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Porous-Medium Scaling of CO$_2$ Plume Footprint Growth

Fernando Alonso-Marroquin, Christian Tantardini

Abstract

Building on porous-medium-type nonlinear diffusion, we compare analytical Barenblatt-type similarity solutions with plume's radii from digital analysis of published seismic monitoring images, to quantify field-scale CO$_2$ plume-footprint growth. Using an area-based equivalent radius extracted from time-lapse plume maps at Sleipner, Aquistore, and Weyburn--Midale, we obtain effective plume-growth exponents that are broadly compatible with slow porous-medium scaling in axisymmetric geometry. We then interpret the plume as a vertically segregated CO$_2$ layer of thickness $b(r,t)$ within an aquifer of thickness $H$, and derive closed-form expressions for the normalized thickness $b(r,t)/H$, the compact-support plume edge $R(t)$, and a transient inner core radius $a(t)$ that marks the region where the plume occupies the full aquifer thickness. In the shut-in case, the core radius decreases with time and eventually vanishes, after which the plume recovers the pure Barenblatt regime; under constant injection, the model predicts an injection-controlled core and a plume edge that grows with the square-root law. This framework provides a physically transparent baseline for comparing plume-radius evolution, internal plume structure, and core development across sites, and establishes a consistent route for incorporating non-local effects by fractional derivatives in future extensions.

Porous-Medium Scaling of CO$_2$ Plume Footprint Growth

Abstract

Building on porous-medium-type nonlinear diffusion, we compare analytical Barenblatt-type similarity solutions with plume's radii from digital analysis of published seismic monitoring images, to quantify field-scale CO plume-footprint growth. Using an area-based equivalent radius extracted from time-lapse plume maps at Sleipner, Aquistore, and Weyburn--Midale, we obtain effective plume-growth exponents that are broadly compatible with slow porous-medium scaling in axisymmetric geometry. We then interpret the plume as a vertically segregated CO layer of thickness within an aquifer of thickness , and derive closed-form expressions for the normalized thickness , the compact-support plume edge , and a transient inner core radius that marks the region where the plume occupies the full aquifer thickness. In the shut-in case, the core radius decreases with time and eventually vanishes, after which the plume recovers the pure Barenblatt regime; under constant injection, the model predicts an injection-controlled core and a plume edge that grows with the square-root law. This framework provides a physically transparent baseline for comparing plume-radius evolution, internal plume structure, and core development across sites, and establishes a consistent route for incorporating non-local effects by fractional derivatives in future extensions.

Paper Structure

This paper contains 12 sections, 86 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Governing density equation for a condensable gas: from Global Buckley--Leverett to q-PME
  4. Nonlinear constitutive closure and the resulting q-PME
  5. Self-Similar Barenblatt Solution in d Dimensions for the q-PME
  6. Vertically averaged sharp-interface reduction and plume-thickness profile
  7. Composite plume profile with a transient full-thickness core
  8. Results
  9. Evolution of finite-core plume profiles under constant injection and after shut-in
  10. Field evidence for CO2 plume-radius scaling R_eq(t) proportional to t^beta
  11. Conclusions
  12. Direct reduction of the GBL transport equation to the axisymmetric q-PME

Figures (3)

  • Figure 1: Normalized cross-section of the composite plume profile. The shaded upper region represents the injected CO$_2$, while the lower region represents brine. The dashed vertical line marks the radius $a(t)$ of the inner full-thickness core, where $b(r,t)=H$, and the solid vertical line marks the plume edge $R(t)$, where the plume thickness vanishes. For $a(t)>0$, the profile consists of a full-thickness core joined continuously to a Barenblatt tail.
  • Figure 2: Normalized plume-thickness profiles $b(r,t)/H$ for the finite-core model. (a) Constant injection: the plume develops an inner region of full thickness, $b=H$ for $r\le a(t)$, followed by a Barenblatt-type outer tail that vanishes at the plume edge $r=R(t)$. As time increases, both the core radius $a(t)$ and the plume edge $R(t)$ move outward. (b) Injection paused: after shut-in, the inner full-thickness core shrinks with time, while the outer plume continues to spread. Once $a(t)$ reaches zero, the finite core disappears and the solution crosses over to the pure Barenblatt regime.
  • Figure 3: Equivalent plume radius $R_{\mathrm{eq}}(t)$ obtained by digitizing published plan-view monitoring images from Sleipner, Aquistore, and Weyburn--Midale. Symbols denote extracted values of $R_{\mathrm{eq}}(t)$, and lines denote least-squares power-law fits of the form $R_{\mathrm{eq}}(t)=A\,t^\beta$. The data are shown on log--log axes to emphasize scaling behavior.