Vertical transmissibility assessment from pressure transient analysis with integration of core data and its impact on water and miscible water-alternative-gas injections

Abstract

The major key uncertainty of complex carbonate reservoirs are the vertical transmissibility across the tight dense (stylolite) layers and areal distribution of high permeability streaks (HKS), which have major impact on reservoir management, well locations, and well completion design in water and miscible water-alternative-gas (miscible WAG) injection process. The present study presents interpretation methodology of vertical transmissibility through assessment of horizontal to vertical permeability ratio (K_v/K_h) from various dynamic data. The K_v/K_h range assessment was done after integration with whole core data and pressure transient data. The impact of K_v/K_h on water and miscible WAG injection processes has also been investigated. The result shows that good vertical communication between the bulk of the porous sub-units and all across stylolite layers except one stylolite layer which acts as field wide barrier. In addition, simulation result of water and miscible WAG injection with higher order of estimated K_v/K_h ratio (0.2 to 1 as found in good permeability porous layers of most of the carbonate reservoirs) indicates no major impact on water cut (WCT), gas oil ratio (GOR), water breakthrough (WBT), gas breakthrough (GBT), and expected ultimate recovery (EUR) in homogeneous area, while oil recovery acceleration with lower WCT/GOR and slightly early WBT/GBT time in heterogeneous area due to gravity or viscous effect suppressed by heterogeneity effect. However, the lower order of K_v/K_h ratio (~ < 0.05) provide delay in WBT/GBT and lower WCT / GOR production due to viscous dominant flow which results in lower gravity-viscous number.

Appendices

Appendix-1: Analytical model to estimate K _v/K _h from horizontal well tests

The gradient intercept method given by Falade et al. (1998)can be utilized in the absence of late time radial flow regime or not fully developed of early time linear flow regime to estimate vertical permeability. Falade et al. (1998) described that the early time radial flow regime line and early time linear flow regime line intersects at a unique point in a horizontal well test. Figure 3a shows the real field well test data in which the late time radial flow was not developed in horizontal well test due to required pro-long shut-in time; therefore, K_v/K_h was calculated using intersection of early radial regime and early linear flow regime and early radial flow regime. The intersection point can be identified in the log-log diagnostic plot as shown in Fig. 3a. Therefore, the derivative of early time radial flow equation and early time linear flow equation will be equalized at this unique intersection point as described below Eq. (1):

$$ t\frac{dp}{dt}=\frac{70.6\ q\mu Bo}{L_w\sqrt{K_y{\mathrm{K}}_v}}=\left(\frac{4.064\ q Bo}{H}\sqrt{\frac{\mu }{\varnothing {C}_t{K}_y{L_w}^2}}\right)\sqrt{t} $$

(1)

The above equation simplify for vertical permeability (Kz) as below Eq. (2):

$$ {k}_v=301.8\left[\frac{\varnothing \mu {c}_t{H}^2}{t_{erf}}\right] $$

(2)

The above equation estimates the vertical permeability (K_v) using the time (t_erf) at which early time radial flow and linear flow intersect. For a horizontal well which drilled in mid-way of the reservoir at H_w distance (H = 2H_w), the above equation becomes Eq. (3):

$$ {k}_v=1207.2\left[\frac{\varnothing \mu {c}_t{H}_w^2}{t_{erf}}\right] $$

(3)

The generalized form of the above equation after consideration of Kuchuk (1995) concept (using time to felt near and far boundaries) is given by Eq. (4) as below:

$$ {k}_v=1207.2\left[\frac{\varnothing \mu {c}_t{H}^2}{t_{erf}}\right]\mathit{\operatorname{Min}}\left[{H}_{Dw}^2,{\left(1-{H}_{Dw}\right)}^2\right] $$

(4)

The above equation estimates the vertical permeability (K_v) using horizontal well dimensionless location, i.e., well eccentricity (H_DW = H_w/H) and t_erf, while H_w is a distance from no-flow boundary from horizontal length position {min (h1, h2)} and h₁ and h₂ are the distance from no flow boundaries to the horizontal well as shown in Fig. 3a.

On the other hand, the horizontal permeability can be calculated from early radial flow regime near wellbore in vertical plane which provides geometric mean of vertical and horizontal permeability ($Kbar=\sqrt{K_y{K}_v}\Big)$ using below Eq. (5):

$$ {\left(t\frac{dp}{dt}\right)}_{er}=\frac{70.6\ q\mu Bo}{L_w\sqrt{K_y{K}_v}} $$

(5)

From Eqs. 4 and 5, the K_v/K_h can be estimated from below Eq. (6):

$$ \frac{k_v}{k_h}={\left(17.099\ \frac{\varnothing {C}_t{L}_w{H}^2}{qBo}\frac{{\left(t\frac{dp}{dt}\right)}_{er}}{t_{er f}}\mathit{\operatorname{Min}}\left[{H}_{Dw}^2,{\left(1-{H}_{Dw}\right)}^2\right]\right)}^2;\left({H}_{Dw}=\frac{H_w}{H}\ \right) $$

(6)

Appendix 2: Analytical model to estimate K _v/K _h from vertical interference test

Early flow regime

The early radial flow regime is usually short and unlikely to observe in most tests because it might be masked by the tool storage effect in early time. However, this flow regime will be analyzed from normal radial flow equations to calculate horizontal permeability (K_h) using downhole shut-in tool with consideration that during this period the pressure response behaves as if the formation thickness is equal to the length of the open zone (h_p) between two straddle packers.

