# A New Pseudo Steady-State Constant for a Vertical Well with Finite-Conductivity Fracture

^{1}

^{2}

^{*}

## Abstract

**:**

_{Dps}

_{s}is defined as the difference between the dimensionless wellbore pressure and dimensionless average pressure of a reservoir with a PSS flow regime. As an important parameter, b

_{Dps}

_{s}has been widely used for decline curve analysis with Type Curves. For a well with a finite-conductivity fracture, b

_{Dps}

_{s}is independent of time and is a function of the penetration ratio of facture and fracture conductivity. In this study, we develop a new semi-analytical solution for b

_{Dps}

_{s}calculations using the PSS function of a circular reservoir. Based on the semi-analytical solution, a new conductivity-influence function (CIF) representing the additional pressure drop caused by the effect of fracture conductivity is presented. A normalized conductivity-influence function (NCIF) is also developed to calculate the CIF. Finally, a new approximate solution is proposed to obtain the b

_{Dps}

_{s}value. This approximate solution is a fast, accurate, and time-saving calculation.

## 1. Introduction

_{Dps}

_{s}is the Pseudo Steady-State (PSS) constant [24]. This PSS constant b

_{Dpss}has been widely used to define the appropriate dimensionless decline rate in many currently used production-decline rate analysis models [20,25,26,27,28].

_{Dps}

_{s}can be obtained by running a long-term numerical simulation [20,21,23]. Pratikno et al. [20] presented a numerical solution for the PSS constant for a well with a finite-conductivity fracture in a circular closed reservoir (Figure 1).

_{Dps}

_{s}is usually approximated as an analytical expression [20,21,23]. Pratikno et al. [20] proposed the following approximate expression:

_{fD}), is the additional pressure drop caused by the effect of fracture conductivity. We define f(C

_{fD}) as the conductivity-influence function (CIF). Thus, the PSS constant b

_{Dps}

_{s}can be expressed as the sum of the pressure drop of a well with an infinite-conductivity fracture and the conductivity-influence function (CIF). Note that the CIF in Equation (4) is only a function of the fracture conductivity.

_{fD}) using regression for a circular reservoir.

_{Dps}

_{s}. The corresponding conductivity-influence function for an elliptical reservoir was presented by Amini et al. [21]:

_{Dps}

_{s}calculation.

- (1)
- The assumption of elliptical flow is an approximate model of the circular boundary [29]. For a fractured well, the real pressure front should be a circle instead of an ellipse during the late-time flow regime.
- (2)
- When C
_{fD}trends to infinity, i.e., the infinite-conductivity fracture, Equations (4), (6), and (7) cannot meet the following condition, meaning the limit of conductivity-influence function f(C_{fD}) should be zero.$$\underset{{C}_{fD}\to \infty}{\mathrm{lim}}f\left({C}_{fD}\right)=0$$ - (3)
- For the existing approximate models [20,21,23], the conductivity-influence function f(C
_{fD}) is only relative to the fracture conductivity. For different penetration fracture ratios I_{x}, the distributions in the flow field around the fracture are different, which affects the value of conductivity-influence function (Figure 2). Thus, CIF is not only a function of conductivity, but also related to penetration ratio.

_{Dps}

_{s}by use of the PSS function instead of the transient-pressure function [20]; (2) based on the results from the semi-analytical method, a new conductivity-influence function (CIF) is introduced considering the effect of penetration ratio and fracture conductivity has been established; and (3) a new normalized conductivity-influence function (NCIF) is introduced to calculate the value of b

_{Dps}

_{s}.

## 2. Mathematical Model

#### 2.1. Basic Assumptions

_{e}. The flow in the reservoir and fracture is assumed to be single phase and isothermal with a slightly compressible Newtonian fluid. The penetrate ratio I

_{x}is defined as:

#### 2.2. Semi-Analytical Model for b_{Dpss} Calculation

#### 2.2.1. Flow Model of the Reservoir

#### 2.2.2. Flow Model of the Fracture

#### 2.2.3. Semi-Analytical Solution for b_{Dpss}

_{Dpss}can be obtained by solving Equations (17) and (18) with the Gauss elimination method.

