KSCE Journal of Civil Engineering (0000) 00(0):1-9 Copyright ⓒ2016 Korean Society of Civil Engineers DOI 10.1007/s12205-016-0514-5

Tunnel Engineering

pISSN 1226-7988, eISSN 1976-3808 www.springer.com/12205

TECHNICAL NOTE

Novel Method for Calculating the Effective Stress Coefficient in a Tight Sandstone Reservoir Yinghao Shen*, Guohua Luan**, Haiyong Zhang***, Qian Liu****, Junjing Zhang*****, and Hongkui Ge****** Received March 31, 2016/Revised September 12, 2016/Accepted October 27, 2016/Published Online December 12, 2016

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Abstract Determining the influence of effective stress on rock deformation is essential for geotechnical stability analysis in oil and gas production. There is no universal effective stress coefficient for all rock properties, and different values of effective stress coefficient apply for different physical quantities (Gurevich, 2004). Although the effective stress law and its application is not new, frequently overlooked or misapplied. Hence, a novel method was proposed for measuring and calculating the effective stress coefficient in this work. Firstly, pore compressibility under different confine pressure values was measured using reservoir fluid or experimental fluid. Secondly, effective stress was calculated by comparing pore compressibility under different confine pressure and then, the range of effective stress coefficients was determined eventually. Finally, the reliability of the proposed method was validated via the stresssensitive curves of tight sandstone core samples and by comparing the results with those of two other calculation methods for the effective stress coefficient. This work suggests that the stress-sensitive curves of the two core samples from the same location and with similar physical properties have given similar effective stress coefficient (η = 0.201) calculated using the proposed method, which indicates that the calculation is reasonable. The comparison of the proposed method with other methods also indicate that the proposed technique is reliable. Keywords: effective stress coefficient, tight sandstone, laboratory test, pore volume, stress sensitivity ··································································································································································································································

1. Introduction Effective stress within a rock pore system can change because of depressurization during the production life cycle of a reservoir as pore pressure changes. The effective stress coefficient is an important parameter in the geotechnical mechanics of porous media, particularly in determining the effective stress of the rock skeleton. Consequently, experimental measurements and calculations of the effective stress coefficient have been performed by numerous researchers. Terzaghi (1924 and 1936) presented a formula for calculating effective stress in soil media as follows: σ eff = σ c − σ p

(1)

where σeff is the effective stress of soil media; and σc and σp are the confine pressure and the pore pressure, respectively. However, the Terzaghi effective stress law does not apply to fluid in saturated porous rocks (Biot, 1941). Biot (1941) proposed the effective stress coefficient η to

modify the effective stress law. The effective stress (σeff) as defined by Biot, is the difference between the overburden stress (σc) and a fraction called the effective stress coefficient (η) of the pore pressure (Pp) (Biot, 1941). σ eff = σ c − η Pp

(2)

It indicates that the pore pressure counteracts the overburden stress. The effective stress coefficient indicates that fluid accounts for the percentage of the total cross-sectional area in porous media. It is a key parameter that quantifies the contribution of pore pressure to effective stress (Bear, 1972). The contact area among grains is small for granular soil; hence, considering that the cross section is nearly occupied with fluid and that the corresponding η is approximately equal to 1 is possible. However, for rocks composed of debris matter or characterized by cementation, grain contact is higher and assuming that the occupation of the major part of the cross section by fluid is inappropriate. Thus, the corresponding η will be less than 1.

*Research Assistant, China University of Petroleum, Beijing, China (Corresponding Author, E-mail: [email protected]) **Engineer, CNPC Research Institute of Safety and Environment Technology, Beijing, China (E-mail: [email protected]) ***Engineer, CNOOC EnerTech-Drilling and Production Co., Tianjin, China (E-mail: [email protected]) ****Engineer, CNPC Research Institute of Petroleum Exploration and Development, Beijing, China (E-mail: [email protected]) *****Engineer, ConocoPhillips Company, Houston, Texas, USA (E-mail: [email protected]) ******Professor. China University of Petroleum, Beijing, China (E-mail: [email protected]) −1−

Yinghao Shen, Guohua Luan, Haiyong Zhang, Qian Liu, Junjing Zhang, and Hongkui Ge

