Data-Driven approach

Data-driven approaches are founded on the causality between well production and the 4D seismic responses (Huang et al., 2012). In this technique, known information (i.e. pressure and saturation values) at well locations is calibrated to magnitudes of the 4D seismic signals around the wells, to build training samples. Correlations established with the training set are then used to estimate pressure and saturations at the well locations and also away from well locations which have not been sampled. This circumvents the need for a rock-physics model or detailed core analysis as the estimation procedure is driven by field observations. Thus, the computation is unbiased with respect to the many assumptions implicit in model-driven methods. One short- coming of the data-driven process when compared to model-driven methods that generate 3D volumetric changes in pressure and saturation, is that it has so far been applied using 2D maps. However, the quick computation time, ease and accuracy (as it is determined by measured data) with which the inversion process can be implemented, the 2D maps from the data-driven method are rather useful for timely integration and reservoir management decisions.

MacBeth et al. (2006) define the following engineering-consistent normalised linear approximation for any 4D seismic attribute ∆𝐴,

∆𝐴(𝑥, 𝑦) 𝐴̅𝑏 ≈ 𝐶𝑆 ∆𝑆_𝑜(𝑥, 𝑦) 𝑆̅𝑜𝑖 + 𝐶𝑃 ∆𝑃(𝑥, 𝑦) 𝑃̅𝑖 ( 1-13)

where (𝑥, 𝑦) is the seismic bin location; 𝑆̅_𝑜𝑖 and 𝑃̅_𝑖 are the field’s average initial oil saturation and average initial pore pressure, respectively; and 𝐴̅_𝑏 is the average baseline seismic map. The weighting coefficients 𝐶_𝑆 and 𝐶_𝑃 are assumed constant (i.e. spatially invariant), to be determined by calibration to production data. For any two or more attributes that are observed to act independently and with a different relation to pressure and saturation effects, the coefficients 𝐶_𝑆 and 𝐶_𝑃 obtained from these attributes will also be different, thus, the method can be used to invert for pressure and saturation changes across the field. For a single 4D seismic attribute, Equation 1-13 is organised into matrix form to determine 𝐶_𝑆 and 𝐶_𝑃 in a least squares inversion which uses oil saturation, ∆𝑆_𝑜 and pressure, ∆𝑃 changes measured (or estimated) at chosen wells:

[ ∆𝐴(𝑥, 𝑦)₁ 𝐴̅𝑏 ∆𝐴(𝑥, 𝑦)2 𝐴̅_𝑏 .. ∆𝐴(𝑥, 𝑦)_𝑛 𝐴̅_𝑏 ] = [ ∆𝑆_𝑜(𝑥, 𝑦)₁ 𝑆̅𝑜𝑖 ∆𝑆𝑜(𝑥, 𝑦)2 𝑆̅_𝑜𝑖 .. ∆𝑆_𝑜(𝑥, 𝑦)_𝑛 𝑆̅_𝑜𝑖 ∆𝑃(𝑥, 𝑦)1 𝑃̅_𝑖 ∆𝑃(𝑥, 𝑦)₂ 𝑃̅_𝑖 .. ∆𝑃(𝑥, 𝑦)𝑛 𝑃̅_𝑖 ] [_𝐶𝐶𝑆 𝑃] (1-14)

where, 1. . 𝑛 denote the well locations; the rest of the parameters are the same as above. In an application to the Cormorant field (Figure 1-12), Floricich (2006) enhances this approach by performing a multiattribute analysis which determines the most suitable subset from a number of 4D seismic attributes. These include but are not limited to: far amplitude, near amplitude, full amplitude, intra-reservoir time-shift (time-shifts computed in a window below the reservoir minus the time-shifts computed in a window above the reservoir) and instantaneous frequency. To do this, Equation 1-14 is applied simultaneously to different combinations of multiple attributes and the combination with the lowest cross-validation error is selected for the inversion. Principal component analysis is applied to filter out redundancy and interdependencies among the attributes, the results from which are used to determine the most representative 𝐶𝑆 and 𝐶𝑃. It was

found that AVO seismic attributes (near, mid and far) and 4D time-shifts yielded the best combination.

Figure 1-12 Application of the multiattribute analysis and Bayesian inversion using the engineering- consistent approximation, Equation 1-13, on the Cormorant field. The 4D amplitude map response (left), the inverted oil saturation change, in fraction (middle) and the inverted pressure change , in psi (right). In the furthest left image, pressure increase masks the underlying fluid saturation, but the inversion is able to separate the two. Modified after Floricich (2006).

