Fast Forward Modeling Method for Gamma LWD Using 1D Equivalent Integral in High Inclination or Horizontal Well

During logging-while-drilling (LWD) geosteering, comparing the forward gamma logging response with the real one is one key technology to determine the real formation mode. As the Monte Carlo method is too slow to meet the real-time need of forward modeling, it is also difficult to deduce analytical expressions for gamma logging response in high inclination or horizontal wells. Therefore, it is a challenge to find a fast forward modeling method for real-time applications of gamma LWD. The author once invented a fast forward modeling method using three-dimensional (3D) partition integral in high inclination or horizontal wells, but the algorithm is somewhat complex to apply. Therefore, the author invents a new one-dimensional (1D) integral fast forward modeling method, which overcomes the difficulty of real-time gamma forward modeling. The principle of this method is to take the gamma ray flux received by the detector in an inclined or horizontal well as a 1D equivalent integral problem under vertical well conditions, including three key points. (1) First, take the total gamma rays received by the detector, equivalent to that from sphere space with effective detection radius r0. Within the sphere space, obtain the integral expression φ(r0, z) of the gamma ray flux under vertical well conditions, which is a function of the distance z between the formation and detector center. (2) Then, using the numerical integral results of φ(r0, z) to fit a double exponential function varying with the distance z. After normalization, get the 1D longitudinal equivalent contribution coefficient gz of the formation’s gamma ray under vertical well conditions. (3) Finally, using the effective detection radius r0 of the gamma ray detector, determine the main contribution sphere space of gamma rays from the formation. Moreover, calculate the distance z between the detector and the formation within the sphere. Then, along the formation normal direction, using the 1D gz as the contribution coefficient, integrate the gamma ray flux of formation contribution and get the forward value. Figure 1a shows the forward results at different angles between wellbore trajectories and formation model, which using the 1D equivalent integral method and the 3D sphere space partition integral algorithm individually. In Fig. 1a, the dashed line in step shape represents the formation gamma ray model; other different types of lines are obtained using a 3D integral, and scattered points with different shapes are obtained using a 1D integral. We can see that at the same angle between formation and wellbore trajectory, the forward results of the two methods overlap together, just with an average relative error of about 1%, verifying the correctness of the 1D method. Figure 1b is a demonstration using the 1D fast forward modeling for bed-boundary identification. The target layer is a shale gas zone with mud interlayers. The upper and left sides show the LWD curves indexed on horizontal displacement and vertical depth of the wellbore trajectory. RGR (blue color) is the real gamma LWD curve, and FGR (red color) is the fast forward gamma curve. The lower right area shows the vertical profile of the formation model and wellbore trajectories; the red one represents the actual trajectory, and the blue one represents the designed trajectory, which is along the maximum horizontal displacement direction of the wellbore. The FGR shape is consistent with the RGR overall, indicating the correctness and effectiveness of the method for real-time formation model identification and adjustment. These results show that the 1D equivalent integral method has high efficiency and is more convenient than the 3D one, without the complex division of 3D sphere space, can meet the real-time requirements of forward modeling, and provides an algorithm foundation for fast inversion.