The stress distribution of rock matrix and microfracture media is determined by effective stress, and thus, accurately determining the range of the effective stress coefficient and then calculating real effective stress are critical to characterize the stress-sensitive features of a low permeability reservoir. To improve the calculation accuracy and efficiency of the effective stress coefficient, numerous scholars have proposed methods based on the concept of effective stress. Biot (1941) and Gurevich (2004) presented a conventional method to calculate the effective stress coefficient of porous media, and its formula was given by K

η = 1−

K

fr

K =

ε

fr

(4)

p

v

where σp is the stress of rock skeleton, εv is the pore volume variation of rock. K = K （1 − s

C=

fr

K

f

φ

（ ⋅ 1−

C ） M

K K

fr

s

）（1 +

(5) K

f

φK

（1 − φ ⋅ s

K K

fr

） ）

(6)

s

where Kf is the pore fluid bulk modulus. M=

CK K −K s

s

(7) fr

As shown in Eq. (4), Kfr can be determined through a triaxial stress experiment, whereas Ks can be obtained through iterative formulation using Eqs. (5) to (7) (Chen, 2000). The disadvantages of this method are as follows. First, the iterative calculation process for obtaining the effective stress coefficient is complex. Second, the calculation error is considerable (Chen, 2000). Bodaghabadi and Moosavi (2008) proposed a simple method to calculate the effective stress coefficient η, as follows: η = 1−

E Es

k λ = a1 + a2 pc + a3 p p + a4 pc2 + a5 pc p p + a6 p 2p

(8)

where E is the Bulk elastic modulus, and Es is the elastic modulus of solid parts in porous media. The two parameters can be determined easily through a conventional uniaxial stress test in a laboratory. According to the correlation between elastic modulus and porosity, the effective stress coefficient can be defined as a function of porosity. However, for samples from different regions, the relationship between elastic modulus and porosity should be determined specifically using this method (Bodaghabadi and Moosavi, 2008). Moreover, numerous data points are required to ensure that the regression equation will be accurate. Each reservoir will have own η value. Li and Xiao (2008) derived a calculation formula of the effective stress coefficient based on the relationship between permeability

(9)

In Eq. (9), the permeability k is (1) power-law transformed and (2) fit in a least-squares sense by a quadratic surface in both pc and pp (Warpinski, 1992). Where k is the permeability, pc is the confine pressure, and pp is the pore pressure and the ai are parameters to be fit. The power law (λ) employed is determined through a maximum-likelihood approach (Warpinski, 1993). Given this empirical function of the form k(pc, pp), Bernabe's formula for the tangent η (Bernabe, 1986) can be employed as

(3)

s

where Kfr is the Bulk modulus of a rock, and Ks is the bulk modulus of solid materials. And in Eq. (3), σ

and confine pressure/pore pressure. This formula, called the Warpinski model, and is given as follows:

η=−

a2 + a4 pc + 2a5 p p ∂k ∂k / =− ∂p p ∂pc a1 + 2a3 pc + a4 p p

(10)

The general procedures of this method are as follows: 1) to determine the conversion factor λ for converting the corresponding permeability, 2) to fit coefficient ai and then analyze coefficient ai using F test and significance test, 3) to predict the range of permeability, and 4) to calculate the effective stress coefficient η using Eq. (10) under different pc and pp values. The disadvantages of this method are as follows; First, the conversion coefficient should be initially determined and then the converted permeability should be fitted. In addition, the calculation process is complex and exhibits poor accuracy (Zheng, 2008). Qiao et al. (2012) indicated that the effective stress coefficient was a function of permeability, pore pressure, and confine pressure based on the assumption that permeability would follow the effective stress law. However, finding permeability variation is difficult when rocks are tight. In literature there are numerous methods presented for measuring and calculating the effective stress coefficient. Although most of these methods are applicable to rock mechanics (Biot, 1941; Gurevich, 2004; Li and Xiao, 2008; Dassanayake, 2015), their calculation process is time consuming and require a considerable number of experiments. Therefore, determining the effective stress coefficient of porous media rapidly and accurately with high accuracy is difficult, and numerous errors may be occur in the results. A novel method is proposed for measuring and calculating the effective stress coefficient in this work. This method was combined with existing laboratory equipment in the field of seepage mechanics to overcome the disadvantages of current methods, including the complex calculation process with considerable errors. In this method, initially, the pore compressibility of rocks under different confine pressure values was measured using reservoir fluid or experimental fluid. Second, effective stress was calculated by comparing pore compressibility under different confine pressure values, then the range of the effective stress coefficient was determined. Eventually, the reliability of the proposed method was validated via the stress-sensitive curves of tight sandstone core samples, as well as by comparing with two other calculation methods for the effective stress coefficient.