Because the calibration or training data set from the wells are unlikely to cover the entire range of variations in rock properties and production of the reservoir, Floricich et al. (2005) tries to compensate for this by formulating the problem in a Bayesian scheme to account for uncertainties. This outputs probabilities for ∆𝑆₀ and ∆𝑃 at each seismic bin location.

Based on a case study from the Schiehallion field, Floricich et al. (2006) proved that the methodology is not restricted to a two-fluid phase system and the approximation is not required to be linear. Incorporating gas saturation changes leads to the non-linear equation for the change in seismic attribute, ∆𝐴

∆𝐴 ≈ 𝑎(𝑒𝑏∆𝑆𝑔_{− 1) + 𝑐∆𝑆}

𝑤+ 𝑑∆𝑃2+ 𝑓∆𝑃 , (1-15)

4D differences Oil saturation changes Pressure changes

Cormorant field hardening 0 0.60 0.40 0 400 800 softening 0.25 ₁₂₀₀

where ∆𝑆𝑔 , ∆𝑆𝑤 and ∆𝑃 are the changes in gas saturation, water saturation and pore

pressure, respectively. Similarly, the weighting coefficients 𝑎, 𝑏, 𝑐, 𝑑, and 𝑓 are assumed constant, and are again obtained by calibration at the wells.

Like Landrø’s (2001) approach and subsequent model-driven cases, Floricich’s (2006) data-driven method assumes that the weighting coefficients are invariant across the reservoir. However, they also note that this assumption is invalid, as reservoirs are not homogenous. Although, accounting for uncertainties by utilising a Bayesian scheme is appropriate, weighting coefficients are assumed constant with no physical meaning of their significance. However, Floricich’s (2006) approach demonstrated that they are specific to the 4D seismic attribute used in calibrating them.

For thin reservoirs, Falahat et al., (2012) reports that the principal parameters controlling mapped 4D seismic signatures are not the pressure and saturation changes per se, but these changes scaled by the corresponding thickness (or pore volume) of the reservoir volume that these effects occupy. By pore-volume scaling and based on the principle of superposition, they derive a three-fluid phase approximation in the form,

∆𝐴(∆𝑃, ∆𝑆𝑔, ∆𝑆𝑤)

≈ 𝑎ℎ_𝑝[𝜙𝑁𝑇𝐺]∆𝑃 + 𝑏ℎ𝑔[𝜙𝑁𝑇𝐺]∆𝑆𝑔

+ 𝑐ℎ_𝑤[𝜙𝑁𝑇𝐺]∆𝑆𝑤 ,

(1-16)

where the square bracketed terms refer to the effective porosity, 𝜙𝑁𝑇𝐺, averaged over the depth range affected by the particular change at each seismic bin location.Thus, the mapped seismic response is dependent on the changes averaged over the total pore volume. The thicknesses occupied by pressure, water saturation and gas saturation changes, within the reservoir interval are given by ℎ𝑝, ℎ𝑤 and ℎ𝑔, respectively. The

coefficients 𝑎, 𝑏, and 𝑐 are constants obtained around well locations by calibration using pore-volume scaled maps of pressure, (ℎ_𝑝[𝜙𝑁𝑇𝐺]∆𝑃), gas, (ℎ𝑔[𝜙𝑁𝑇𝐺]∆𝑆𝑔) and water

saturation changes (ℎ𝑤[𝜙𝑁𝑇𝐺]∆𝑆𝑤) from a history-matched simulation model.This is

similar to Equation 1-13 where the 𝐶_𝑆 - 𝐶_𝑃 coefficients now represent the combined terms in Equation 1-16 which weight the pressure, water and gas saturation changes

individually by accounting for the pore volumes. On the Schiehallion field,Falahat et al. (2012) perform a least squares inversion using a number of observed 4D seismic attribute maps (full angle stacks, gradient stacks, envelope weighted frequency and

Figure 1-13 Application of the pore-volume scaled least-squares deterministic inversion using the linear approximation in Equation 1-16, on the Schiehallion field. (a) the observed 4D seismic amplitude response (b) the inverted pore-volume scaled pressure change (c) the inverted pore-volume scaled gas saturation change (d) the inverted pore-volume scaled water saturation change. Highlighted areas are for comparison, modified after Falahat et al. (2012).