Spherical flow

The spherical flow is the dominant flow regime in early time with straddle packer configuration and identify by -1/2 slop from log-log plot of pressure derivative versus elapsed time as shown in Fig. 4b. This flow regime represents the spherical permeability (K_sp), i.e., geometric mean of three directional permeability as given by below Eq. (7):

$$ {K}_{sp}={\left({K}_x{K}_y{K}_v\right)}^{\frac{1}{3}}={\left({K_h}^2{K}_v\right)}^{\frac{1}{3}} $$

(7)

The flow is perfectly spherical to a well of radius r_w in an isotropic medium. Then according to Joseph and Koederitz (1985), the analysis in systems possessing simple anisotropy (unequal horizontal and vertical permeability components) can be also done without significantly affecting the radial coordinate. In that case, permeability (k) should be content and represent spherical region permeability (K_sp). Therefore, the radial flow equation can be represented by below Eq. (8):

$$ \frac{1}{r^2}\frac{d}{dr}\left({r}^2\frac{dp}{dr}\right)=\frac{\phi \mu {c}_t}{K_{sp}}\frac{dp}{dt} $$

(8)

The spherical region of singularity is called a continuous “spherical sink” can be visualized as a sphere, which corresponds physically to a wellbore. Hence, the cylindrical wellbore of radius r_w must be represented by a fictitious spherical wellbore of radius r_sw given by Eq. (9) originally suggested by Moran and Finlklea (1962):

$$ {r}_{sw}=\frac{h_p}{2\ln \left({h}_p/{r}_w\ \right)}; \mathrm{hp}\ge r\mathrm{w} $$

(9)

According to Joseph and Koederitz (1985), with above assumptions, the spherical source solution of Eq. (8) for long time can be given by below Eq. (10) in terms of dimensionless pressure derivative and dimensionless time:

$$ {P_D}^{\prime }=\frac{1}{2\surd \pi }{t_D}^{3/2} $$

(10)

Where the dimensionless pressure and dimensionless time for spherical flow suggested by Joseph and Koederitz (1985) in the field units are specified from below Eqs. (11) and (12):

$$ {P}_D=\frac{4\uppi {K}_{sp}{r}_{sw}}{q\mu}{\left(\Delta P\right)}_{sp} $$

(11)

$$ {t}_D=\frac{K_{sp}{r_{sw}}^2}{\phi \mu {c}_t{r}^2}\left({t}_{sp}\right);\kern0.5em r\ge {r}_{sw} $$

(12)

After substituting the dimensionless terms in Eq. (10) from Eqs. (11) and (12), the constant spherical permeability can be calculated from below Eq. (13):

$$ {K}_{sp}={\left(1227\frac{qB\mu}{{\left(t\ast \frac{dp}{\mathrm{d}t}\right)}_{sp}}\sqrt{\frac{\phi \mu {c}_t}{t_{sp}}}\right)}^{2/3} $$

(13)

The above equation can be reframed after taking log as below Eq. (14)

$$ \mathit{\log}{\left(t\ast \frac{dp}{dt}\right)}_{sp}=-\frac{1\ }{2\ }\log \left({t}_{sp}\right)+ constanttermaslog\left(1227\frac{qB\mu}{K_{sp}}\sqrt{\frac{\phi \mu {c}_t}{K_{sp}}}\right) $$

(14)

This expression shows that a plot of measured t*dp/dt versus time on a log-log graph will yield a straight line of slope –½ when spherical flow is dominant.

Late radial flow regime

The late radial flow will be generated after spherical flow and identify through zero slop from log-log plot of pressure derivative versus elapsed time as shown in Fig. 4b. Once the radial flow to the wellbore is established, the derivative becomes constant. This late time radial flow represents the geometric mean of horizontal directional permeability in x and y direction as shown in below eq. (15):

$$ \left({K}_h=\sqrt{K_x{K}_y}\right) $$

(15)

The normal radial flow equation applies considering the total formation thickness (h) to calculate horizontal permeability in late radial flow regime as shown in below Eq. (16) according to Tiab (1993):

$$ {K}_h=\frac{70.6\ q\mu B}{h{\left(t\ast \frac{dp}{dt}\right)}_{lr}} $$

(16)

The K_v/K_h can be calculated from Eq. (7), (13), (15), and (16) as below Eq. (17):

$$ \frac{K_v}{K_h}={\left(17.3796\frac{{\left(t\ast \frac{dp}{dt}\right)}_{lr}}{{\left(t\ast \frac{dp}{dt}\right)}_{sp}}\sqrt{\frac{\phi \mu {c}_t{h}^2}{t_{sp}}}\right)}^{2/3} $$

(17)

Where lr and spare the suffix correspond to spherical flow and late radial flow regimes. Time (t) and pressure derivative (dp/dt) can be read from any point in corresponding flow regime.

• Interpretation methodology of vertical transmissibility through assessment of permeability ratio

• The impact of permeability ratio on water and miscible WAG injection processes

• Early WBT/GBT time was observed in heterogeneous area due to gravity or viscous effect

• Lower order of K_v/K_h ratio provides delay in WBT/GBT and lower WCT/GOR production