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KSCE Journal of Civil Engineering

Novel Method for Calculating the Effective Stress Coefficient in a Tight Sandstone Reservoir

Given its simple testing procedure, the method proposed in this work allows us to calculate the effective stress coefficient more rapidly and accurately compared with the other two methods.

2. Proposed Method 2.1 Testing Principle The principle of the proposed method is as follows. Reservoir rocks with similar mechanical properties exhibit similar relationship between the pore compression coefficient and the corresponding effective stress. Hence, the values of the pore compression coefficients are close to one another under the same effective stress conditions for rocks with similar physical properties from the same coring location. Through laboratory tests, effective stress was calculated by comparing the pore compression coefficient curves under different confine pressure values. Eventually, the range of the effective stress coefficient was determined. 2.2 Samples and Apparatus The tight sandstone core samples used in this work were obtained from the Changqing Oilfield in the Ordos Basin. The samples were collected from an outcrop (Fig. 1). Grain size distributions primarily consisted of fine–medium and very fine– Fig. 3. The Experimental Device Diagram and Equipments: (a) Overburden Pressure Porosity Measurement Device Diagram, (b) The Experimental Equipments

fine sizes. The mineral content was mainly quartz and feldspar. The average pore radius was 25 µm–30 µm. The average throat radius was 3 µm–5 µm (Fig. 2). An automatic displacement device (AFS-300) manufactured by Core Laboratories (USA) was used in this work (Fig. 3(a) and 3(b)). The drained liquid volume of cores (pore volume reduction) are measured as the confine pressure increased, in order to determine the relation between the pore compression coefficient and confine stress. The accuracy of the AFS-300 device is described as follow. Flow rate ranges from 0.01 ml/min to 50 ml/ min; pressure ranges from 0.06895 MPa to 68.95 MPa, with a zero pressure drift of ±0.5%; and temperature drift is ±0.015%/°C (full range: -1°C to 70°C). The fluid used in the tests was standard brine (density: 1.055 g/ cm3, viscosity: 1.2797 mPa·s). Laboratory tests were conducted under different confine pressure values according to the SY/T 5358-2002 industrial standard.

Fig. 1. Core Samples Used in the Tests

Fig. 2. The Scanning Electron Microscopy (SEM) analysis of the Pore and Throat Radius Vol. 00, No. 0 / 000 0000

2.3 Methods On the basis of the previous testing principle, the procedure for determining the effective stress coefficient could be divided into four simple steps: a) to measure the relationship between the pore compression coefficient CP and effective stress σeff (namely, CP = f (σeff)), b) to measure the reduced value of pore volume ΔVP with increasing confine pressure, c) to calculate the corresponding reduced value of pore volume under different effective stress

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Yinghao Shen, Guohua Luan, Haiyong Zhang, Qian Liu, Junjing Zhang, and Hongkui Ge

Table 1. Parameters of the Core Samples Used in the Tests Core Sample Y-b1 Y-b2 Y-b3 Y-b4

Length (cm) 6.138 6.176 6.180 6.162

Diameter (cm) 3.775 3.772 3.77 3.772

Permeability (10−3 µm2) 0.1252 0.1797 0.1133 0.1272

Table 2. Pore Compressibility Coefficients of Core Samples

Porosity (%) 12.471 11.347 12.257 12.378

Pore compression coefficient (MPa−1) Y-b1 Y-b2 Y-b3 0.00069390 0.0007480 0.0006585 0.00042988 0.0004759 0.0003978 0.00025894 0.0003296 0.0003138 0.00013709 0.0001833 0.0001458 0.00007785 0.0000814 0.0000788

