Abstract
Effective permeability is a key parameter for evaluating reservoirs and their productivity. With the wide application of resistivity logging tools in drilling, the advantages of resistivity logging in response to the dynamic invasion process of drilling fluids have become increasingly prominent. We analyzed the variation law of the measured resistivity data for different permeability formations at different times. In this study, we proposed an effective permeability modeling method based on the time-lapse resistivity logging data. First, based on the resistivity measurement data at different times, the dynamic resistivity profile of the reservoir was obtained through joint inversion; we then obtained the invasion depth and invasion zone resistivity of the drilling fluid at different times, along with the original formation resistivity. Subsequently, combined with parameters such as the soaking time, fluid viscosity, and saturation change of the drilling fluid, we obtained the phase permeability curve of the reservoir and dynamic effective permeability of the fluid near the wellbore. This study provides basic parameters for subsequent formation analyses and productivity prediction and substantially improves the reservoir evaluation technology from static to dynamic.
1. Introduction
The seepage characteristics of the reservoir are key parameters that control the decision-making for its completion, oilfield development, and reservoir management. The most important parameter for evaluating the seepage characteristics of a reservoir is the effective permeability and can be obtained through testing. However, the cost of testing is extremely high; therefore, the effective permeability is generally extrapolated by the absolute permeability of the laboratory, such as core analysis, log comparison, and well testing [1–6]. The existing methods for obtaining effective permeability have certain limitations in practical applications. For example, core analysis provides an average permeability equivalent to a few cubic inches of samples, and the sample size is too small to describe a reservoir in general terms. However, the scale of formation testing is several thousand cubic feet and cannot provide a detailed description of the reservoir heterogeneity. In view of the aforementioned limitations, an increasing number of scholars at home and abroad have begun to characterize the microscopic pore structure and seepage characteristics of reservoirs through numerical simulation techniques through digital cores, especially in the field of unconventional reservoirs such as shale oil and shale gas [7–13]. With the development of logging and drilling technology, many scholars have begun to pay attention to the multi-angle correlation between the logging data and reservoir permeability. In the early stages, foreign scholars proposed a method of inversion to obtain the absolute permeability of the reservoir by combining the measured data from three types of tools, such as resistivity logging, neutron logging, and density logging, through the history matching method [14–17]. To ensure the reliability of the formation permeability parameters obtained through resistivity logging and other data, scholars began to use nuclear magnetic resonance (NMR), formation testing, pump sampling, and other data to correct the inversion results and estimate the relative permeability curve and capillary pressure of the reservoir curve [18–21]. Through a comparative study, we observed that the main aim of previous research was to invert the permeability of the reservoir by comparing the measured and simulated curves; however, less attention has been paid to the dynamic resistivity profile of the reservoir after considering the drilling fluid invasion and reservoir rocks. The quantitative relationship between the physical parameters, especially the dynamic change in the resistivity profile of the reservoir, is caused by the difference in logging time and reservoir permeability.
The Dongfang Block in western South China Sea is a high-pressure gas field. The well sections of the target layers were all measured using the Schlumberger ARC675 electromagnetic wave resistivity logging tool (ARC675); the resistivity logging is commonly affected by the drilling fluid. Considering the phenomenon of low invasion effect, repeated logging was conducted in each well logging operation, and there was abundant time-lapse resistivity logging data. Considering the actual measurement of electromagnetic wave resistivity while drilling in this area, the author combined the vector finite element method and front-line solution method to solve the forward model of the logging-while-drilling (LWD) resistivity response. The dynamic resistivity profiles of the formation, including the drilling fluid invasion depth, invasion zone resistivity at different times, and the undisturbed formation resistivity, were combined with the changes in formation drilling time, fluid viscosity, and saturation to obtain discrete phase permeability curve characteristic points. The functional relationship between saturation and relative permeability was obtained by fitting the least squares algorithm. Finally, the dynamic effective permeability of the fluid near the wellbore was obtained according to the saturation curve at different times.
2. Method of Joint Inversion of Time-Shift Resistivity
The combined inversion results based on the resistivity logging data at different times can accurately reflect the changing process of the dynamic invasion of the reservoir drilling fluid. First, the vector finite element method was used according to the Maxwell equation of electromagnetic wave response. This method began from the formation model (including the layered structure of the wellbore) to simulate the measurement response of electromagnetic wave resistivity while drilling in the nonuniform formation and to establish and verify a forward modeling method. On this basis, three sets of formation models were constructed as the solution parameters, namely, the radius of the invasion zone (Ri), the resistivity of the invasion zone (Rxo), and the formation resistivity (Rt). The drilling fluid invasion depth, resistivity of the invasion zone, and true resistivity of the formation were obtained at different times, and the dynamic resistivity profile was obtained to provide basic data for the subsequent calculation of the effective permeability.
