Introduction

Jh equation of Mutalik et al. (1988) is well known and presented by many authors (Duda et al. (1991), Economides et al. (1991b), Economides et al. (1991a), Frick and Economides (1993), Thomas et al. (1998), Penmatcha et al. (1999), Saavedra and Joshi (2002), Cho (2003), Jackson et al. (2011), Adesina et al. (2011), Ouyang (2015), Cetkovic et al. (2016), Wu et al. (2018), Luo et al. (2018), and Zhang et al. (2020)). In reservoir simulators, Jh equation of Mutalik et al. (1988) has been used to compute well productivity index (Krawchuk et al. 2006; Pinzon et al. 2007; Huebsch et al. 2008; Haidar et al. 2008; Brown and Tiwari 2010; Wood et al. 2010; and Leone et al. 2015). The Jh equation is defined as follows:

$${J}_{\mathrm{h}}=\frac{0.007078 k h/{\mu }_{o} {B}_{o}}{\mathrm{ln}\left(\frac{\sqrt{A/\pi }}{{r}_{\mathrm{w}}}\right)-2.1243-\mathrm{ln}\left(L/4 {r}_{\mathrm{w}}\right)+\mathrm{ln}\sqrt{30.8828/{C}_{A}} }$$
(1)
$${C}_{A}=\frac{8.9832 {x}_{e} {y}_{e} }{{L}^{2}}\mathrm{exp}\left[2\left(2 \pi {\left({T}_{\mathrm{DA}}\right)}_{\mathrm{PSS}}- {\left({P}_{\mathrm{D}}T\right)}_{\mathrm{PSS}}\right)\right]$$
(2)

where Jh is the horizontal well productivity (STB/day.psi), k is the reservoir permeability (md), h is the reservoir thickness (ft), µo is the oil viscosity (cp), Bo is the oil formation volume factor (bbl/STB), A is the drainage area (ft2), rw is the well radius (ft), L is the length of horizontal well (ft), xe is the reservoir length in the X-direction (ft), the horizontal well is placed along the X-direction, ye is the reservoir length in the Y-direction (ft), and (PDT)PSS is the total dimensionless pressure at PSS.

The Jh equation [Eq. (1)] is for rectangular bounded reservoir at depletion stage (i.e., PSS). In CA equation [i.e., Eq. (2)], the (PDT)PSS was obtained by the PTM:

$$\begin{gathered} P_{{\text{D}}} T = 2\pi \mathop \smallint \limits_{0}^{{T_{{{\text{DA}}}} }} \left\{ { \left[ {1 + \frac{{8 x_{e} }}{\pi L} \mathop \sum \limits_{n = 1}^{\infty } \frac{1}{n}\exp \left( {\frac{{ - n^{2} \pi^{2} y_{e} T_{{{\text{DA}}}} }}{{x_{e} }}} \right)\cos \left( {\frac{{n \pi L x_{{{\text{wD}}}} }}{{4 x_{e} }}} \right) } \right.} \right. \hfill \\ \left. {\cos \left( {\frac{{n \pi L \left( {x_{{{\text{wD}}}} + x_{D} } \right)}}{{4 x_{e} }}} \right) \sin \left( {\frac{n \pi L}{{4 x_{e} }}} \right)} \right] \times \left[ {1 + 2\mathop \sum \limits_{n = 1}^{\infty } \exp \left( {\frac{{ - n^{2} \pi^{2} x_{e} T_{{{\text{DA}}}} }}{{y_{e} }}} \right) } \right. \hfill \\ \left. {\cos \left( {\frac{{n \pi L y_{{{\text{wD}}}} }}{{4 y_{e} }}} \right) \cos \left( {\frac{{n \pi L \left( {y_{{{\text{wD}}}} + y_{D} } \right)}}{{4 y_{e} }}} \right)} \right] \times \left[ {1 + 2} \right. \hfill \\ \left. {\left. {\mathop \sum \limits_{n = 1}^{\infty } \exp \left( { - 4 n^{2} \pi^{2} \frac{{x_{e} y_{e} }}{{h^{2} }} T_{{{\text{DA}}}} } \right) \cos \left( {n \pi z_{{{\text{wD}}}} } \right) \cos \left( {n \pi z_{{\text{D}}} } \right)} \right] } \right\} {\text{d}}T_{{{\text{DA}}}} \hfill \\ \end{gathered}$$
(3)