Confine stress σ (MPa) c

6.90 13.79 20.69 27.59 34.48

coefficients ηi (variation range: 0–1) via Eq. (1) and the relationship CP = f (σeff), and d) comparing the calculated reduced values of pore volume (via procedure c) and measured (via procedure b) reduced values of pore volume under different effective stress coefficients. The specific procedures for determining the effective stress coefficient are described as follows. 1) The rock samples used in the tests were from the same coring location and coring direction under the same effective stress conditions to make sure that the rocks have similar physical properties as much as possible (Table 1). In Table 1, the permeability of the core sample Y-b2 used in the present study is different with other three specimens, and the measurement error or the cracks/micro-fractures generated in the core sample Y-b2 may be the key factor. In that the pressure ranges of the AFS-300 device are from 0.06895 MPa to 68.95 MPa with a zero pressure drift of ±0.5%, and also the diagenesis and the tensile stress may generate cracks or fractures leading to a larger permeability of the core sample Y-b2. But the permeability difference has little influence on the measurement of the relationship between the pore compression coefficient CP and effective stress σeff (namely, CP = f (σeff)). The average relative error of the fitting data points used in the fitting function (CP = f (σeff)) is 6.17%, within the experimental allowed error ranges (Fig. 4). 2) Air is extracted from each core sample to achieve a vacuum state, and then, the core is saturated with testing fluid. Pore pressure is maintained at atmospheric pressure. Then, triaxial

confine pressure is gradually increased. As the confine pressure increased, the drained liquid volume of cores (pore volume reduction) Y-b1, Y-b2, and Y-b3 are measured. The relation between the pore compression coefficient and confine stress can be determined using Eq. (11); Table 2 presents the pore compression coefficient of the tested cores under different confine pressure. ⎛ 1 ΔV ⎞ Cp = ⎜ ⎜ V ΔP ⎟⎟ ⎝ p ⎠P

(11)

where CP is the pore compression coefficient of a rock, MPa−1; VP is the pore volume of a rock, cm3; ΔV is the drained liquid volume of the core as confine pressure changes, cm3; and ΔP is the variation of confine pressure, MPa. 3) The confine pressure σc is equal to the effective stress σeff due to that the pressure at the outlet end of core is atmospheric pressure (it means the pore fluid pressure Pp is zero in that there is no backup pressure)(σeff = σc−η · 0 = σc, see Eq. (1)). Fig. 4 shows the Log-log plot of the pore compressibility against confine pressure, a linear relation has been obtained between the two parameters. The fitting function is shown in Eq. (12): C p = f (σ eff

)

=1.25 ×10−3 − 8.00354 ×10−4 × lg10 (σ eff

)

(12)

4) The saturated core sample is placed into the core holder (Fig. 3(a)). Then, the displacement pump is opened to fill the pipelines with fluid. The valves at both the upstream and downstream ends are closed, and pore pressure is kept at a constant value of Pc. Subsequently, the triaxial confine pressure is increased (to reduce testing error, the increase interval is 1 MPa). Consequently, the corresponding pore pressure increases. Then, the added value Pc-add is measured when pore pressure achieves a steady state. For example (Fig. 5), the original confine pressure of core Y-

Fig. 4. Pore Compressibility Coefficient vs. Effective Stress Curve (here, σeff = σc) −4−

Fig. 5. The Procedure to Get the Increase in Pore Pressure KSCE Journal of Civil Engineering

Novel Method for Calculating the Effective Stress Coefficient in a Tight Sandstone Reservoir

Table 3. Corresponding CP and ΔVp Under Different η Values η

i

σ (MPa) P : 13.81 MPa P : 14.01 MPa 15.8530 16.5297 14.4715 15.1285 13.0900 13.7272 11.7086 12.3259 10.3271 10.9246 8.9456 9.5233 7.5641 8.1220 6.1826 6.7207 4.8012 5.3194 3.4197 3.9182 eff-i

p

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

p

CPi (MPa−1)

ΔVpi (ml)

4.08E-05 4.53E-05 5.02E-05 5.56E-05 6.17E-05 6.86E-05 7.67E-05 8.64E-05 9.85E-05 1.14E-04

0.0015 0.0018 0.0019 0.0021 0.0022 0.0024 0.0026 0.0028 0.0031 0.0034 Fig. 6. Calculation Process of the Effective Stress Coefficient

b4 is 17.24 MPa. After closing the valves (the total closed volume between the ends is 20.062 ml), pore pressure is 13.81 MPa. When the confine pressure is increased to 17.93 MPa, pore pressure reaches 14.01 MPa after it achieves a steady state (the procedure is shown in Fig. 5 in order to get the increase value in pore pressure (Pc-add - Pc)). 5) Given that the liquid compression coefficient CL is known (both the upstream and downstream ends are closed), the reduced pore volume ΔVP value can be obtained using Eq. (13) with the increase in pore pressure, as follows: ΔVp = V ⋅ ( Pc − add − Pc ) ⋅ CL