2.1. Method of Joint Inversion of Time-Shift Resistivity
Measurement theory of electromagnetic wave resistivity while drilling involves solving Maxwell’s equations under the given boundary conditions [22]. The electromagnetic field in the logging problem satisfies the following Maxwell equation:
According to Equations (1) and (2), we deduced that the vector wave equation satisfied by the electric field is where is the complex dielectric constant and is the dielectric constant of the formation.
In layered structured media, the numerical simulation of electrical logging tools is vital for obtaining the tool parameters in complex environments. Several numerical methods have been successfully applied to simulate complex media, including the finite difference time domain (FDTD), finite element method (FEM), and numerical mode matching (NMM). The author first established a physical model of the layered stratigraphic structure including the wellbore (as shown in Figure 1), discretized the stratigraphic model space according to the electromagnetic field distribution law, and then established a two-dimensional discrete grid space (as shown in Figure 2). The vector finite element method and the front-line solution method were combined to solve the aforementioned Maxwell equation to realize the forward modeling of the measurement response of the electromagnetic wave resistivity while drilling in the nonuniform formation. To verify the reliability of the forward modeling model and algorithm, the Schlumberger’s ARC675 instrument was used as an example to conduct relevant forward modeling simulations, and the results were verified with the Schlumberger calibration plate for accuracy.
Figures 3(a), 3(b), 4(a), and 4(b) show the comparison results of the borehole calibration chart for the ARC675 tool. In the chart, the abscissa is the apparent resistivity (Rps and Rad are the apparent resistivities of the phase difference and amplitude ratio, respectively), and the ordinate is the true value of the formation and apparent resistivity. The ratio between resistivity represents the correction factor. Compared with the Schlumberger borehole calibration chart, the data error of each source distance chart was less than 3%, indicating that the forward modeling algorithm was reliable.
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2.2. Time-Shift Resistivity Joint Inversion
Joint inversion refers to the inversion of geological model parameters using two or more types of logging data with different physical mechanisms [23]. The resistivity data collected at different times have different invasion depths and resistivity values of the invasion zone due to the different soaking times of the drilling fluid. However, the resistivity of the undisturbed formation is unique, and the time-lapse resistivity joint inversion can more accurately reflect the geological model parameters because it increases the amount of effective information for the specific detection targets. Compared with the static resistivity profile of iterative inversion, the dynamic resistivity profile of the reservoir was obtained by inversion by considering the influence of logging time on the resistivity profile.
The time-lapse resistivity joint inversion is based on the damped least-squares method and organically unifies the resistivity logging response at different times and the dynamic profile of the formation resistivity for inversion. First, the least-squares objective function is constructed, assuming that is the actual measured logging curve data and f(x) is the response function of the tool to the formation model, and the objective function is assumed to be in the form of the sum of squares: where is the number of logging curves and is the multivariate nonlinear response function for the parameters. The response function is related to the working principle of the tool, structure of the tool, and formation model. This is the parameter to be inverted, including the resistivity measurement at time T1. A total of five sets of formation model parameters can be obtained through the foregoing resistivity forward model to solve for f(x): namely, the intrusion zone radius Ril, intrusion zone resistivity Rxol at time T2, intrusion zone radius Rih, intrusion zone resistivity Rxoh when the resistivity is measured at T2, and the formation resistivity Rt.
Marquette’s algorithm was used to solve the least-squares problem. Gauss proposed a linearization method to expand into a Taylor series near and omit the quadratic term and the above quadratic terms. The formula used is as follows:
It is abbreviated as where is called the Jacobian matrix.
The linear approximate expression of the function after substituting Equation (6) into Equation (4) is as follows:
The purpose of this equation is to solve the minima of the function ; thus, should satisfy the following conditions: wherein, in the formula:
Therefore,
To prevent the appearance of the local optimal solution and improve the convergence of the solution, we introduced a damping factor , and Equation (13) is transformed into where is the identity matrix.
In practical problems, when the observed value is given, the appropriate model and initial values are selected; , , , and can be obtained.
Subsequently, is considered the initial value, and the calculation is repeated to find until the absolute value of component of is less than the error given in advance.