where PDT is the total dimensionless pressure, TDA is the dimensionless time based on drainage area, xwD, ywD, and zwD are the dimensionless well location coordinates, xD is the X-coordinate for a point in the porous media. (xD=0) at the horizontal well center, and (xD= 1 and −1) at the well tips. yD is the Y-coordinate for a point in the porous media. (yD=rWD) if the point is at the well wall, zD is the Z-coordinate for a point in the porous media. (zD= zWD) if the point is at the well wall

PTM [i.e., Eq. (3)] is based on 3D instantaneous source functions with Newman product method under homogenous and isotropic rectangular bounded reservoir at constant well flow rate and uniform flux wellbore condition of Gringarten and Ramey (1973). It is assumed that horizontal well is analogous to a vertical well of a vertical fracture. The equivalent infinite conductivity wellbore condition was obtained by solving PDT at xD equal to (0.732) (Mutalik et al. (1988) and Gringarten et al. (1974)). “Appendix 1” presents a summary of PTM.

When PDT slope with time becomes constant at PSS and is equal to (2π), both the (TDA)PSS and (PDT)PSS have been obtained to use for CA calculation. CA at PSS also becomes constant for a given well and reservoir configuration.

PTM and their reported CA tables have identified three issues. Firstly, PTM as defined in Eq. (3) contains several dimensional parameters (i.e., xe, ye, h, and L) and this makes the solution not only awkward, but also difficult to use PTM to expand the reported CA tables to any other well and reservoir configurations. Secondly, PTM has not been verified in terms of PDT which may lead to errors in the reported CA tables. The third issue is the discrepancy in the assumed input data of the reported CA tables. The first assumption of the input data (i.e., rw = 0.225 ft and ye = 933.33 ft) produces variable rWD where (rWD = 2rw/L) depends on penetration ratio (PR) and (xe/ye). The second assumption of the input data shows that rwD is constant and is equal to (10–4) for any well and reservoir configurations. These three limitations cast a doubt on the representativeness and usefulness of the reported CA tables for their respective configurations or other configurations for which simply interpolation or extrapolation of those tables is required as the correlative models previously presented by Bahadori (2012), Bahadori et al. (2013), and Ahmadi et al. (2015).

The main purposes of this study are in four parts. (1) To overcome the first weakness in PTM (i.e., the dimensional parameters) by recasting PTM into a fully dimensionless form (i.e., GPTM) and to verify GPTM in terms of PDT. (2) To develop efficient algorithm to compute CA and Jh accurately for any well and reservoir configuration. Because CA depends on the all-configuration parameters, unlimited tables of CA can be generated; therefore, the MATLAB code is provided. (3) To quantify the accuracy of the reported CA tables in their both assumed input data (i.e., constant and variable rWD), and to determine the consequences of any identified errors in the reported CA tables when they are used to estimate Jh. (4) To generate new CA tables for a wider spectrum of well and reservoir configurations.

The generalized pressure transient model (GPTM)

The PTM is improved to fully dimensionless form (i.e., to GPTM) by applying the following dimensionless parameters into Eq. (3):