(13) 3

where the total closed volume V (m ) between the ends is the sum of the pore volume VP (m3) and the dead volume VD (m3) of the pipelines (V = VP+ VD). The compression coefficient of standard brine is 4.511 × 10−4 MPa−1, and the reduced pore volume ΔVP value determined using Eq. (13) is 1.803×10−3 ml. 6) Different effective stress coefficients ηi (variation range: 0– 1) are assumed according to Eq. (1). The corresponding effective stress is σeff-i, and the pore compression coefficient CPi can be calculated using the function CP = f (σeff ). Finally, the corresponding variation value of pore volume ΔVPi can be obtained using Eq. (11) as shown in Table 3. 7) The corresponding reduced values of pore volume ΔVPi are calculated under different compression coefficients CPi using Eq. (11). Then, the values of ΔVP and ΔVPi are compared to determine the effective stress coefficient η. The detailed calculation processes of the effective stress coefficient η are given as follows. Firstly, the reduced value of pore volume ΔVP is determined to be 1.803 × 10−3 ml through a laboratory test, as confine pressure increases from 17.24 MPa to 17.93 MPa. Secondly, through the procedure 6) in Section 2.3, the corresponding variation value of pore volume ΔVPi can be calculated using Eq. (11), the results can be seen in Table 3. Thirdly, compare the measured reduced value of pore volume ΔVP and the calculated reduced value of pore volume ΔVPi. In the fifth column of Table 3, we can see that the measured reduced Vol. 00, No. 0 / 000 0000

value of pore volume ΔVP (1.803 × 10−3 ml, based on the laboratory test) is within the certain ΔVPi region (0.0018-0.0019 ml, calculated through the procedure 6)). And hence, the corresponding effective stress coefficient ηi ranges from 0.2 to 0.3. Further, through assuming more accurate effective stress coefficients ηi (for instance, ηi = 0.201,0.202,0.203, …,etc.), the specific effective stress coefficient value is obtained (η = 0.201). Figure 6 shows the calculation process of the effective stress coefficient.

3. Result Analysis 3.1 Result Verification The AFS-300 automatic displacement device used in this work which can simulate reservoir conditions (i.e., reservoir pressure and temperature). Stress-sensitive experiments were conducted under reservoir conditions. Two groups of tests were designed to analyze the influence of selecting the effective stress coefficient on stress-sensitive curves. 1) According to the reservoir condition of the Changqing Oilfield, confine pressure is set to 6000 Psi (41.38 MPa; the actual confine pressure is approximately 42 MPa) and pore pressure is set to 3000 Psi (17.24 MPa; the actual pore pressure is approximately 16.7 MPa). To simulate the production life cycle of a reservoir, pore pressure was decreased gradually until atmospheric pressure was achieved. 2) Pore pressure was maintained at atmospheric pressure, and then, triaxial confine pressure was increased gradually. The physical properties of the two core samples (L2-17 and L2-18) from the same coring location and coring direction are provided in Table 4. Subsequently, the permeability variation of cores (k/ki, where Table 4. Parameters of the Core Samples Used in the Experiment

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Core sample L2-17 L2-18

Length Diameter Permeability (cm) (cm) (10−3 µm2) 8.144 3.83 1.3 8.15 3.82 1.833

Porosity (%) 10.895 10.904

Core type Outcrop core Outcrop core

Yinghao Shen, Guohua Luan, Haiyong Zhang, Qian Liu, Junjing Zhang, and Hongkui Ge

Table 5. Effective Stress Coefficients Under Different Confine Pressure Values σ (MPa) 17.2300 27.5800 41.3700 c

P (MPa) 14.1 14.1 14.1

η 0.4–0.5 0.3–0.4 0.2–0.3

p

C (MPa−1) 4.51×10−4 4.51×10−4 4.51×10−4 L

Table 4). However, the finding is not in accordance with conventional knowledge, and the selection of the effective stress coefficient may be the key problem. The procedures to fit η into permeability determination are given as follows. 1) measure the permeability k under different confine pressure with the triaxial confine pressure increased gradually, while the pore pressure was maintained a constant; 2) the effective stress σeff can be calculated by Eq. (1); and 3) the relationship between the permeability variation k/ki (where k, ki is the permeability under different effective stress and the initial permeability, respectively) and the effective stress σeff can be obtained. Second, the stress-sensitive curves were obtained with the effective stress coefficient η = 0.201, as shown in Fig. 7(b). The features of the two stress-sensitive curves of cores L2-17 and L218 are similar, which indicates that the selection of the effective stress coefficient (η = 0.201) is appropriate.