2.3. Joint Inversion Convergence and Confidence
A three-layer stratigraphic model (Figure 5) was used to examine the convergence and confidence of the joint inversion. The stratigraphic model is rotationally symmetric about the well axis and is uniform. The model parameters are set as follows: The formation thickness is 4.0 m, the formation interface values are 1004.0 and 1008.0 m, the borehole diameter is , the mud resistivity is , the upper and lower surrounding rock formation resistivity values are , the radius of the invasion zone while drilling in the target layer is , the resistivity of the invasion zone while drilling is , and the formation resistivity is . The sampling point range and sampling interval are 1000.0 to 1012.0 m and 0.2 m, respectively; a total of 61 points were used.
The resistivity joint inversion of the model was conducted according to the above model parameter settings, and the results are shown in Figure 6. The first trace in the figure is the invasion depth trace, and the blue dotted line is the inversion result of the model invasion radius. The third trace is the resistivity trace, and the red dotted line is the model formation true resistivity inversion result. The green dotted line in the RXOL curve is the model inversion of intrusive zone resistivity. The remaining curves show the phase difference and attenuation resistivity results obtained by inversion.
We analyzed the errors of the invasion radius, invasion zone resistivity, and true formation resistivity obtained by model inversion. As presented in Table 1, the relative errors of the three parameters were within 1.5%, and the inversion exhibited good convergence. This result converges to the true value.
3. Construction Method of the Characteristic Permeability Curve
The time-lapse resistivity joint inversion considers the logging time factor; therefore, the inversion obtains the dynamic resistivity profile of the reservoir. The change in the dynamic profile of the reservoir was mainly caused by the invasion of mud filtrate into the formation to replace the movable oil and gas in it with the passage of logging time. The change in the formation resistivity reflects the change in the formation fluid saturation, and the change in saturation reflects the intrusion of the mud filtrate. The intrusion amount of mud filtrate per unit time is related to the permeability of each phase fluid; therefore, the effective permeability of the reservoir can be obtained by further analysis based on the dynamic resistivity profile of the joint inversion.
Figure 7 shows the basic flow chart of the effective permeability construction. First, based on the dynamic resistivity profile of the reservoir, the resistivity is converted into saturation parameters using Archie’s formula, including the water saturation in the invasion zone and that in the original formation. The saturation change amount of the intrusion zone at the intrusion depth is as follows:
Assuming that the porosity of the reservoir remains unchanged during mud invasion, the change in volume of the fluid water in the rock can be obtained by combining the definitions of porosity and saturation (Equation (19)). Combined with the specific logging time (i.e., the drilling fluid soaking time), the flow formula (Equation (20)) can be calculated, namely, the invasion amount of the mud filtrate per unit time (): where is the formation porosity, is the rock volume of the reservoir, is the logging interval, _x is the change in water saturation, is the drilling fluid invasion depth at the interval, and is the reservoir thickness.
By combining the initial pressure of the formation and the pressure difference of the drilling string, the permeability of the water phase can be deduced by substituting Equation (20) into Darcy’s formula (21). As we considered the real logging environment when calculating the saturation change, namely, the simultaneous existence of multiphase fluids (water, oil, or gas) in the rock, the calculated permeability at this time is the effective permeability of the water phase. where is the effective permeability of the water phase, denotes the wellbore radius, denotes the viscosity of water, denotes the pressure of the drilling string, and denotes the initial pressure of the undisturbed formation.
Assuming that there are only two-phase fluids in the reservoir, and that they are incompatible and incompressible, because the total reservoir fluid saturation remains unchanged, the amount of change in water saturation in the invasion zone and that in oil and gas saturation remains the same. In a complementary relationship, the intrusion of mud filtrate in the reservoir is equal to the displacement of oil and gas, and the effective permeability of oil and gas can be obtained similarly. The relative permeability of each fluid phase can be obtained by calculating the ratio of the effective permeability to the absolute permeability.
The effective permeability calculated above was constructed based on the dynamic resistivity profile of the reservoir, and the dynamic resistivity profile of the reservoir was obtained based on the time-lapse resistivity curve using the joint inversion technique. Therefore, the constructed phase-permeability curve is a discrete characteristic phase permeability curve. It is necessary to obtain a continuous functional relationship between saturation and relative permeability using data fitting technology. The basic functional relationship between saturation and relative permeability is modified by the Brooks-Corey model as follows:
Among them, , , , and represent the undetermined coefficients to be solved; and represent the relative permeability endpoint values of the aqueous and non-aqueous phases; and represent the exponential coefficients; and represents the effective saturation, which is defined as: where is the irreducible water saturation and is the residual oil and gas saturation.