#### 2.3. Conductivity-Influence Function (CIF)

_{Dpss}for different I

_{x}and C

_{fD}using the semi-analytical model in Section 2.2. (2) Calculate the pressure drop of the infinite-conductivity fracture f(r

_{eD}) for different I

_{x}with Equations (3) and (9). (3) Calculate the difference of b

_{Dpss}and f(r

_{eD}) with Equation (2) and the value of conductivity-influence function (CIF) can be obtained.

_{fD}is less than 10. Additionally, fracture conductivity function tends to be zero when C

_{fD}is greater than 300. Table 1 presents the value of the conductivity-influence function for I

_{x}= 0.001–1 and C

_{fD}= 0.1–1000.

#### 2.4. New Approximate Solution of Pseudo Steady-State Constant b_{Dpss}

_{Dpss,FC}is the function of the penetration ratio, and the conductivity-influence function is the function of the penetration ratio and conductivity.

_{x}= 0). As shown in Equation (9):

_{Dpss}can be approximately replaced by the dimensionless-pressure difference between the finite-conductivity fracture and infinite-conductivity fracture at the radial flow regime for the infinite reservoir [15].

_{x}= 0 (blue) and I

_{x}= 1 (red). Two regression equations for CIF are obtained as follows.

_{x}= 0:

_{x}= 1, the hydraulic fracture fully penetrates the reservoir.

_{x}= 0.001–1 and C

_{fD}= 0.1–1000 with the method in Section 2.2 and Section 2.3. Ten values were calculated for each logarithmic period. Thus, 1200 values of CIF were obtained. The values of NCIF can be calculated using Equation (28).

_{x}. Notably, all data fall in the same straight line in logarithmic coordinates. This means that the NCIF is solely dependent on penetration ratio I

_{x}.

_{x}= 0, the NCIF is equal to 0 in Equation (29).

_{Dpss}is presented in Equation (32).

## 3. Results

_{Dpss}obtained by our semi-analytical method, Pratikno et al.’s method and our approximate model. As shown in Table 2, the maximum relative error is 0.4% at I

_{x}= 0.1 and 1.93% at I

_{x}= 1 between our approximate model and semi-analytical model. However, huge differences between Pratikno et al.’s method and our semi-analytical method can be observed at large penetration ratio, for example, relative error 16.57% at C

_{fD}= 0.631 and I

_{x}= 1.

_{fD}is greater than 10 for all penetration ratios. The lines cross the centers of circles at low penetration ratios, such as I

_{x}values less than 0.3. For these cases, the differences among the three models can be ignored. Our approximate solutions (blue lines) match very well with semi-analytical solutions (circles with cross) for all penetration ratios and fracture conductivities. The red lines deviate from the circles and blue lines and huge differences between Pratikno et al.’s solutions and our solutions are noticeable at low conductivity (C

_{fD}< 10) with a high penetration ratio (I

_{x}> 0.5).

## 4. Conclusions

_{fD}is greater than 300. Additionally, CIF has obviously differences, especially when C

_{fD}is less than 10; the CIF decreases with increasing penetration ratio. (2) Based on the PSS function [11], a new semi-analytical model was proposed to directly calculate the b

_{Dpss}, which consumes less time. Besides, our method is more accurate due to simultaneously considering both fracture conductivity and penetration ratio. (3) A new conductivity-influence function (CIF), considering the effect of penetration ratio and fracture conductivity, was developed. A normalized conductivity-influence function (NCIF) was also developed to calculate the value of CIF. Our work provides a fast, accurate, and time-saving method to evaluate b

_{Dpss}for a well with a finite-conductivity fracture in a circular closed reservoir.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Schematic of the flow field around a fracture in a circular reservoir at late-time regime: (

**a**) a short fracture with a low penetration ratio and (

**b**) a long fracture with high penetration ratio.

**Figure 3.**Conductivity-influence function with fracture conductivity (C

_{fD}) and penetration ratio (I

_{x}).

