Development of Analysis Method for Geothermal Pressure Interference Test Using Kalman Filtering

卡尔曼滤波已被应用于开发方法来分析多井，多流量问题的压力干扰测试。在这项研究中考虑了两种不同的油藏系统：无限大的多孔油藏和存在水力边界的多孔油藏。在上述情况下，观测井中观测到的压力变化可以用叠加原理在线源解上表示。这些解析解是非线性的参数来确定：transmittance和storativity。为此，采用了两种不同的线性化方法：一种是取普通对数，另一种是用泰勒级数展开式逼近非线性函数。该分析方法依赖于线性化方式的特点是：1）对于对数情况，即使初始估计值与最可能值之间的偏差大几个数量级，估计值在估计过程中也不会发散; 2）对于Taylor展开式，在初始估计值与最概然值相差一个数量级的条件下，可以在估计过程的早期估计出合理的参数值。因此，与上述方法相比，可以更快速地估计。

Kalman filtering has been applied to develop methods to analyze pressure interference tests of multiple-well, multiple-rate problems. Two different reservoir systems are considered in this study : an infinite porous reservoir and a porous reservoir with a presence of a hydraulic boundary. Pressure change observed in an observation well under the situations above can be expressed by applying the principle of superposition to the line source solution. These analytical solutions are nonlinear with respect to parameters to be identified : transmissivity and storativity. Therefore, two different methods are adopted to linealize the solutions : one is taking the ordinary logarithm, and another is to approximate the nonlinear function with Taylor's series expansion. Then, Kalman filtering is formed using these linealized equations, and two parameters are estimated at every moment when observation pressure value is obtained.Characteristics of the analysis method depending on the way of linealization are :1)For the logarithmic case, estimated values never diverge during the estimation procedure even if the discrepancy between the initial estimates and the most probable values is greater by several orders of magnitude. Therefore, this method is particularly effective when no prior information on the parameters is available.2)For Taylor's expansion case, reasonable values of parameters can be estimated at early stage of the estimation procedure under conditions that the initial estimates are given within a difference of one order of magnitude compared with the most probable values. Thus, more quick estimation is possible compared with the method above.