4. Oilfield Application Examples
Well xx adopted a drill bit diameter of 8.375 in; the drilling fluid type was PDF-THERM; the mud resistivity was 0.23 at 23.89 °C; the mud filtrate resistivity is 0.09 at 27.2 °C; and the reservoir lithology was gray mudstone and siltstone. Schlumberger’s EcoScope was the electromagnetic wave resistivity instrument used while drilling, including two time-lapse resistivity measurement datasets of real-time measurement during the drilling and upward remeasurement. The reservoir porosity was 14-19%, and the permeability was 5-62 mD. The resistivity curves of different probing depths measured in real time during drilling were separated to some extent, indicating that a certain amount of drilling fluid invaded the formation. Compared with those measured in real time, the resistivity curves of different probing depths remeasured, while drilling exhibited a greater decrease in resistivity. This shows that the drilling fluid invaded deeper and the change in resistivity corresponded well with the permeability of the reservoir, as shown in Figure 8.
4.1. Reservoir Dynamic Resistivity Profile Inversion
Figure 9 shows the combined inversion results of the main and remeasured resistivity logging in a well, including the invasion depth and invasion zone resistivity measured in real time during drilling, invasion depth and invasion zone resistivity of remeasurement while drilling, and true resistivity of the formation. The first trace in the figure represents the invasion depth trace, including the gamma curve and the inversion depth of the mud invasion at different times. LXO_T1_INV represents the mud invasion depth of joint inversion measured in real time, and LXO_T2_INV represents the remeasured mud invasion depth of the joint inversion. The third and fourth traces show a comparison of the phase difference and amplitude-specific resistivity curves measured in real time, along with the joint inversion results of the resistivity of the invasion zone and true resistivity of the formation, respectively. Traces 5 and 6 show the comparison of the phase difference and amplitude-specific resistivity curves obtained by the remeasurement using the combined inversion results of the resistivity of the invasion zone and true resistivity of the formation, respectively. ERXO_T1_INV represents the intrusion zone resistivity of joint inversion measured in real time, ERXO_T2_INV represents the remeasured intrusion zone resistivity of the joint inversion, and ERT_IINV represents the true resistivity of joint inversion. The seventh trace represents the neutron density porosity curve. The above joint inversion results showed that the inversion depth curve has a good corresponding relationship with the GR lithology curve, as well as with the neutron density curve. The remeasured mud filtrate of the combined inversion of the reservoir with large porosity and good permeability has a greater invasion depth, which demonstrates the rationality of the inversion results.
4.2. Determination of Permeability Characteristic Scattering Points and Construction of Permeability Curve
The dynamic resistivity profile of the reservoir is based on the “layer-to-layer” joint inversion, which expresses the variation law of the electrical profile of the reservoir after drilling-fluid invasion at different times. Therefore, the obtained permeability is the characteristic point of the effective permeability in the near-wellbore zone after drilling fluid invasion at a certain time. Assuming that the porosity of the reservoir remains unchanged during the process of mud filtrate invasion, the pressure difference between the pressure of the drilling fluid column and that of the original formation is experienced on both sides of the formation invasion zone, and the resistivity and saturation are converted using Archie’s equation. The cementation index and porosity index adopt regional empirical values such as and ; the coefficient ; the formation water resistivity , mud filtrate resistivity , and viscosity of formation water are , , and , respectively; and the gas viscosity under formation conditions is . Table 2 lists the average values of the resistivity, invasion depth, porosity, permeability, and other curves of each target feature layer.
Combined with the above reservoir physical parameters and joint inversion results, Equations (16), (22), (23), and (24) can be used to obtain the characteristic phase dispersion point of the target interval; and a characteristic scatter point phase permeability curve relationship diagram is obtained (Figure 10). Based on the data shown in Figure 10, the functional relationship between the water saturation and the relative permeability of each fluid phase can be obtained by fitting Equations (28) and (29).
The fitting function relationship between the formation water saturation and relative permeability of water is
The fitting function relationship between the formation water saturation and relative permeability of gas is where represents the effective saturation, which is defined as where represents the irreducible water saturation and represents the residual gas saturation. During actual data processing, the and values need to be selected in combination with the core analysis results. We conducted a comprehensive analysis of the core NMR and high-temperature and high-pressure gas-water permeability experimental results of the actual application case wells to obtain the irreducible water saturation of the treated well section; and are both 0.2.
