Time-Dependant Responses of High-Definition Induction Log and Case Studies

The process of drilling mud filtrate invading into a reservoir is time dependant. It causes dynamic invasion profiles of formation parameters such as water saturation, salinity, and formation resistivity. Thus, the responses of a high-definition induction log (HDIL) tool are time dependent. The logging time should be considered as an important parameter during logging interpretation for the purposes of determining true formation resistivity, estimating initial water saturation, and evaluating a reservoir. The timedependentHDIL responses are helpful for log analysts to understand the invasion process physically. Field examples were illustrated for the application of present method.


Introduction
Resistivity log is an effective method to estimate the critical properties of a reservoir in petroleum exploration. Induction log is an important tool to measure the formation resistivity. In addition to traditional dual-induction log tools [1], new array type induction devices [2], such as array induction tool (AIT) and high-definition induction log (HDIL), can provide more messages. The AIT device was designed with 10-, 20-, 30-, 60-, and 90-inch (25, 51, 76, 152, and 229 cm) depths of investigation, and the HDIL tool provides six depths of investigation, 10, 20, 30, 60, 90, and 120 inches (25, 51, 76, 152, 229, and 305 cm), respectively. The HDIL instrument is a typical multiarray induction logging tool that measures the formation resistivity simultaneously with six arrays.
However, the resistivity measurement is affected by invasion process strongly. When a reservoir was opened, drilling mud filtrate poured into permeable and porous formations; thus, an invaded zone was formed. The behaviors of the invaded zone are very different from the original formation. The physical understanding of an invasion process is essential for logging interpretation and reservoir evaluation. Conventional invasion model is the static step-invasion profile [3] that presumes that the resistivity varies sharply at the boundary between invaded zone and uncontaminated formation. In fact, the displacement of moveable native fluids in a reservoir by mud filtrate is a percolation process. The formation-and fluid-related parameters do not vary in a stepinvasion style. In addition, a realistic invasion process is time dependant. At the beginning of bit penetration, the rate of invasion is rapid. With the lapse of time, mud cake is built at the wall of borehole. With the building of mud cake and the extension of invading geometry area, the invasion rate decreased.
The dynamic invasion process has a severe impact on logging tools. Logging at various time stages will lead to different measurement results, that is, time-dependant resistivity responses. The dynamic invasion process and its influences on resistivity-logging tools have been investigated internationally for traditional resistivity log tools [4][5][6] and new AIT devices [7,8]. The present work focused on the influence of dynamic invasion process on HDIL measurements.
The theoretical model of dynamic invasion and the geometry factor theory for resistivity log were used to calculate the time-dependant responses of a HDIL device. The present results can give a reasonable interpretation for the difference of resistivity responses at various measurement times. In addition, comparing the dynamic model with static step-invasion style, the dynamic invasion profiles and timedependant logging responses can provide more information for log analysts to evaluate a reservoir and to understand the complexity of the invasion process.
The present model was used in a field located at west of China. Examples of field applications were illustrated to determine the true-formation resistivity and initial saturation and furthermore to evaluate a reservoir.

Dynamic Invasion Model
The time-dependent resistivity-logging model is based on dynamic invasion theory. During drilling, mud filtrate pours into the formation radially and displaces the native fluids in porous volumes under the pressure differential between borehole and formation. We supposed that the reservoir is thick and the displacement between filtrate and hydrocarbon is immiscible.
The invasion process is also related to formation parameters such as absolute permeability , formation porosity , native-water salinity , mud-filtrate salinity mf , relative permeability and relative permeability , original saturation and original saturation , formation pressure and formation pressure , fluid viscosity and fluid viscosity in formation, and fluid flow capacity and fluid flow capacity . We used subscript for oil phase and for water phase, respectively.
At the beginning of invasion, mud cake is built up promptly at the wall of borehole. With the extension of invading geometry area and the building of mud cake, the invasion rate slows down with time. The dynamic invasion process satisfied fluid flow equations [9] as in the following steps.
First, both saturation and pressure in reservoir are functions of invasion time and radial distance from the borehole. They can be solved from flow equations: In order to solve (1), the relationships + = 1 and − = were used. is capillary pressure. It can be obtained from special core analysis.
Second, since the mud salinity mf is different from the salinity of water in formation, the mixing of mud filtrate with native water causes the variation of salinity distribution with time during invasion process. The variation of water salinity is generated from the following dispersion equation after and were obtained from (1): Third, the petrophysics formulas [4,5] were used to generate the water resistivity and formation resistivity They are functions of both and . They can be determined from above-solved saturation and salinity.
where is the temperature ( ∘ C) in the reservoir, both and are constants, is the cementation factor, and is the saturation exponent.