0 hardening softening 0 min max 0 max 0 max 1 km 1 km 1 km 1 km

(a)

(b)

(c)

(d)

reservoir thickness in time). Since the reservoir’s pore-volume is not known exactly, the results do not yield absolute values of pressure and saturation but a scaled alternative (Figure 1-13), which they show can be semi-quantitatively compared with corresponding pore-volume scaled maps from the simulation model, assuming the simulation model is correct. Landa et al. (2015) also apply Falahat et al.’s (2012) approach in a Bayesian inversion scheme on an offshore turbidite field undergoing water-flood production.

To provide further insights into how pore pressure and saturation changes may combine to affect the 4D seismic signatures, Alvarez and MacBeth (2014) investigated the physical reservoir properties that may control the weighing coefficients presented in MacBeth et al. (2006). By performing extensive sensitivity tests using rock-physics forward modelling, a slight modification to Equation 1-13 expressed as absolute differences, gives

∆𝐴(𝑥, 𝑦, 𝜃) = 𝐶𝑠(𝑥, 𝑦, 𝜃)∆𝑆𝑤(𝑥, 𝑦) − 𝐶𝑃(𝑥, 𝑦, 𝜃)∆𝑃(𝑥, 𝑦) (1-17)

and,

∆𝑡(𝑥, 𝑦, 𝜃) = 𝑡_𝑖( 𝐶𝑠(𝑥, 𝑦, 𝜃)∆𝑆𝑤(𝑥, 𝑦) − 𝐶𝑃(𝑥, 𝑦, 𝜃)∆𝑃(𝑥, 𝑦)) (1-18)

where (𝑥, 𝑦) is the seismic bin location; ∆𝐴 and ∆t are the 4D seismic amplitude and 4D seismic time-shift at a given incidence angle, 𝜃, respectively; and 𝑡_𝑖 is the initial two-way-time of the reservoir surface. The 4D seismic data could be angle/offset stacks (i.e. AVO stacks) or CMP gathers (i.e. AVO gathers). Note that the sensitivity coefficients/ parameters, 𝐶_𝑆 and 𝐶_𝑃 are different for amplitudes and time-shifts, and are also dependent on the AVO data from which they are derived.

Alvarez and MacBeth (2014) confirm what is suspected, that the weighting coefficients, 𝐶_𝑠 and 𝐶_𝑃 are indeed related to the in-situ reservoir rock and fluid properties, with effective porosity being a major controlling factor on the magnitudes of the spatially varying 𝐶𝑠 and 𝐶𝑃 (Figure 1-14). Other reservoir parameters (initial

reservoir pressure and saturation, top reservoir contrast, etc.) were found to have minimal influence and acted as groups of parameters rather than as individual contributors to form the 4D seismic signal. In a synthetic application using the

Schiehallion field model, Alvarez (2014) first determines a single 𝐶_𝑠 and 𝐶_𝑃 for each AVO amplitude stack (near, mid and far) by calibration to production data, and then accounts for their lateral variability by weighting using maps of porosity and overburden stress computed from the field’s fluid-flow simulation model.

Figure 1-14 Rock-physics modelling insights for an oil-water sandstone reservoir. Effects of pressure increase and water saturation increase compete against each other. For a range of porosity, 𝜙 , cross- plots are given for (a) amplitude change, ∆A for the near offset stack (angle, ϴ = 0° - 10°) versus pressure change, ∆P (at a fixed water saturation value of 0.6) (b) the same as in (a) but for the far offset stack (angle, ϴ = 25° - 35°). The weighting coefficients 𝐶_𝑆 and 𝐶_𝑃 depend on the reservoir’s geology, indicated here by porosity, which are intrinsic to the sensitivity of the offset dependent amplitude changes to pressure and water saturation effects. Pressure increase effects are stronger in the near offset and weaker in the far offset which is the opposite for water saturation effects, and the 𝐶_𝑆⁄ 𝐶_𝑃 ratio, also change according to offset and porosity. As the effects of pressure and water saturation increases compete, the horizontal black indicates the magnitudes of pressure changes that will need to occur, to override water saturation changes and how porosity affects this margin at the near and far offset. Modified after Alvarez and MacBeth (2014).

Saturation dominates Pressure dominates Porosity NEAR Saturation dominates Pressure dominates Porosity FAR

(a)

(b)