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A Simple Scaling Approach to the Spontaneous Clearing Time of Water Block

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Abstract

Hydraulic fracturing causes water to invade reservoir rock. This invaded water is called a water block as it lowers the effective permeability of the rock to oil. Field and laboratory results show that the water block is temporary. We hypothesize that (a) the clearing is caused by capillary-dominated advection of water, and (b) the clearing takes place when the water saturation drops at the fracture face. We test this hypothesis with laboratory experiments, analytic solutions, and numerical simulations to a capillary-driven model. In particular, we determine how the clearing time scales with the permeability of the rock and the amount of water invaded, and this scaling matches experimental observations. From this match, we develop an equation that estimates clearing time and test how the leading coefficient of this equation depends on rock constitutive parameters (e.g., capillary pressure, relative permeability).

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Availability of data and materials

The data sets collected in this study are available from the corresponding author on request.

Abbreviations

\(\alpha\) :

The scaling coefficient for the water block clearing time, 1 [min/cm] = 3.2*105 (\(\mathrm {min}\sqrt{\mathrm {mD}}/\mathrm {cm}^2\))

\(\lambda\) :

The pore-size-distribution parameter in the pressure-saturation curve, (dimensionless)

\(\lambda _o\) :

The oil mobility, (\(\frac{1}{\mathrm {Pa}\,\mathrm {s}}\))

\(\lambda _t\) :

The total mobility, (\({\frac{1}{\mathrm {Pa}\,\mathrm {s}}}\))

\(\lambda _w\) :

The water mobility, (\({\frac{1}{\mathrm {Pa}\,\mathrm {s}}}\))

\(\phi\) :

The porosity of the rock, (\(\mathrm {m}^3/\mathrm {m}^3\))

\(\sigma _{ow}\) :

The water/oil interfacial tension, (dynes/cm)

\(\tau\) :

The dimensionless temporal variable, (dimensionless)

\(\tau _w\) :

The dimensionless clearing time of the water block, (min/min)

\(\xi\) :

The dimensionless spatial variable, (dimensionless)

A :

The cross-sectional area, (\({\mathrm {cm}^ 2}\))

\(D^*\) :

The dimensionless part of the capillary dispersion term, (dimensionless)

\(D_0\) :

The scaling part of the capillary dispersion term, (\({\mathrm {m}}^2/{\mathrm {s}}\))

\(f_w\) :

The water fractional flow, (\({\frac{\mathrm{m}^3/\mathrm{s}}{\mathrm{m}^3/\mathrm{s}}}\))

J :

The Leverett J-function, (psi/psi)

\(J_0\) :

The coefficient in the Leverett J-function, (psi/psi)

K :

The absolute permeability, (mD)

\(k_{ro}\) :

The relative permeability of oil, (mD/mD)

\(k_{ro}^{\circ }\) :

The end-point relative permeability of oil, (mD/mD)

\(k_{rw}\) :

The relative permeability of water, (mD/mD)

\(k_{rw}^{\circ }\) :

The end-point relative permeability of water, (mD/mD)

\(L_w\) :

The superficial invasion length, (cm)

\(n_{o}\) :

The Corey exponent for oil relative permeability, (dimensionless)

\(n_{w}\) :

The Corey exponent for water relative permeability, (dimensionless)

\(P_c\) :

The water/oil capillary pressure, (psi)

\(S_w\) :

The water saturation, (\({\mathrm {m}^3/\mathrm {m}^3}\))

\(S_{or}\) :

The residual oil saturation, (\({\mathrm {m}^3/\mathrm {m}^3}\))

\(S_{w0}\) :

The initial water saturation, (\({\mathrm {m}^3/\mathrm {m}^3}\))

\(S_{wj}\) :

The average water saturation in the invaded region, (\({\mathrm {m}^3/\mathrm {m}^3}\))

\(S_{wr}\) :

The residual water saturation, (\({\mathrm {m}^3/\mathrm {m}^3}\))

t :

The independent temporal variable, (s)

\(t_w\) :

The clearing time of the water block, (min)

\(u_t\) :

The total volumetric flux, (\(\mathrm {m}^3/\mathrm{s}\))

\(u_w\) :

The water volumetric flux, (\(\mathrm {m}^3/\mathrm{s}\))

\(V_{inv}\) :

The volume of water invasion, (mL)

x :

The independent spatial variable, (cm)

\(x_{inv}\) :

The real water invasion length, (cm)

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Funding

This work was funded by Equinor Gulf of Mexico LLC through the Equinor Fellowship Program at The University of Texas at Austin. We would also like to acknowledge the support from the Shell-UT Unconventional Research (SUTUR) and Foundation CMG.

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Authors and Affiliations

  1. Hildebrand Department of Petroleum and Geosystems Engineering, The University of Texas at Austin, Austin, USA

    Xiao Luo, Quoc Nguyen & David DiCarlo

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  1. Xiao Luo
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Contributions

XL contributed to conceptualization, methodology, investigation, writing original draft, visualization. QN contributed to resources, supervision, writing—review and editing, funding acquisition. DD contributed to resources, writing—review and editing, supervision, funding acquisition.

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Correspondence to David DiCarlo.

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Luo, X., Nguyen, Q. & DiCarlo, D. A Simple Scaling Approach to the Spontaneous Clearing Time of Water Block. Transp Porous Med 137, 1–19 (2021). https://doi.org/10.1007/s11242-020-01529-3

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