Hemi-Ellipsoidal Trap Filling

The domed shape of the topographic trap for Layer 8 suggests that it can be approximated as the cap of a hemi-ellipsoid (Figure 6.3). The benefit of approximating the trap as a three-dimensional geometric shape is that a simple mathematical relationship can be found between the volume of fluid filling the cap and the planform area of the fluid layer’s base.

Boait et al. (2012) used this relationship to estimate the volume of CO2 in all layers at the Sleipner field. In their study, this model generally produced a poor fit to the planform area with time. However, the change in planform area with time has been reassessed in this study, and also includes information from the 2010 survey. The new measurements for Layer 8 suggests that this model might be able to provide insight into the filling of this layer.

Boait et al. (2012) used the planform area of Layer 8 to estimate the eccentricity of the ellipse. The height of the dome was assumed to be 20 m based on approximate thicknesses of the sand layer for each CO₂ horizon (10–40 m). In this study, the topography of the Layer 8 caprock is used to find the best-fitting hemi-ellipsoidal trap. Using this simplified geometry, the volume of fluid required to match the observed areal extent can be constrained.

The equation for a hemi-ellipsoidal trap is given by [(x − x₀) cos θ + (y − y₀) sin θ]²

a² + [(x − x₀) cos θ − (y − y₀) sin θ]²

b² +(z − z₀)²

c² = 1, (6.1) for positive x, y and z, where x₀, y₀ and z₀ refer to the location of the ellipsoid’s centre in space, a, b and c are the size of the principal axes and θ refers to the rotation of the ellipsoid about the z axis. By fitting a, b, c, x₀, y₀, z₀ and θ to four cross sections through the topography extracted from the seismic images, a hemi-ellipsoid that approximates the topography of the Layer 8 trap can be constructed (Figure 6.3b). Example cross-sections through the topography and best-fit hemi-ellipsoidal trap are shown in Figure 6.3c-d. For Layer 8, the best-fitting principal axes are a = 775 ± 15 m, b = 1000 ± 15 m and c = 22 ± 1 m.

Figure 6.3: Topography of Layer 8 caprock. (a) Topography of caprock extracted from seismic reflection images. Red lines = transects used to fit hemi-ellipsoid. (b) Hemi-ellipsoid used to approximate topography of Layer 8 caprock. (c) Cross section through trap from X to X⁰. (d) Cross section through trap from Y to Y⁰. Black line = caprock topography. Red line = hemi-ellipsoidal trap.

Following the analysis of Boait et al. (2012), if the cap of this ellipsoid is filled with fluid to

The volume of fluid, V , trapped in the cap of this ellipsoid is given by integrating (6.2) in the vertical direction between 0 and h,

V (h) = πφS_CO2abc h²

c² − h³ 3c³



, (6.3)

where φ is the porosity and S_CO2 is the CO₂ saturation within the reservoir. If volume is approximated as V = C(t − t₀)ⁿ, where t₀ is the initiation time of layer filling and C and n are constants, then the relationship between the layer’s basal area and time is given by

t = t₀+ πφS_CO2abc

Figure 6.4 shows planform area against time for Layer 8. Fitting (6.4) to the data with two free parameters, initiation time, t₀, and flux, C, with n = 1 (i.e. constant input flux) yields a reasonable fit to the data, with t₀ = 1999.3 ± 0.1 and C = 1.2 ± 0.1 × 10⁵ m³ yr⁻¹ (Figure 6.4). This fit to the area at early and late times is inhibited by the constant input flux constraint. However, this model fits the planform area within its estimated uncertainty.

The fit exhibited by this model suggests an improvement could be made by allowing the flux to vary through time.

Allowing the flux to vary as a function of time produces an excellent fit to the change in area with time, where t₀ = 1998.7 ± 0.3, C = 5 ± 1.7 × 10⁴ m³ yr⁻ⁿ and n = 1.4 ± 0.1 (Figure 6.4). Permitting a small flux at early times allows the initial area growth to be slower, while at late times the area growth can increase faster than for the constant flux case.

Figure 6.4: Layer 8 area against time. Black line = constant flux model. Red line = increasing flux model.

Due to the resolution of the seismic reflection surveys, the observed plume edge occurs when the CO₂ layer is ∼0.75 m thick (Bickle et al., 2007). The assumption of a flat base on these layers means that the volume estimated using these areal measurements is likely to be an underestimate. To address the issue of resolution at the plume edge, a layer of CO₂ 0.75 m thick is added to the estimated base of Layer 8 at each time step. This correction adds a volume of CO₂ equal to 0.75φS_CO2A to the total volume estimate for each year. While this correction does not account for the extreme edges of the plume, these regions are not expected to contribute significantly to the overall volume. The volume of CO₂ in Layer 8 calculated from these two models with the added correction is shown in Figure 6.5. Both of these methods show approximately similar trends, and suggest that by 2010 the volume of CO₂ in Layer 8 is ∼ 1.8 × 10⁶ m³.

This method exploits a simple geometric relationship between volume and area for an el-lipsoid. The assumptions involved in the method, such as the shape of the trap being approximately hemi-ellipsoidal and the CO₂ having a flat base, mean that this model is only appropriate for very specific cases. For example, these assumptions would clearly be invalid

Figure 6.5: Layer 8 volume against time for hemi-ellipsoidal model. Blue line = constant flux model. Red line = increasing flux. Shaded areas = estimated uncertainty from

±0.25 m resolution of the edge of the layer.

for Layer 9 based on the results of Chapters 3 and 4, where CO2 does not appear to fill a simple trap. The uncertainties involved in this method are significant, the largest of which come from the estimate of trap shape. First, conversion of the topography extracted from the seismic images from two-way travel time into depth is difficult due to uncertainties in the velocity of seismic waves through the overlying lithologies. Secondly, the topography’s complex shape extracted from the seismic images cannot exactly be described as a hemi-ellipsoid. A third complication is that the input location of CO2 into this layer is not at the centre of the hemi-ellipsoid. The off-centre location of the migration point means that in the early years, the CO2 in Layer 8 will not simply be ponding beneath a structural trap.

However, as the layer grows, the uncertainty introduced by this approximation decreases as the CO2 layer edge encompasses the input location.

Despite these uncertainties, hemi-ellipsoidal trap filling provides a straightforward and simple way to calculate the volume of CO₂ within the layer without the need for complicated numerical modelling. While uncertainties in this method may be large, this approximation is likely to be an underestimate due to the assumption that the CO₂ layer has a flat base.