Time-Dependent Responses of HDIL Devices
Since the radial distribution of formation resistivity varies with time, the apparent resistivity of HDIL measurement at time is determined by resistivity invasion profile ( , ).
According to the theory of geometry factor for induction log [1], the apparent resistivity of a HDIL device is where ( ) is radial differential geometrical factor. The apparent resistivity readings of a HDIL device, , vary with measurement time . The variation is the physical basis of time-dependant HDIL measurement.
A HDIL device is composed of seven three-coil arrays at 8 operating frequencies (10,30,50,70,90,110,130, and 150 kHz, resp.) [2]. It provides six apparent resistivities, which are usually marked as HDIL10, HDIL20, HDIL30, HDIL60, HDIL90, and HDIL120. They correspond to six detective depths: 10, 20, 30, 60, 90, and 120 inches (25, 51, 76, 152, 229, and 305 cm), respectively. Figure 1 illustrated the radial differential geometrical factor for each array of a HDIL device. The region of peak makes the greatest contribution to apparent resistivity according to (5). Therefore, the HDIL10 array is influenced by Chinese Journal of Engineering 3 the vicinity of borehole strongly and has the shallowest detective depth. The HDIL120 array has maximum investigation depth; hence, it can reflect the character of resistivity far away from the borehole. The detective depth of a HDIL device will increase according to the arrays HDIL10, HDIL20, HDIL30, HDIL60, HDIL90, and HDIL120, as shown in Figure 1.

Time-Dependant HDIL Measurements and Their Applications
The present model of dynamic invasion effects on timedependant HDIL measurement was applied in a reservoir located at west of China.

An
Oil-Bearing Reservoir. The bed was logged at 25 days after the formation was opened. The field logging data for HDIL10, HDIL20, HDIL30, HDIL60, HDIL90, and HDIL120 were 10.0, 10.5, 12.7, 18.5, 22.0, and 28.0 ohm⋅m, respectively. History-matching method [4,5] was used to simulate the time-dependant responses of HDIL measurement. The simulation began with an initial estimation of water saturation . In order to reach the best matching with well logging data, the initial water saturation was set to be 0.  Figures 2 and 3 showed the present calculation for radial profiles of water saturation from (1) and water salinity from (2), respectively. The horizontal axis denoted the distance from borehole. Since the diameter of borehole in present simulation was 8 inches, the radial invasion curves were drawn from the wall of borehole.
Since the well was filled with mud during drilling, the water saturation in borehole was high. In Figure 2, the line marked with = 0 means the initial water saturation before invasion occurs ; it corresponds to the value of , 0.32. At the beginning of mud invasion, the well was filled with mud filtrate. After the formation was opened, the filtrate began to pour into the reservoir, so that the water saturation was high at the adjacency of borehole. Figure 2 showed the radial invasion profiles of water saturation at 4, 10, 23, and 50 days, respectively, after the mud filtrate poured into the formation.
In the present case study, the initial water salinity was 40000 mg/L, which was greater than mud salinity of 26700 mg/L, so that the salinity profile near the wellbore was low. With the lapse of time, the low-salinity filtrate invaded into the reservoir so that the radial profile of water salinity changed, as shown in Figure 3.
The changes of both saturation and salinity lead to the alternations of water resistivity and formation resistivity. According to (3), the water resistivity was determined by water salinity. The low-salinity fresh filtrate near the borehole leads to the high water resistivity, as shown in Figure 4. With the lapse of time, this high water-resistivity profile moved into the deeper formation.    Formation resistivity was obtained from (4). It was determined both by time-dependant water saturation and by water resistivity . Hence, the profiles of formation resistivity changed with time as well. They were drawn in Figure 5 for four time stages, 4, 10, 23, and 50 days, in the invasion process.
High water saturation and high water resistivity in the vicinity of borehole caused a low-resistivity range, in which the physical characters were very different from the original formation. After 20 days of the mud invading, the invasion depth reached one meter ( Figure 5). The resistivity distribution in formation does not vary sharply at the boundary between the invaded zone and noninvaded area.
The change of with time leads to the time-dependant responses of the HDIL measurement. The lines in Figure 6 illustrated the calculation results of various HDIL readings with time using (5). The horizontal axis denoted the logging time after the formation was immersed and the vertical axis   was the apparent resistivity of HDIL measurement. The dots in Figure 6 were the logging data. After the formation was opened, the invasion zone moved into the reservoir gradually. When the low-resistivity invasion zone in Figure 5 reached the detective depth of the HDIL array, the logging readings decreased according to (5). Since HDIL10 has the shallowest detective depth and HDIL120 has the deepest investigation depth, the HDIL10 array reached its minimum value and HDIL120 array got its maximum reading. For the present field example, the result, HDIL120 > HDIL90 > HDIL60 > HDIL30 > HDIL20 > HDIL10, was obtained.
The present simulation suggested the true-formation resistivity is equal to 36.9 ohm⋅m, which just corresponded to the value before invasion occurrence, that is, the apparent resistivity at = 0 in Figure 6. It also reflected the initial water saturation setting, = 0.32, according to (4). The low value of initial water saturation illustrated the behaviors of an oilbearing reservoir with high oil saturation, = 1 − = 0.68, for the present field application. For this case application, all of the field HDIL readings were less than the true resistivity, because of the influence of low-resistivity invasion zone, as shown in Figure 5. Therefore, if the logging data were used to estimate the trueformation resistivity directly, error would be introduced. The calculation results of history-matching were also helpful for log analysts to understand the characters of dynamic invasion process physically.

