# Effects of stress-dependent permeability on well performance of ultra-low permeability oil reservoir in China

## Abstract

An experimental investigation of the behaviors of stress-dependent permeability under in situ conditions was conducted and discussed, applying cores from an ultra-low permeability oil reservoir in China. The variation characteristics of formation permeability resulting from pore pressure drawdown and increase were compared. The results indicate that formation permeability at any possible location of the reservoir could be altered in response to the change in stress state caused by both oil production and water injection. A mathematical model of fluid flow in stress-sensitive reservoir was established to evaluate the effect of stress changes on well performances, and an analytical solution method was presented. Several analytical simulations under the conditions of constant wellbore flowing pressure were performed to quantitatively assess the impact of stress sensitivity on single well performance. It is demonstrated that despite the stress-dependent permeability can have an adverse impact on production rate and recovery volume, it may be favorable for water injection. Based on the analysis, a practical and efficient waterflooding program was presented to reduce the influence of permeability damage on reservoir productivity. This program was verified by numerical reservoir simulation to have a combined positive effect for development of ultra-low permeability oil reservoir.

## Keywords

Stress-dependent permeability Ultra-low permeability reservoir Well performance Constant flowing pressure Waterflooding## List of symbols

*ρ*Fluid density (lb/ft

^{3})*ϕ*Formation porosity (%)

- \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {\upsilon }\)
Flow rate (ft/h)

*t*Time (h)

*t*_{a}Pseudo-time (h)

*k*Formation permeability (mD)

*k*_{i}Initial formation permeability (mD)

*μ*Fluid viscosity (cp)

*p*Formation pressure (psi)

*p*_{i}Initial formation pressure (psi)

*p*_{p}Pseudo-pressure (psi)

*V*_{L}Fluid volume (ft

^{3})*V*_{p}Pore volume (ft

^{3})*c*_{L}Fluid compressibility (psi)

*c*_{ϕ}Rock compressibility (psi)

*c*_{t}Total compressibility (psi)

*N*_{p}Cumulative oil production (STB)

*B*Formation volume factor (RB/STB)

*q*_{sc}Production rate (STB/D)

*r*_{w}Wellbore radius (ft)

*r*_{inv}Investigation radius (ft)

*h*Formation thickness (ft)

*γ*Euler’s constant, 1.781

## Introduction

Oil and gas resource embedded in ultra-low permeability reservoirs is an important and aggressively increasing source of hydrocarbon energy in China. One of the problems that we have to consider in developing such reservoirs is the stress-dependent formation properties (permeability and porosity) during the production life cycle of the reservoir. In general, producing from a hydrocarbon reservoir may result in a decrease of fluid pressure and thus a subsequent increase of effective overburden load on porous reservoir rock, which will compact the reservoir rock and alter the detailed pore geometry (as a matter of course, injection into a reservoir will have the opposite situation). If fluid flow properties of the reservoir rocks are highly sensitive to effective stress changes and rock deformation, the reservoir should be considered to be stress-sensitive (Chin et al. 2000a).

The characteristics of permeability decrease with increased confining stress have been well demonstrated for a great variety of reservoir rocks in the literature. According to a comprehensive study presented by Davies and Davies (2001), the rock permeability behaves in an exponential manner with the net confining stress variation in most cases, and the greatest variation of permeability occurs dominantly at low pressure (0–3000 psi). In this low-pressure range, rocks can lose between 10 and 99% of their original permeability. Pore geometry is the fundamental control on stress-dependent permeability in sandstone reservoirs. It has been proved that formations with pore distribution of smaller radio are very sensitive to compressive stress. Besides, the impact of stress on property alteration generally increases with the tightness of the reservoir rock.