$${L}_{\mathrm{D}}=\frac{L}{2h}$$
(4)
$${x}_{e\mathrm{D}}=\frac{2{x}_{e}}{L}$$
(5)
$${y}_{e\mathrm{D}}=\frac{2{y}_{e}}{L}$$
(6)
$$\begin{gathered} P_{{\text{D}}} T = 2\pi \mathop \smallint \limits_{0}^{{T_{{{\text{DA}}}} }} \left\{ { \left[ {1 + \frac{{4 x_{{e{\text{D}}}} }}{\pi } \mathop \sum \limits_{n = 1}^{\infty } \frac{1}{n}\exp \left( {\frac{{ - n^{2} \pi^{2} y_{{e{\text{D}}}} T_{DA} }}{{x_{{e{\text{D}}}} }}} \right)\cos \left( {\frac{{n \pi x_{{{\text{wD}}}} }}{{2 x_{{e{\text{D}}}} }}} \right) } \right.} \right. \hfill \\ \left. {\cos \left( {\frac{{n \pi \left( {x_{{{\text{wD}}}} + x_{{\text{D}}} } \right)}}{{2 x_{{e{\text{D}}}} }}} \right)\sin \left( {\frac{{n \pi x_{{{\text{wD}}}} }}{{2 x_{{e{\text{D}}}} }}} \right)} \right] \times \left[ {1 + 2\mathop \sum \limits_{n = 1}^{\infty } \exp \left( {\frac{{ - n^{2} \pi^{2} x_{{e{\text{D}}}} T_{{{\text{DA}}}} }}{{y_{{e{\text{D}}}} }}} \right) } \right. \hfill \\ \left. {\cos \left( {\frac{{n \pi y_{{{\text{wD}}}} }}{{2 y_{{e{\text{D}}}} }}} \right) \cos \left( {\frac{{n \pi \left( {y_{{{\text{wD}}}} + y_{{\text{D}}} } \right)}}{{2 y_{{e{\text{D}}}} }}} \right)} \right] \times \left[ {1 + 2} \right. \hfill \\ \left. {\left. {\mathop \sum \limits_{n = 1}^{\infty } \exp \left( { - 4 n^{2} \pi^{2} x_{{e{\text{D}}}} y_{{e{\text{D}}}} L_{{\text{D}}}^{2} T_{{{\text{DA}}}} } \right) \cos \left( {n \pi z_{{{\text{wD}}}} } \right) \cos \left( {n \pi z_{{\text{D}}} } \right)} \right] } \right\} {\text{d}}T_{{{\text{DA}}}} \hfill \\ \end{gathered}$$
(7)

where xeD is the dimensionless reservoir length in the X-direction and yeD is the dimensionless reservoir length in the Y-direction

For anisotropic reservoir solution, the dimensionless parameters are presented in “Appendix 2.”

The following meaningful mathematical relationship should be noted in the well and reservoir configuration:

\(\frac{{y}_{\mathrm{wD}}}{{y}_{e\mathrm{D}}}=\frac{ {y}_{\mathrm{w}}}{{y}_{e}}\) Used for centrally and off-centrally well location (OCWL) in the Y-direction.

\(\frac{{x}_{\mathrm{wD}}}{{x}_{e\mathrm{D}}}=\frac{ {x}_{\mathrm{w}}}{{x}_{e}}\) Used for centrally and OCWL in the X-direction.

\({z}_{\mathrm{wD}}=\frac{{z}_{\mathrm{w}}}{h}\) Used for centrally and OCWL in the Z-direction

\(\frac{{x}_{e\mathrm{D}}}{{y}_{e\mathrm{D}}}=\frac{{x}_{e}}{{y}_{e}}\) Dimensionless rectangular reservoir aspect ratio.

\(\mathrm{PR}=\frac{L}{2{x}_{e}}\) PR is also equal to (1/xeD) [i.e., the inverse of Eq. (5)].

Note that OCWL is used to indicate the offset of the well relative to the central well location (i.e., OCWL = 1) at a respective coordinate direction for the purpose of classifying the configurations only.

The efficient algorithm to solve Eq. (7) is developed and implemented in MATLAB code and is given in “Appendix 3.”