**Figure 4.**Regression of conductivity-influence function when the penetration ratio (I

_{x}) equals 0 and 1.

**Table 1.**Conductivity-influence function (CIF) of different penetration ratio (I

_{x}) and fracture conductivity (C

_{fD}).

C_{fD} | I_{x} | ||||||||
---|---|---|---|---|---|---|---|---|---|

0.001 | 0.05 | 0.01 | 0.05 | 0.1 | 0.3 | 0.5 | 0.8 | 1 | |

0.1 | 2.924955 | 2.923983 | 2.924917 | 2.923983 | 2.921063 | 2.889918 | 2.82763 | 2.675801 | 2.535652 |

0.125893 | 2.694126 | 2.693183 | 2.694089 | 2.693183 | 2.690353 | 2.660159 | 2.599771 | 2.452575 | 2.316702 |

0.158489 | 2.473614 | 2.472703 | 2.473578 | 2.472703 | 2.46997 | 2.440814 | 2.382503 | 2.240371 | 2.109172 |

0.199526 | 2.259775 | 2.258901 | 2.25974 | 2.258901 | 2.256277 | 2.228293 | 2.172325 | 2.035904 | 1.909976 |

0.251189 | 2.050804 | 2.049971 | 2.050771 | 2.049971 | 2.047474 | 2.02083 | 1.967543 | 1.837656 | 1.71776 |

0.316228 | 1.846441 | 1.845657 | 1.84641 | 1.845657 | 1.843302 | 1.818188 | 1.767959 | 1.645526 | 1.532512 |

0.398107 | 1.647651 | 1.64692 | 1.647622 | 1.64692 | 1.644726 | 1.621328 | 1.574532 | 1.460467 | 1.355176 |

0.501187 | 1.456232 | 1.45556 | 1.456206 | 1.45556 | 1.453543 | 1.432024 | 1.388986 | 1.284082 | 1.187247 |

0.630957 | 1.274395 | 1.273785 | 1.274371 | 1.273785 | 1.271955 | 1.252433 | 1.213389 | 1.118219 | 1.03037 |

0.794328 | 1.104337 | 1.103791 | 1.104315 | 1.103791 | 1.102154 | 1.084688 | 1.049755 | 0.964606 | 0.886007 |

1 | 0.9479 | 0.947418 | 0.947881 | 0.947418 | 0.945973 | 0.930557 | 0.899725 | 0.824572 | 0.7552 |

1.258925 | 0.806339 | 0.805919 | 0.806322 | 0.805919 | 0.80466 | 0.791229 | 0.764366 | 0.698889 | 0.638448 |

1.584893 | 0.680231 | 0.67987 | 0.680217 | 0.67987 | 0.678786 | 0.667223 | 0.644099 | 0.587732 | 0.535701 |

1.995262 | 0.569498 | 0.56919 | 0.569486 | 0.56919 | 0.568267 | 0.558422 | 0.538731 | 0.490735 | 0.446431 |

2.511886 | 0.473515 | 0.473256 | 0.473505 | 0.473256 | 0.472478 | 0.464176 | 0.447573 | 0.407101 | 0.369743 |

3.162278 | 0.391259 | 0.391042 | 0.39125 | 0.391042 | 0.390391 | 0.383452 | 0.369573 | 0.335742 | 0.304514 |

3.981072 | 0.321455 | 0.321276 | 0.321448 | 0.321276 | 0.320736 | 0.314979 | 0.303467 | 0.275404 | 0.2495 |

5.011872 | 0.262718 | 0.26257 | 0.262712 | 0.26257 | 0.262125 | 0.257382 | 0.247898 | 0.224778 | 0.203437 |

6.309573 | 0.213648 | 0.213526 | 0.213643 | 0.213526 | 0.213162 | 0.209279 | 0.201513 | 0.182584 | 0.16511 |

7.943282 | 0.17291 | 0.172811 | 0.172906 | 0.172811 | 0.172514 | 0.169353 | 0.16303 | 0.147618 | 0.133391 |