As for conventional reservoirs, we have had a clear knowledge of the behavior of flow-reducing properties of formation rocks and the inherent controlling mechanism. Through analytical, numerical, or coupled flow models, the combined effects of stress, fluid flow, and reservoir property changes on well performance have been also widely illustrated in the past decades (Vairogs et al. 1971; Raghavan et al. 1972; Vairogs and Rhoades 1973; Samaniego et al. 1977; Evers and Soeiinah 1977; Ostensen 1986; Chin et al. 2000a, b; Samaniego and Villalobos 2003; Lei et al. 2007). There is a broad consensus that the stress-dependent permeability of matrix or natural fractures may have a significant impact on the performance of both the individual well and the reservoir. In order to evaluate reservoirs with stress-dependent permeability accurately, many techniques for quantifying key reservoir properties controlling storage and flow, calculating hydrocarbons in place, establishing recovery and forecasting production have been developed as well (Samaniego et al. 1979; Samaniego and Cinco 1980, 1989; Han and Dusseault 2003; Raghavan and Chin 2004; Chen et al. 2008; Xiao et al. 2009). In addition, with the extensive development of unconventional reservoirs (ex. coalbed methane, shale gas/oil, ultra-low permeability oil reservoir) around the world, the subject of stress-dependent permeability is also of great interest because the ultra-tight matrix and natural/generated fractures are more susceptible to stress-state changes. Some researchers (Thompson et al. 2010; Okouma et al. 2011; Cho et al. 2013; Clarkson et al. 2013; Qanbari and Clarkson 2013a, b) have chosen to include stress-sensitive effects for more accurate assessments of the production potential of such reservoirs.

Virtually, all the investigations on the stress-sensitive phenomenon mentioned above are mainly concentrated on the permeability decline rule and the influence on fluid flow into a production well. To our knowledge, little research has paid attention to the behavior of formation permeability variation when the reservoir rock is subject to increasing pore pressure due to fluid injection. Because of the extremely small pore throat, the correspondingly ultra-low permeability and lack of natural energy, artificial waterflooding is the preferred development technique for ultra-low permeability oil reservoir in China. Thus, compared to other stress-sensitive reservoirs, ultra-low permeability oil reservoir has its unique characteristics: the pore pressure will experience both decrease and increase during development. It is expected that the permeability will change in a more complex manner from the perspective of the whole reservoir.

It is the objective of this work to use experimental data and mathematical models to evaluate the interaction between the stress state and fluid flow and its influence on well performance of an ultra-low permeability reservoir. In this paper, we first demonstrate the results of an experimental study on permeability changes using natural cores prepared from Changqing oilfield in China. Then, we present the basic governing equations under unsteady-state condition for fluid flow in stress-sensitive reservoir and develop an analytical method to solve the nonlinear problem. Based on the analytical solution derived we present several theoretical studies to reveal the complex characteristics of permeability changes and the corresponding production performances of the reservoir. Finally, on the basis of the above research, an optimum water injection schedule was recommended to reduce the enormous consequence of rock deformation on the development of ultra-low permeability oil reservoir.

## Experimental study

Petrophysical characterizations of cores

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
---|---|---|---|---|---|---|---|---|

Depth (ft) | 6759 | 6972 | 7044 | 6762 | 7051 | 7054 | 7064 | 7060 |

Diameter (in) | 0.977 | 0.974 | 0.983 | 0.978 | 0.983 | 0.979 | 0.980 | 0.975 |

Length (in) | 2.522 | 2.430 | 2.068 | 2.634 | 2.210 | 2.474 | 2.474 | 2.350 |

Porosity (%) | 13.26 | 11.99 | 17.52 | 13.72 | 16.51 | 17.43 | 11.56 | 13.36 |

Initial permeability (mD) | 0.362 | 0.141 | 0.113 | 0.052 | 0.055 | 0.115 | 0.315 | 0.266 |

The objective of experimental study is to discover the change rules of permeability variation caused by production and injection, and to generate simple but rigorous data for reservoir simulation. The detailed mechanisms that control changes in formation permeability for different rock types are not discussed.

