts

D Dimensionless

e External boundary

eff Effective

f Fluid

fb Flowback

fr Fracture

i Initial

inj Injection

l Liquid

lk Leak-off

r Reservoir/rock

s Solid

sc Standard condition

sf Sand face

v Variable rate

w Well

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