PDT by GPTM of this study [i.e., Eq. (7)] is verified against its counterpart by the horizontal well pressure transient model of Ozkan and Raghavan (1988) for a square bounded reservoir (i.e., xe/ye = 1). LD is equal to (10), and PR = (0.2) and (1), for a well that is centered in x-, y-, and z-directions [i.e., (xwD/xeD = 1), (ywD/yeD = 1), and (zwD = 0.5), or (OCWL = 1)]. The wellbore condition is uniform flux (xD = 0), and rwD is equal to (2 × 10–4). Figure 1 shows the identical results of PDT by both the models.

Fig. 1
figure 1

Comparison of PDT by GPTM and by the pressure transient model of Ozkan and Raghavan (1988) for a centrally well located in a square reservoir

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For further verification on GPTM, PDT of this study is compared with PDT of the pressure transient model of Gringarten et al. (1974) for vertical fracture well under square bounded reservoir. (LD = 103) (i.e., to make the thickness of the fracture to be zero), PR = (0.5) and (1), (OCWL = 1), (xD = 0), and (rwD = 10–4). Figure 2 shows the identical results of PDT by both the models.

Fig. 2
figure 2

Comparison of PDT by GPTM and by the pressure transient model of Gringarten et al. (1974) for vertical fracture well under square reservoir

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In the both previous verifications, the time intervals of TDA in Eq. (7) are to be taken: (0–10–9), (10–9–10–8), (10–8–10–7), (10–7–10–6), (10–6–10–5), (10–5–10–4), (10–4–10–3), (10–3–10–2), (10–2–10–1), (10–1–1), (1–10), and (10–100), and within each of these intervals its TDA increment is taken to be (10–9), (10–9), (10–9), (10–9), (10–9), (10–8), (10–7), (10–6), (10–5), (10–4), (10–3), and (10–2), respectively. These TDA intervals and their increments are sufficient to yield accurate PDT. It is found that the first-time logarithmic cycle (FTLC) [i.e., (0–10–9)] plays a crucially important role in determining the accuracy of PDT at any period even at PSS and then in CA calculation as to be shown in the next sections.

The CA equation [i.e., Eq. (2)] can be written in general form by substituting Eqs. (5) and (6) into Eq. (2):

$${C}_{A}=2.2458 {x}_{e\mathrm{D}} {y}_{e\mathrm{D}}\mathrm{exp}\left[2\left(2 \pi {\left({T}_{\mathrm{DA}}\right)}_{\mathrm{PSS}}- {\left({P}_{\mathrm{D}}T\right)}_{\mathrm{PSS}}\right)\right]$$
(8)

For vertical fracture well (i.e., when (LD > 103)), a comparison between the CA(s) by this study and their counterparts by Earlougher (1977) and Hagoort (2009) is presented in Table 1. For FTLC equal to (0–10–9) or (0–10–5), excellent matching is found with CA(s) by Hagoort (2009) at any PR and with CA(s) by Earlougher (1977) at just high PR.

Table 1 Comparison of CA by this study, Earlougher (1977) and Hagoort (2009)
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In what follows, the reported CA tables by PTM for selected well and reservoir configurations are examined, and their well and reservoir configurations can be classified into four cases in terms of OCWL. Table 2 lists the four cases of the reported CA tables. The parameters in Cases (1) to (3) correspond to tables (4), (7), and (8) in Mutalik et al. (1988) paper, respectively. Note that parameters in Case (4) are deduced from Fig. 12 in Mutalik et al. (1988) paper.

Table 2 The four cases of the reported CA tables by Mutalik et al. (1988)
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The accuracy of the reported C A tables by PTM (case 1)

Although the accuracy of the reported CA tables by PTM for other cases has been examined, here only the results for the Case 1 are presented for both assumed input data (i.e., the discrepancy issue). For each CA by PTM, the counterpart CA is calculated by GPTM. The relative errors of the reported CA tables are determined [i.e., CA% Err = (((CA)PTM − (CA)GPTM)/((CA)GPTM) × 100)] for the both assumed input data.

