# Four-Dimension Seismic Analysis in Carbonate: A Closed Loop Study
## 2 Materials and Methods
### 1D Well-Based Time-Lapse Feasibility Study  
The petro-elastic model was built (Figure 1a) with the following main steps:  
* Mix mineral properties. * Add pores using effective media theories. * Calculate fluid properties. * Perform fluid substitution. Fluids are introduced into pore space using the equations of Gassmann [24], using the results of the last three steps. Gassmann's relations are often used to predict the modulus of saturated rock from the modulus of dry rock, and vice versa. The most common in situ problem is to predict the changes that occur when one fluid is replaced by another that has both mineral and dry bulk moduli and fluid properties. Here, we used the normal Gassmann equation to introduce the effective fluid into the empty pores included in the matrix. The Gassmann substitution of fluid was used to calculate the effective saturated properties (Figure 1b). The research activities were divided into three phases. The first phase involved a well-based (1D) time-lapse model using a production well with gassy carbonate at the crest of the structure and an injection well at the flank of the structure. The 4D modeling covered a range of possible production scenarios expected in the field, including changes in water saturation due to water injection and changes in gas saturation due to pressure changes. Feasibility of the 4D approached was investigated by running an appropriate petro-elastic model and a 1D time-lapse with production and injection data to investigate the detectability of 4D signals and determine how these could relate the modeling to actual 4D seismic. Petrophysical analysis confirmed that two minerals should be used for the reservoir to model the matrix. The elastic properties used for these minerals were standard values, but they were also derived from data analysis during petrophysical interpretation, as was their volume fraction. In the area of the reservoir under study, we determined the average porosity by experimenting with statistics from several wells within the main reservoir zone. The overall average porosity was about 15 PU, with the K-frame being relatively loose because the reservoir zone is a mixture of dolomite and calcite. These elastic values are listed in Table 1. In addition, this study used the Voigt-Reuss-Hill model [26], to mix these minerals:  
The reservoir fluids and the dry rock elastic properties, computed from the PEM, were combined in the [PERSON]'s (1951), equation to calculate the saturated elastic moduli at different reservoir conditions:  
* In-situ (pre-production). * Water displacing gas to different saturations: 10, 50 and 85%. * CO\({}_{2}\) displacing gas to different saturations: 10, 50 and 85%. The expected 4D field impacts were evaluated by a well-based (1D) fluid substitution feasibility study. This 1D model tested three possible production scenarios that could be expected in the field. Changes in water saturation due to water injection and three scenarios \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mineral** & **Density (g/cc)** & **Bulk Modulus (GPa)** & **Shear Modulus (GPa)** \\ \hline Calcite & 2.71 & 69.3 & 32 \\ Dolomite & 2.87 & 74.5 & 35.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Reservoir elastic properties. Figure 1: (**a**) Physics model for carbonate rocks after [25], and (**b**) fluid substitution saturated Properties after [24]. of gas saturation in this gas-bearing reservoir due to pressure changes were modeled. The predicted 4D response was evaluated using synthetic seismic angles between zero and 45 degrees obtained from P-wave, S-wave, and density logs from a production well and an injection well. A 30 Hz Ricker wavelet was fitted to generate the synthetic gathers corresponding to the dominant frequency of the reservoir-level seismic data. In the gas-producing well, density was reduced at 10% water flooding compared to 50% and 85% water flooding, while the change in VP/VS was very small compared to the change in density.