Fig. 7. Permeability Ratio vs. Net Confine Pressure Curve: (a) η = 1, (b) η = 0.201

k, ki is the permeability under different effective stress and the initial permeability, respectively) L2-17 and L2-18 was measured under different experimental designs. First, stress-sensitive curves were obtained according to Eq. (1) (effective stress coefficient η = 1), as shown in Fig. 7(a). As shown in Fig. 7(a), the two permeability stress-sensitive curves of cores L2-17 and L2-18 varied significantly and were completely different (η = 1). The features of the permeability stress-sensitive curves of cores L2-17 and L2-18 should have been similar with similar physical properties (the porosity values are close and the magnitude of permeability is small as shown in

3.2 Range of the Effective Stress Coefficient On the basis of the proposed method, the range of the effective stress coefficient under different confine pressure values was calculated based on the laboratory tests (The detailed calculation processes are the same as the procedures 1) to 7) in Section 2.3). The physical properties of rock samples used in the tests can be seen in Table 1. The results are presented in Table 5. The effective stress coefficient clearly tends to be smaller as confine pressure increases when pore pressure is constant. This finding is consistent with the results of Bernabe (1986 and 1987) and Ojala and Fjaer (2007). In their work, these authors determined that the effective stress coefficient depended on the applied stresses and exhibited larger reduction with increasing confine pressure.

4. Influence of the Effective Stress Coefficient This section analyzes the influence of the effective stress coefficient on the stress sensitivity of matrix cores and microfracture cores. The parameters of the tight sandstone core samples used in the

Table 6. Physical Properties of Sandstones with Low Permeability Sample Y9-1 Y5-3 Y5-5a Y5-5b Y5-8

Length (cm) 4.192 4.126 4.019 4.019 8.570

Diameter (cm) 2.530 2.540 2.541 2.541 3.820

Permeability (10−3µm2) 0.3488 0.1598 0.9487 1.3211 4.0200

Porosity (%) 9.1300 8.4780 10.68 10.22 10.52 −6−

Core type Matrix Matrix Matrix Microfracture Microfracture

Microfracture conductivity (%) ---28.18 36.14

KSCE Journal of Civil Engineering

Novel Method for Calculating the Effective Stress Coefficient in a Tight Sandstone Reservoir

tests are provided in Table 6. Shear stress microfracture cores were fabricated via a triaxial stress experiment using core samples from the Changqing Oilfield in the Ordos Basin. The mechanics principles of simulating shear stress microfractures via a triaxial stress experiment is given by Chen (2008).

4.2 Influence on the Stress Sensitivity of Microfracture Core The microfracture tight reservoir has two kinds of rock media: matrix and microfracture. The stress-sensitive features are

4.1 Influence on the Stress Sensitivity of Matrix Core The overburden pressure of the Chang6 Strata in Changqing Oilfield is 42 MPa, and the initial fluid pressure of this strata is 16.7 MPa (fluid pressure ranges from 5 MPa to 16.7 MPa). As shown in Fig. 7, effective stress is high under reservoir conditions when η = 0.201, which leads to an obvious reservoir compaction effect. Moreover, variation in fluid pressure cannot be completely transformed into variation in rock skeleton stress. When η = 0.201, the effective stress of the rock skeleton in the actual exploitation path will not change sharply (Fig. 8). The stress-sensitive effect only slightly influences variation in permeability.

Fig. 8. Reservoir Development Path Under Different Effective Stress Coefficients

Fig. 10.Stress-sensitive Curves of Cores Y5-5a and Y5-5b: (a) Matrix Core Y5-5a, (b) Microfracture Core Y5-5b

Fig. 9. Matrix Core Stress-dependent Curve Under Different Effective Stress Coefficients

Fig. 11. Stress-sensitive Curves of the Microfracture Core Under Different η Values

Vol. 00, No. 0 / 000 0000

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Yinghao Shen, Guohua Luan, Haiyong Zhang, Qian Liu, Junjing Zhang, and Hongkui Ge

Table 7. Comparison of the Effective Stress Coefficients Obtained using Three Different Methods Core sample

Length (cm)

Diameter (cm)