A Water Zone.
A water-bearing formation shows different characters. Figure 7 illustrated a simulation result of field example for a water zone. This bed was located in the same well, with the reservoir simulated in Figures 2∼ 6. The measurement results for HDIL10, HDIL20, HDIL30, HDIL60, HDIL90, and HDIL120 were 3.52, 3.30, 3.25, 3.18, 3.17, and 3.19 ohm⋅m, respectively, after the formation was invaded for 30 days.
During the history-matching, the main input data were the same as in Figures 2∼6 for the same well. The timedependant HDIL responses were illustrated using lines and the logging data were marked by dots, respectively, in Figure 7.
The present simulation suggested that the initial water saturation was equal to 0.8, and the true-formation resistivity was 3.26 ohm⋅m, which corresponded to the apparent resistivity at = 0 in Figure 7, that is, before invasion occurrence. The present history-matching suggested the reservoir was a water zone with high water saturation, 0.8.
If a reservoir is filled by movable water, it has high water saturation. When native saline water with low resistivity was replaced by fresh mud filtrate with high resistivity, a highresistivity invasion zone formed near to the borehole. It influenced the responses of shallow detective HDIL arrays much more, so that the apparent resistivities of shallow detective arrays were usually greater than the values measured from the arrays of deeper investigation depths after the formation was immersed for a long time period, as shown in Figure 7.

Conclusions
There is not a sharp boundary between invaded zone and uncontaminated formation. The invasion profiles of formation parameters, such as water saturation, salinity, and resistivity, show complex shapes. It is important that the process of drilling mud filtrate invading into a reservoir is time related, so that the invasion profiles vary with time as well. This dynamic invasion style is very different from the traditional static step-invasion model. The profiles of formation parameters vary with time. It leads to that the apparent resistivities from the measurement of a high-definition induction log device are time-dependant . The different logging data measured at various time stages will cause confused log interpretations, so that errors or incorrect evaluation would be introduced.
The logging time is suggested to be recorded as a parameter in order to get a reasonable interpretation for HDIL measurements. The time-dependant apparent resistivities of a HDIL device can be obtained using the dynamic invasion model and resistivity-logging theory.
The history-matching method can be used to obtain the true-formation resistivity and initial fluid saturation in a reservoir from the present simulation of time-dependant HDIL measurement. Site applications in a well located at west of China proved the validity of the present method in log interpretation and reservoir evaluation.
Time-dependant log is a useful technique in geophysical exploration. It has successful applications such as rock physics model describing [10] and cross-hole electrical resistivity tomography monitoring [11].