10 | 0.139273 | 0.139193 | 0.13927 | 0.139193 | 0.138953 | 0.136392 | 0.131272 | 0.118789 | 0.107268 |

12.58925 | 0.111637 | 0.111572 | 0.111634 | 0.111572 | 0.111379 | 0.109316 | 0.105189 | 0.095131 | 0.085847 |

15.84893 | 0.089036 | 0.088984 | 0.089034 | 0.088984 | 0.088829 | 0.087175 | 0.083866 | 0.075802 | 0.068357 |

19.95262 | 0.070637 | 0.070596 | 0.070636 | 0.070596 | 0.070472 | 0.069152 | 0.066513 | 0.060078 | 0.054139 |

25.11886 | 0.055731 | 0.055698 | 0.05573 | 0.055698 | 0.0556 | 0.054552 | 0.052457 | 0.047349 | 0.042633 |

31.62278 | 0.043716 | 0.043691 | 0.043715 | 0.043691 | 0.043613 | 0.042785 | 0.04113 | 0.037096 | 0.033371 |

39.81072 | 0.034091 | 0.03407 | 0.03409 | 0.03407 | 0.034009 | 0.033359 | 0.032058 | 0.028887 | 0.02596 |

50.11872 | 0.026434 | 0.026418 | 0.026433 | 0.026418 | 0.02637 | 0.025861 | 0.024844 | 0.022363 | 0.020074 |

63.09573 | 0.020396 | 0.020383 | 0.020395 | 0.020383 | 0.020346 | 0.01995 | 0.019157 | 0.017225 | 0.015441 |

79.43282 | 0.015687 | 0.015677 | 0.015687 | 0.015677 | 0.015648 | 0.015341 | 0.014725 | 0.013223 | 0.011838 |

100 | 0.012068 | 0.012061 | 0.012068 | 0.012061 | 0.012038 | 0.011799 | 0.011321 | 0.010155 | 0.009078 |

125.8925 | 0.009341 | 0.009335 | 0.009341 | 0.009335 | 0.009318 | 0.009131 | 0.008758 | 0.007849 | 0.007009 |

158.4893 | 0.007342 | 0.007337 | 0.007342 | 0.007337 | 0.007324 | 0.007177 | 0.006883 | 0.006166 | 0.005504 |

199.5262 | 0.005936 | 0.005933 | 0.005936 | 0.005933 | 0.005922 | 0.005803 | 0.005567 | 0.004991 | 0.004459 |

251.1886 | 0.005012 | 0.005009 | 0.005012 | 0.005009 | 0.005 | 0.004902 | 0.004706 | 0.004228 | 0.003786 |

316.2278 | 0.004479 | 0.004476 | 0.004478 | 0.004476 | 0.004468 | 0.004383 | 0.004213 | 0.003799 | 0.003417 |

398.1072 | 0.004259 | 0.004257 | 0.004259 | 0.004257 | 0.00425 | 0.004172 | 0.004017 | 0.003639 | 0.00329 |

501.1872 | 0.004292 | 0.004289 | 0.004292 | 0.004289 | 0.004283 | 0.004208 | 0.004059 | 0.003695 | 0.003359 |

630.9573 | 0.004525 | 0.004523 | 0.004525 | 0.004523 | 0.004516 | 0.00444 | 0.00429 | 0.003923 | 0.003584 |

794.3282 | 0.004917 | 0.004914 | 0.004916 | 0.004914 | 0.004907 | 0.004828 | 0.00467 | 0.004285 | 0.00393 |

1000 | 0.005431 | 0.005429 | 0.005431 | 0.005429 | 0.005421 | 0.005336 | 0.005167 | 0.004753 | 0.004372 |

Semi-Analytical Solutions (This Study) | H.Pratikno et al. Solutions (2003) SPE 84287 | Approximate Solutions (This Study) | Relative Error between H.Pratikno et al. Solutions and Semi-Analytical Solutions, % | Relative Error between Semi-Analytical Solutions and Approximate Solutions, % | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