### Experimental procedure

All the cores were cut cylindrically into 1 in diameter sections, and the length of the cores varied from 2.0 to 2.6 in. After the cores were cleaned with tobuence for several days, the displacement experiment was carried out using an AFS-300 displacement system developed by Core Laboratories. Three high-pressure Isco pumps were used to generate flows of fluid through the cores, and control confining and back pressure, respectively. According to the results of well-log and well-testing analysis, the overburden pressure (*P* _{o}) of Chang 6 formation is about 6090 psi and the initial pore pressure (*P* _{ i }) is about 2420 psi. To simulate the in situ formation stress state, the confining pressure was set at 6090 psi and the back pressure was increased from 2420 to 4620 psi or decreased from 2420 to 250 psi. The fluid used in these experiments was standard brine with a viscosity of 1.003 mPa s. Two sets of experiments, step-down pore pressure and step-up pore pressure, were performed. Flow rate was maintained at 0.01 mL/min to avoid any damage due to the high flow rate. The experiment procedures are described as follows.

First, a vacuum pump was used to pump air and other impurities out of the cores. After the core was saturated with brine and weighted, it was set in core holder and then the confining pressure was set to the overburden pressure. When this process was performed, the confining pressure was maintained constant, while the back pressure was adjusted to a given value (initial pore pressure). The displacement pump was started to inject brine to the core and when the flow was stable, the flow rate and the inlet and outlet pressure were recorded, and core permeability was calculated at this pressure level. Then, the back pressure was gradually increased or decreased and the displacement procedure was repeated. The values of permeability and pore pressure of every state were calculated.

### Experimental results

The values of absolute permeability used in core analysis in this study vary between samples. To compare all data, it is necessary to normalize values of permeability at each measure point. We use permeability at initial formation pressure (*p* _{ i }) as the reference value to study the effect of stress state on formation permeability for each core.

## Mathematical model and solution

To quantify the effect of stress sensitivity of permeability on well responses, we developed a transient flow model. The basic assumptions usually made about the formation and fluid properties in well test theory are applied. With respect to the stress-sensitive behavior, we assume that the overburden pressure is constant during the life cycle of production, and thus the variation in permeability due to stress change can be described as a single value function of pore pressure.

### Governing equations

The equations governing isothermal single-phase fluid flow in a deformable porous medium with stress-dependent permeability are derived based on mass conservation principles and Darcy’s law, as follows:

Fluid state equation:

Formation rock state equation:

*f*(

*p*) is defined as

Equation 10 is a partial differential equation for single-phase flow of slightly compressible fluid in a reservoir with stress-dependent permeability.

### Analytical solution

Analytical solutions provide an advantageous method for analyzing and modeling well test or production data, which are primarily developed for linear problems of a constant viscosity and compressibility fluid flowing in formations with constant porosity and permeability. However, the diffusivity equation (Eq. 10) is strongly nonlinear due to the incorporation of stress-dependent permeability. In this study, we defined two pseudo-parameters considering stress-dependent permeability to linearize the diffusivity equations and then presented an analytical solution method as follows:

*r*

_{inv}is the radius of investigation, and

*γ*= 1.781 is Euler’s constant.

Using Boltzmann transformation, Eq. 16 can be solved under constant rate or constant flowing pressure inner boundary condition for a well centered in an infinite circular reservoir.

*q*

_{sc}(

*t*

_{a}) gives the following equation:

*q*

_{sc}as a function

*t*rather than

*t*

_{a}. An iterative approach for obtaining

*q*

_{sc}(

*t*) using the equations derived above is presented in Fig. 3.

## Results and discussion

In the following, we will briefly illustrate the fluid flow behavior of a reservoir with stress-dependent permeability and set the stage for our discussion. Because production rate is of vital concern from a reservoir engineering view point, we first examine the change in well productivity in detail. Then, we discuss the response of injecting water into a stress-sensitive reservoir. Finally, an effective development method that would permit us to reduce or eliminate the influence of stress sensitivity for ultra-low permeability reservoir is evaluated.

### Fluid flow behavior of single well

Reservoir and fluid properties used for simulation

| | | | | | | |
---|---|---|---|---|---|---|---|

0.33 | 50 | 13.72 | 0.052 | 2420 | 1.0 | 1.34 | 1.17 × 10 |

*P*

_{ wf }= 500, 1000, 1500, 2000 psi) are compared in this study, as shown in Figs. 4 and 5. Note that incorporation of stress-dependent permeability reduces the production rates to varying degrees, depending on the level of wellbore flowing pressure. For non-stress-sensitive reservoir, additional wellbore pressure drawdown will increase oil production by a similar value compared to the previous pressure drawdown. That is, an linear increase in production rate is created as a result of the reduced wellbore flowing pressure. Whereas, for stress-sensitive reservoir, additional pressure drawdown will result in a relatively lower increase in oil production. Although the values of production rates are impacted by the stress-sensitive permeability, the general character of each production rate curve is not changed between the constant permeability case and stress-dependent permeability case.