4.3. Effective Penetration Rate and Its Damage Analysis
Based on the aforementioned relationship curve of saturation and relative permeability, combined with the dynamic resistivity profile of the reservoir inversion and the logging permeability, a continuous effective permeability curve can be obtained; this includes the original formation (before drilling fluid invasion) and the effective permeability of the gas and water phases at time 1 of real-time logging (drilling fluid immersion for 1 h) and time 2 of re-logging (drilling fluid immersion for 3 h). KEG and KEW in Figure 11 represent the effective permeability curves of the gas and water phases of the undisturbed formation, respectively; KEG1 and KEW1 represent the effective permeability curves of the gas and water phases at time 1 of real-time logging; KEG2 and KEW2 represent the effective permeability curves of the gas and water phases at time 2 of re-logging; TM, TM1, and TM2 represent the comprehensive fluid effective permeability curve of the original formation, real-time logging time 1, and re-logging time 2, respectively, namely, the fluid flow capacity. PDR represents the permeability damage ratio of the reservoir, which is defined both in the original formation and during the re-logging measurement; comprehensive permeability ratio and PDR values less than 1 in a well at time 2 indicate that the reservoir has been benignly improved; a PDR value equal to 1 indicates that the reservoir is free of pollution; and PDR values greater than 1 and larger indicate a strong pollution degree of the reservoir. The third trace is the electromagnetic wave resistivity curve during drilling and the joint inversion result at time 1 of real-time logging; the fourth trace is the electromagnetic wave resistivity curve during drilling and the joint inversion result at time 2 of re-logging; the fifth trace shows the effective permeability of the gas phase at different times in the near-wellbore zone; the sixth trace represents the water phase effective permeability in the near-wellbore zone at different times; the seventh trace represents the comprehensive fluid mobility capacity of the near-wellbore zone at different times; and the eighth trace represents the coefficient of the reservoir PDR.
From the above calculation results, we observed that the gas layer sections (3126-3137 m and 3165-3175 m) are affected by the drilling fluid invasion; only the effective permeability of the gas phase of the well zone gradually decreases and that of the water phase gradually increases. The effective permeability of the water phase near the wellbore was almost zero before drilling fluid invasion, indicating that the saturated water in the formation was irreducible. In the gas-water interval (3200-3208 m) affected by the drilling fluid invasion, the variable law of the effective permeability of the gas and water phases in the near-wellbore zone is consistent with that of the pure gas interval; however, the variation amplitude is smaller than that of the pure gas interval. The effective permeability of the gas and water phases in the near-wellbore zone before the drilling fluid invasion in the same interval of gas and water is not zero, indicating that it contains both movable water and gas. The permeability and PDR curve shows that the 3126-3137 m interval of the gas reservoir is seriously polluted, the 3165-3175 m interval is slightly polluted, and the 3200-3208 m interval of the gas-water reservoir is slightly improved. From the comprehensive permeability curve of the reservoir fluid, the effective permeability of the reservoir before and after drilling fluid invasion in the pure gas interval decreases, while that of the reservoir in the same gas and water interval increases. In particular, as the water saturation of the reservoir increases, the variation law of the integrated fluid in the reservoir shows that the permeability decreases first and increases subsequently (TKr curve in Figure 12).
5. Conclusions
(1)The time-lapse resistivity logging data and reservoir dynamic and static permeability have multi-angle correlations. The characteristic phase permeability curve calculation model can be derived using Darcy’s formula, the resistivity dynamic invasion profile, and the water saturation model. By combining the regional interpretation model, we realized an effective permeability evaluation of reservoirs based on time-lapse resistivity logging. The calculation results are more representative than the effective permeability parameters obtained by conventional core analysis and more accurately reflect the seepage characteristics of the reservoir(2)By comprehensively analyzing the dynamic resistivity profile of the reservoir, and combined with the calculated effective permeability parameters of the reservoir before and after drilling fluid invasion, we obtained the permeability damage ratio of the reservoir. This ratio can provide guidance for evaluating reservoir pollution and testing the formation(3)This study did not consider the differences in the relative permeability curves of different reservoirs when calculating the effective permeability. To improve the accuracy of the effective permeability calculation results, it is necessary to analyze the variation law of the characteristic relative permeability curves of the reservoirs in the study area and select the appropriate relative permeability according to the reservoir-type and curve parameters
Data Availability
The well data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.