C_{fD} | Ix = 0.1 | Ix = 0.5 | Ix = 1 | Ix = 0.1 | Ix = 0.5 | Ix = 1 | Ix = 0.1 | Ix = 0.5 | Ix = 1 | Ix = 0.1 | Ix = 0.5 | Ix = 1 | Ix = 0.1 | Ix = 0.5 | Ix = 1 |

0.1000 | 5.1692 | 3.5727 | 2.9135 | 5.1960 | 3.6909 | 3.3237 | 5.1787 | 3.5801 | 2.9210 | 0.52 | 3.31 | 14.08 | 0.18 | 0.21 | 0.26 |

0.1259 | 4.9454 | 3.3517 | 2.7003 | 4.9595 | 3.4544 | 3.0872 | 4.9480 | 3.3523 | 2.7020 | 0.29 | 3.06 | 14.33 | 0.05 | 0.02 | 0.06 |

0.1585 | 4.7252 | 3.1350 | 2.4928 | 4.7319 | 3.2268 | 2.8596 | 4.7276 | 3.1350 | 2.4945 | 0.14 | 2.93 | 14.72 | 0.05 | 0.00 | 0.07 |

0.1995 | 4.5091 | 2.9230 | 2.2915 | 4.5115 | 3.0064 | 2.6392 | 4.5139 | 2.9248 | 2.2953 | 0.05 | 2.85 | 15.18 | 0.11 | 0.06 | 0.17 |

0.2512 | 4.2979 | 2.7163 | 2.0972 | 4.2976 | 2.7924 | 2.4253 | 4.3051 | 2.7201 | 2.1031 | 0.01 | 2.80 | 15.64 | 0.17 | 0.14 | 0.28 |

0.3162 | 4.0924 | 2.5160 | 1.9111 | 4.0902 | 2.5850 | 2.2179 | 4.1009 | 2.5205 | 1.9179 | 0.05 | 2.74 | 16.05 | 0.21 | 0.18 | 0.35 |

0.3981 | 3.8940 | 2.3234 | 1.7340 | 3.8902 | 2.3851 | 2.0179 | 3.9024 | 2.3270 | 1.7405 | 0.10 | 2.66 | 16.37 | 0.22 | 0.16 | 0.37 |

0.5012 | 3.7040 | 2.1396 | 1.5672 | 3.6989 | 2.1938 | 1.8267 | 3.7112 | 2.1415 | 1.5726 | 0.14 | 2.53 | 16.55 | 0.19 | 0.09 | 0.34 |

0.6310 | 3.5242 | 1.9663 | 1.4118 | 3.5181 | 2.0130 | 1.6458 | 3.5296 | 1.9659 | 1.4157 | 0.17 | 2.37 | 16.57 | 0.15 | 0.02 | 0.27 |

0.7943 | 3.3560 | 1.8048 | 1.2687 | 3.3493 | 1.8442 | 1.4770 | 3.3598 | 1.8023 | 1.2713 | 0.20 | 2.18 | 16.42 | 0.11 | 0.14 | 0.21 |

1.0000 | 3.2008 | 1.6563 | 1.1386 | 3.1939 | 1.6888 | 1.3216 | 3.2036 | 1.6522 | 1.1405 | 0.21 | 1.96 | 16.08 | 0.09 | 0.25 | 0.17 |

1.2589 | 3.0597 | 1.5217 | 1.0218 | 3.0530 | 1.5479 | 1.1807 | 3.0623 | 1.5169 | 1.0238 | 0.22 | 1.72 | 15.55 | 0.09 | 0.32 | 0.19 |

1.5849 | 2.9333 | 1.4015 | 0.9186 | 2.9271 | 1.4220 | 1.0548 | 2.9364 | 1.3966 | 0.9210 | 0.21 | 1.46 | 14.83 | 0.10 | 0.35 | 0.27 |

1.9953 | 2.8219 | 1.2957 | 0.8285 | 2.8162 | 1.3111 | 0.9439 | 2.8259 | 1.2912 | 0.8318 | 0.20 | 1.18 | 13.94 | 0.14 | 0.35 | 0.40 |