These simulation results show that reducing the bottomhole pressure to increase the production rate may actually result in a lower increase in production than expected because of the permeability reduction near the wellbore. This also indicates that stress-dependent permeability may be a consideration for attempts to correct the lower-than-expected production rates in many reservoirs. A knowledge of permeability at different values of in situ stress can be used to determine the relationship between production rate and wellbore pressure and therefore evaluate the formation damage resulting from the rapid drawdown of near-wellbore pressure.

*P*

_{ wf }= 3000, 3500, 4000, 4500 psi) are compared, as shown in Figs. 7 and 8. In contrast to the case of production well, the stress-dependent permeability enhances the injection rate, but not significantly for all four cases. There are minimal differences in water injection rate between each stress-sensitive case and non-stress-sensitive case, particularly between the cases of low injection pressure. The cumulative injection volume after 300 days is only increased by 1.3% for the lowest bottomhole flowing pressure case and 3.4% for the highest bottomhole flowing pressure case.

### Waterflooding Performance

Cumulative oil production N_{p} under different water injection timing

Run | Advanced time (days) | | Increment of | | Increment of | | Increment of |
---|---|---|---|---|---|---|---|

1 | 0 (base case) | 151.1 | 0.00 | 267.7 | 0.00 | 377.2 | 0.00 |

2 | 30 | 167.8 | 11.01 | 285.4 | 6.60 | 394.1 | 4.50 |

3 | 60 | 181.8 | 20.29 | 299.6 | 11.89 | 407.6 | 8.06 |

4 | 90 | 193.0 | 27.75 | 310.9 | 16.12 | 418.3 | 10.91 |

5 | 120 | 201.6 | 33.43 | 319.5 | 19.33 | 426.4 | 13.06 |

At this stage, we have investigated and understood the behavior of stress-dependent permeability, as well as its influence on the performance of the individual well in an ultra-low permeability reservoir. It is important to point out that, stress sensitivity has not only negative effects but also positive connotations for some reservoirs, depending on rock types and well-producing conditions. Reducing bottomhole pressure to obtain rapid production rates can result in a significant reduction of near-wellbore permeability in stress-sensitive reservoir. However, it has been revealed that advanced water injection will provide remediation due to its two important positive roles as noted earlier. Since the injection timing and volume are functions of economics and individual reservoir properties, the optimization job should be conducted in terms of the situation of particular reservoir and hence it is not illustrated in this study.

## Conclusions

In this paper, stress-dependent permeability and its effect on the performance of wells in ultra-low permeability reservoir were discussed. The conclusions of this study are as follows: (1) We investigated the change behaviors of permeability under the condition of both pore pressure drawdown and increase through laboratory experiments. Based on the experimental results, it is reasonable to say that the process of oil production and water injection may have profound effects on formation permeability. (2) On the basis of the theory of fluid mechanics in porous media, a flow mathematical model considering stress-dependent permeability was established to reveal the dynamic flowing characteristics of stress-sensitive reservoir during oil production and water injection. (3) With an analytical solution of a conceptual infinite reservoir model, effects of stress-dependent permeability on well performance under constant flowing pressure conditions were examined in detail. Results showed that although the impact of stress on permeability is disadvantageous during production, it may be favorable during water injection. (4) Advanced water injection is a practical development method for ultra-low permeability reservoirs. Starting injecting water before production could not only impart significant additional energy for production but also slow down the reduction rate of permeability, which is a combined active effect for reservoir development.

## Notes

### Acknowledgements

This work is supported by the National Science and Technology Major Project (Project No. 2011ZX05009-004). The authors would express their appreciation to the Project for contribution of research fund. Thanks are also due to Professor Wei Liu for valuable discussion on this paper.

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