Permeability (10−3µm2)

Porosity (%)

Y 6-1 Y 6-2 Y 6-7 Y5-6 Y5-9 Y5-11

7.640 6.386 4.050 4.017 6.640 6.2840

2.544 2.546 2.534 2.538 2.504 2.4840

2.5356 0.1419 0.2848 0.1136 0.2454 0.3144

12.391 9.7580 8.7760 9.9900 9.6856 6.0749

influenced by these two kinds of rock media. Fig. 9(a) shows the stress-sensitive feature of matrix core Y5-5a before being fractured in a triaxial stress experiment. Meanwhile, Fig. 9(b) shows the stress-sensitive feature of microfracture core Y5-5b after being fractured in a triaxial stress experiment. Though a comparison, we can determine that the variation in permeability of the matrix core is smaller than that of the microfracture core because the simple mechanical structure of the microfracture media leads to easier compaction with an increase in net stress. As shown in Fig. 10, the stress-sensitive feature of the microfracture core will be overestimated when the conventional effective stress coefficient η = 1 is applied.

5. Comparison and Discussions The effective stress coefficients were calculated using three methods: the Bodaghabadi method (Eq. (4)), the maximumlikelihood method (Eq. (5)), and the proposed method. The parameters of the tight sandstone core samples and the calculated results are provided in Table 7. The calculated results show that the effective stress coefficients obtained using the three methods are close to one another, which indicates that the accuracy of the proposed method is high. Furthermore, the different features of the three methods should be observed. For the Bodaghabadi method, the relationship between elastic modulus and porosity should be specifically determined for samples from different regions. Numerous data points are required to make the regression equation accurate. This specific relationship does not apply to other reservoirs. Because each reservoir will have own specific relationship. For the maximum-likelihood method, the conversion requires the coefficient to be determined to fit the converted permeability. The calculation process of this method is complex and exhibits poor accuracy. By contrast, the required experimental apparatus for the proposed method is simple, and thus, the effective stress coefficient can be determined using conventional rock permeability testing apparatus. In addition, the procedures for determining the effective stress coefficient are more convenient, fast, and exhibit high accuracy.

6. Conclusions This work proposed a novel method for determining the

Eq. (4) 0.2231 0.2045 0.2119 0.2056 0.2147 0.2093

η Eq. (5) 0.2237 0.2041 0.2123 0.2060 0.2146 0.2089

Proposed method 0.2242 0.2043 0.2120 0.2063 0.2155 0.2092

effective stress coefficient and demonstrated that such coefficient in tight sandstone reservoirs was not equal to 1. When the effective stress coefficient η = 0.201, the stress-sensitive curves of core samples from the same location and with similar physical properties are alike, which indicates that the calculated effective stress coefficient is reasonable. These effective stress parameters depend on confine pressure and pore pressure, and exhibit a larger reduction with increasing confine pressure. The stress-sensitive effect only slightly influences the variation of permeability under the actual exploitation path. The permeability variation of the microfracture core is larger than that of the matrix core with increasing net stress. The stress-sensitive feature of the microfracture core will be overestimated when the conventional effective stress coefficient η = 1 is applied. In addition, the recovery degree of the permeability of the microfracture core is higher during the unloading process than that of the matrix core because of the simple mechanical structure of microfracture media. On the basis of a comparison with two other methods for determining the effective stress coefficient, the results of the proposed method is proven to be accurate. Moreover, the test procedure of the proposed method for determining the effective stress coefficient is simpler, more convenient, and faster than those of the other two methods.

Acknowledgements This study was supported by the National Natural Science Foundation of China (51490652, U1562215 and 51604287), the National 973 Project “Control Mechanism for Formation and Repeated Fracturing Transformation of Artificial Joint Network for Tight Reservoir (2015CB250903)”, the Science Foundation of China University of Petroleum, Beijing (2462015YQ1202), and the Sinopec scientific research project “Micro-nano capillary spontaneous imbibition and unconventional oil and gas reservoir damage mechanism” (P15026).

References Bear, J. (1972). “Dynamics of fluids in Porous Media.” American Elsevier Publishing Company (ISBN-13:978-0-486-65675-5, pp. 59-113) New York. Bernabe, Y. (1986). “The effective pressure law for Chelmsford granite

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KSCE Journal of Civil Engineering

Novel Method for Calculating the Effective Stress Coefficient in a Tight Sandstone Reservoir

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