Tables 3 and 4 show the relative errors in the reported CA tables by PTM for the variable rwD and the constant rwD, respectively. For the first assumed input data (variable rwD), Table 3 shows that in xe/ye = (1) and (2), the CA errors of any PR are in reverse order with LD, while in xe/ye = (5), they are in non-order with LD. The maximum CA error is (50%) at (PR = 1), (LD = 1), and (xe/ye = 2). For the second assumed input data (constant rwD = 10–4), Table 4 shows that in xe/ye = (1) and (2), the CA errors of any PR are in reverse order with LD, while in xe/ye = (5), they are in non-order with LD. The maximum CA error is (1215%) at (PR = 0.2), (LD = 1), and (xe/ye = 1). For any assumed input data, (TDA)PSS by GPTM is noticeably close to its counterpart by PTM and both are independent of LD.

Table 3 Comparison between CA and (TDA)PSS of this study and Mutalik et al. (1988) (when rwD = 0.225 ft and ye = 933.33 ft) (variable rwD)
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Table 4 Comparison between CA and (TDA)PSS of this study and Mutalik et al. (1988) (when constant rwD = 10–4)
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The errors in the reported CA tables are also found in the cases of off-center well location in x- and y-directions for both assumed input data but are not reported here.

To determine the consequences of the identified errors in the reported CA tables of Mutalik et al. (1988) when they are used to estimate Jh, an example problem is applied. The reservoir of (ye = 933.33 ft) and (rw = 0.225 ft) is for the first assumed input data, and the reservoir of (ye = 2640 ft) and (rwD = 10–4) is for the second assumed input data, and all are at (k = 1 md), (µo = 0.5 cp), and (Bo = 1.2 rbbl/STB). The Jh is calculated by using Eq. (1), and the CA(s) by PTM and GPTM is applied by using Tables 3 and 4. For the first assumed input data, PR = (0.4), (1), and (0.8) for xe/ye = (1), (2), and (5), respectively. For the second assumed input data, PR = (0.2) and (0.8) for (xe/ye = 1), and for xe/ye = (2) and (5) they are not reported here. (OCWL = 1) and LD is from (1) to (100). Figures 3 and 4 show a comparison of those Jh(s) and CA(s) for the both assumed input data, respectively.

Fig. 3
figure 3

The Jh comparison by using the reported CA tables and this study CA tables (variable rwD)

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Fig. 4
figure 4

The Jh comparison by using the reported CA tables and this study CA tables (constant rwD = 10–4)

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Figure 3 shows that the reported CA tables of PTM lead to underestimation in the Jh for xe/ye = (1) and (2) and lead to overestimation in the Jh for xe/ye = (5). The relative error in the Jh for each selected xe/ye = (1), (2), and (5) follows the same trend as those in CA table. The maximum underestimation in Jh is (8.29%) corresponding to the error in CA at (xe/ye = 1) and (LD = 1), while the maximum overestimation in Jh is (1037%) corresponding to the error in CA at (xe/ye = 5) and (LD = 100). For both xe/ye = (1) and (2), although the maximum error in Jh (i.e., underestimation) at (LD = 1), a rather unrealistic case in most real fields, from Fig. 3 the error in Jh can still reach (6.38%) underestimation for even a realistic configuration of LD equal to (10). For (xe/ye = 5), the error in Jh (i.e., overestimation) is much more significant and starts from (46.43%) to (1037%) overestimation for LD = (10) to (100), respectively.

Figure 4 shows that the reported CA tables of PTM lead to overestimation in the Jh for both PR = (0.2) and (0.8). The relative error in the Jh for each PR follows the same trend as those in CA table. The maximum overestimation in Jh is (32%) corresponding to the error in CA at (LD = 1) and (PR = 0.2). Although the maximum error in Jh (i.e., overestimation) is at (LD = 1), a rather unrealistic case in most real fields, from Fig. 4 the error in Jh can still reach (7%) overestimation for even a realistic configuration of LD equal to (10).

The source of errors in the reported C A tables

Given that both PTM and GPTM solutions are derived identically, except re-casting some variables into dimensionless forms in the latter model, the source of the errors in the reported CA tables may be originated from the evaluation of the PDT which was not verified by Mutalik et al. (1988).