2.5119 | 2.7249 | 1.2039 | 0.7508 | 2.7198 | 1.2147 | 0.8475 | 2.7301 | 1.2001 | 0.7551 | 0.19 | 0.89 | 12.89 | 0.19 | 0.32 | 0.57 |

3.1623 | 2.6418 | 1.1253 | 0.6847 | 2.6371 | 1.1320 | 0.7648 | 2.6480 | 1.1221 | 0.6899 | 0.18 | 0.60 | 11.71 | 0.24 | 0.29 | 0.75 |

3.9811 | 2.5713 | 1.0587 | 0.6290 | 2.5671 | 1.0620 | 0.6948 | 2.5784 | 1.0560 | 0.6348 | 0.16 | 0.31 | 10.47 | 0.28 | 0.26 | 0.93 |

5.0119 | 2.5121 | 1.0030 | 0.5825 | 2.5068 | 1.0017 | 0.6345 | 2.5198 | 1.0004 | 0.5888 | 0.21 | 0.12 | 8.93 | 0.30 | 0.26 | 1.06 |

6.3096 | 2.4630 | 0.9567 | 0.5441 | 2.4585 | 0.9534 | 0.5862 | 2.4708 | 0.9540 | 0.5505 | 0.18 | 0.35 | 7.73 | 0.31 | 0.28 | 1.15 |

7.9433 | 2.4225 | 0.9186 | 0.5126 | 2.4180 | 0.9129 | 0.5457 | 2.4301 | 0.9155 | 0.5187 | 0.19 | 0.62 | 6.46 | 0.31 | 0.33 | 1.19 |

10.0000 | 2.3894 | 0.8874 | 0.4868 | 2.3847 | 0.8796 | 0.5124 | 2.3966 | 0.8838 | 0.4926 | 0.20 | 0.88 | 5.26 | 0.30 | 0.41 | 1.18 |

12.5893 | 2.3625 | 0.8620 | 0.4659 | 2.3576 | 0.8525 | 0.4853 | 2.3690 | 0.8577 | 0.4712 | 0.20 | 1.11 | 4.17 | 0.28 | 0.51 | 1.12 |

15.8489 | 2.3406 | 0.8415 | 0.4490 | 2.3356 | 0.8305 | 0.4633 | 2.3465 | 0.8364 | 0.4537 | 0.21 | 1.31 | 3.19 | 0.25 | 0.61 | 1.04 |

19.9526 | 2.3230 | 0.8249 | 0.4354 | 2.3179 | 0.8128 | 0.4456 | 2.3281 | 0.8190 | 0.4395 | 0.22 | 1.47 | 2.35 | 0.22 | 0.72 | 0.93 |

25.1189 | 2.3088 | 0.8116 | 0.4244 | 2.3037 | 0.7985 | 0.4314 | 2.3132 | 0.8050 | 0.4280 | 0.22 | 1.61 | 1.64 | 0.19 | 0.83 | 0.83 |

31.6228 | 2.2974 | 0.8009 | 0.4157 | 2.2923 | 0.7872 | 0.4200 | 2.3012 | 0.7936 | 0.4187 | 0.22 | 1.72 | 1.04 | 0.17 | 0.92 | 0.73 |

39.8107 | 2.2883 | 0.7924 | 0.4086 | 2.2832 | 0.7781 | 0.4109 | 2.2916 | 0.7846 | 0.4113 | 0.22 | 1.80 | 0.56 | 0.15 | 0.99 | 0.65 |

50.1187 | 2.2810 | 0.7855 | 0.4030 | 2.2760 | 0.7709 | 0.4037 | 2.2840 | 0.7774 | 0.4054 | 0.22 | 1.86 | 0.18 | 0.13 | 1.05 | 0.59 |

63.0957 | 2.2752 | 0.7801 | 0.3986 | 2.2703 | 0.7652 | 0.3980 | 2.2780 | 0.7717 | 0.4008 | 0.21 | 1.90 | 0.13 | 0.12 | 1.09 | 0.56 |

