背斜地層在擠注條件下之水流與井流量研析

Abstract

背斜地層具有上凸構造，長期以來被視為是合適於進行地質封存之場址。然而，由於背斜構造之不規則邊界，導致推求解析解有其困難度，故迄今尚缺乏文獻就此類問題進行基礎性之探究。因此，本研究根據時變性井流理論，考慮一完全貫穿之直立井的注入情境，建立一個新的數學模式，用以描述在定勢能注液條件下，向上隆起之異質非等向受限含水層內的勢能分佈，並推求解析解。

過程中，本研究提出以階梯形態近似背斜上凸構造之原創性概念，將求解域不規則之幾何形狀，透過階梯高度變化處之垂直假想線的設定，得以分隔為兩個規則形狀之區域。據此，乃可進以利用拉普拉斯轉換與分離變數法，導入邊界條件，分別推求兩個區域之勢能分佈解。再則，依據兩個區域相連界面上的連續條件，藉由傅立葉級數的概念，定義前述勢能分佈解中之未定係數，以推導於拉普拉斯域的解；同時，並利用數值逆轉方法，計算出時間域之勢能分佈值。此外，更基於達西定律與拉普拉斯轉換，直接推得時間域之注入率。使用本解析解進行模擬的結果，顯示在全程注入期間水力傳導係數為影響注入率之關鍵參數，而比儲蓄係數則僅在注入初期有效應，中後期對於注入率所產生之影響幾乎可忽略。

本研究所推得的解析解，亦可用來分析平板形同心雙水層的勢能分佈；若考量水文地質條件為均質等向，且定勢能注液試驗到達穩態時，本解析解變為Thiem equation。綜而言之，本研究針對不規則邊界所發展的勢能分佈與抽注率方程式，不但可模擬含水層因井流造成的勢能時空變化，而且也能作為不同背斜地層之抽注量的評估工具。

It is well known that an anticline aquifer is commonly selected as a geological storage site for fluid injection. However, there is still lack of literatures on analytically solving the problem of flow in anticlines because of the difficulty involved in the irregular shape of domain geometry. In this study, a transient mathematical model is launched for simulating transient head distribution in a heterogeneous and anisotropic anticline formation with a fully penetrating well. A form of step change is proposed to approximate the upper and lower boundaries of the anticline formation. The domain of the host formation is then divided into two regions according to a vertical image line set at the location where the step height changes. By virtue of the properties of Fourier series, the method of separation of variables is first employed. Furthermore, based on Darcy’s law and Laplace inversion using Bromwich integral method, the analytical solution of the model for describing temporal injection rate at wellbore can be obtained. It is found that the hydraulic conductivity is the key factor on affecting injection rate during the injecting time, and the effect of specific storage on the injection rate for middle to late times is almost negligible.

In addition, the solution can also be utilized to analyze head distribution in the reservoir system with two concentric transmissivity zones if the trap of the formation is absent. If the upper boundary becomes flat and the reservoir is homogeneous and isotropic, the solution reduces to the Thiem equation when the injection time is very large. This solution may be used as a primary tool to assess the capacity of fluid injection to various anticline reservoirs.