Figure 5 shows the effect of using FTLC = (0–10–9) and (0–10–5) on the CA accuracy. GPTM is used to calculate CA for square reservoir, (rWD = 10–4), (OCWL = 1), PR = (0.1), (0.5), and (1), and LD from (1) to (103). The maximum error in CA is (3630%) which appears at the lowest PR (i.e., PR = 0.1) and LD (i.e., LD = 1). Notice that the error in CA becomes small and not significant at (LD > 100) and suggests that the FTLC of (0–10–5) would be sufficient for CA accuracy when (LD > 100) for any well and reservoir configuration. This has also been shown previously in Table 1 for vertical fracture well case when (LD > 103).

The pressure drop that occurs at the very early time in the PDT solution, especially when LD is small (i.e., when the reservoir thickness is large), has significant effects on the CA accuracy. The evaluation of PDT in that period requires to use a much shorter FTLC [i.e., FTLC = (0–10–9)] to gain sufficient accuracy.

Due to the absence of the information about the selected FTLC in the calculation of the reported CA tables by PTM, CA tables have errors in reverse order with LD as shown previously for xe/ye = (1) and (2) in Tables 3 and 4. Therefore, the source of errors in the reported CA tables by PTM could be originated and justified from the using of long FTLC in the PDT evaluation. About (xe/ye = 5), no analysis can be achieved.

Additional characteristics of PTM and GPTM

Figure 6 shows the effect of OCWL in the Z-direction on CA for square reservoir, (rWD = 10–4) and PR is equal to (0.1) and (0.9). The effect of zwD (i.e., zD = zwD) on the CA that is calculated by GPTM becomes insignificant when (LD > 35) (CA% error < %5) which is inconsistent with their CA counterparts by PTM [i.e., Fig. 12 in Mutalik et al. (1988) paper] because zwD will be insignificant when (LD > 5). Therefore, zwD should be considered as a major parameter for any correlation.

Fig. 5
figure 5

Effect of FTLC = (0–10–9) and (0–10–5) on the CA accuracy

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Fig. 6
figure 6

The effect of OCWL in the Z-direction on CA

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Table 5 which is presented at the end of the paper shows the effect of the wellbore radius on CA for square reservoir and PR = (0.1) and (1). When rwD increases two times (i.e., from (5 × 10–5) to (10–4)), a significant error occurs when (LD < 15), and when rwD increases (10) to (20) times [i.e., from (5 × 10–4) to (10–3)], a significant error occurs when (LD < 50). This is also inconsistent with their CA counterparts by PTM (i.e., table (9) in Mutalik et al. (1988) paper), the significant error will occur when (LD < 3) if rwD increases 10 times. Therefore, rwD should be considered as a major parameter for any correlation.

Table 5 The effect of rwD on CA
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Conclusions

  1. 1.

    The PTM was improved to GPTM and verified by converting several dimensional parameters to dimensionless forms.

  2. 2.

    An efficient algorithm was developed, and accurate CA and Jh were obtained for any well and reservoir configuration.

  3. 3.

    The accuracy of the reported CA tables was quantified, and the consequences were determined when they were used to estimate Jh.

  4. 4.

    New CA tables and their (TDA)PSS tables were generated for a wider range of well and reservoir configurations. The tables are provided in the Supplementary Material.

  5. 5.

    Because CA depends on the all-configuration parameters, unlimited tables of CA can be generated; therefore, the MATLAB code was provided.

  6. 6.

    Although horizontal wells usually have been drilled with high LD industrially (i.e., LD > 10), it is still a significant error in Jh by (< 7%) that has been determined when using the reported CA of PTM for (xe/ye) = (1) and (2) in the both assumed input data, while for (xe/ye = 5), a significant error in Jh starts from (46.43%) to (1037%) for LD = (10) to (100), respectively, in the first assumed input data (i.e., variable rWD) and expected the same in the second assumed input data (i.e., constant rWD).