79.4328 | 2.2705 | 0.7757 | 0.3950 | 2.2658 | 0.7607 | 0.3935 | 2.2733 | 0.7672 | 0.3972 | 0.21 | 1.93 | 0.36 | 0.12 | 1.10 | 0.55 |

100.0000 | 2.2668 | 0.7722 | 0.3921 | 2.2623 | 0.7572 | 0.3900 | 2.2697 | 0.7638 | 0.3944 | 0.20 | 1.95 | 0.55 | 0.12 | 1.10 | 0.58 |

125.8925 | 2.2639 | 0.7695 | 0.3899 | 2.2595 | 0.7543 | 0.3872 | 2.2670 | 0.7613 | 0.3924 | 0.20 | 1.96 | 0.69 | 0.13 | 1.08 | 0.63 |

158.4893 | 2.2616 | 0.7673 | 0.3881 | 2.2573 | 0.7521 | 0.3850 | 2.2650 | 0.7594 | 0.3908 | 0.19 | 1.97 | 0.80 | 0.15 | 1.04 | 0.71 |

199.5262 | 2.2597 | 0.7655 | 0.3866 | 2.2555 | 0.7504 | 0.3832 | 2.2636 | 0.7581 | 0.3898 | 0.19 | 1.98 | 0.89 | 0.17 | 0.98 | 0.81 |

251.1886 | 2.2582 | 0.7641 | 0.3855 | 2.2541 | 0.7489 | 0.3818 | 2.2626 | 0.7572 | 0.3891 | 0.18 | 1.99 | 0.97 | 0.20 | 0.91 | 0.93 |

316.2278 | 2.2570 | 0.7630 | 0.3846 | 2.2529 | 0.7478 | 0.3806 | 2.2621 | 0.7567 | 0.3888 | 0.18 | 2.00 | 1.04 | 0.22 | 0.83 | 1.07 |

398.1072 | 2.2561 | 0.7621 | 0.3839 | 2.2519 | 0.7468 | 0.3796 | 2.2619 | 0.7565 | 0.3886 | 0.19 | 2.01 | 1.11 | 0.26 | 0.74 | 1.23 |

501.1872 | 2.2553 | 0.7614 | 0.3833 | 2.2511 | 0.7459 | 0.3788 | 2.2619 | 0.7566 | 0.3887 | 0.19 | 2.03 | 1.18 | 0.29 | 0.64 | 1.39 |

630.9573 | 2.2547 | 0.7609 | 0.3828 | 2.2503 | 0.7452 | 0.3780 | 2.2621 | 0.7568 | 0.3889 | 0.20 | 2.06 | 1.26 | 0.33 | 0.54 | 1.56 |

794.3282 | 2.2543 | 0.7604 | 0.3825 | 2.2496 | 0.7445 | 0.3774 | 2.2625 | 0.7572 | 0.3893 | 0.21 | 2.09 | 1.34 | 0.37 | 0.43 | 1.74 |

1000 | 2.2539 | 0.7601 | 0.3822 | 2.2490 | 0.7439 | 0.3767 | 2.2631 | 0.7577 | 0.3897 | 0.22 | 2.13 | 1.44 | 0.40 | 0.32 | 1.93 |

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## Share and Cite

**MDPI and ACS Style**

Cui, Y.; Lu, B.; Wu, M.; Luo, W.
A New Pseudo Steady-State Constant for a Vertical Well with Finite-Conductivity Fracture. *Processes* **2018**, *6*, 93.
https://doi.org/10.3390/pr6070093

**AMA Style**

Cui Y, Lu B, Wu M, Luo W.
A New Pseudo Steady-State Constant for a Vertical Well with Finite-Conductivity Fracture. *Processes*. 2018; 6(7):93.
https://doi.org/10.3390/pr6070093

**Chicago/Turabian Style**

Cui, Yudong, Bin Lu, Mingtao Wu, and Wanjing Luo.
2018. "A New Pseudo Steady-State Constant for a Vertical Well with Finite-Conductivity Fracture" *Processes* 6, no. 7: 93.
https://doi.org/10.3390/pr6070093