An analytical solution is presented for the displacement, strain and stress of a three-dimensional poro-elastic model with three layers, where the three layers are an underburden, a reservoir with a given fluid pressure, and an overburden. The fluid pressure in the reservoir is assumed symmetrical around the z-axis and represented by a Fourier cosine series. The poro-elastic solution is expressed as a superposition of the solutions for each term in the Fourier series. It is shown that the bulk strain in the reservoir layer is proportional to the fluid pressure and that the bulk strain in the underburden and overburden is zero. Using these properties of the bulk strain, a solution is derived for the three-layer model where the fluid flow and mechanics are fully coupled. A particular aim of the model is to study the surface uplift from a given reservoir pressure. The expansion of the reservoir and the uplift of the surface are studied in terms of the wavelengths in the Fourier representation of the pressure. It is shown that the surface uplift can be written in a similar form to the 1D vertical expansion of the reservoir layer, but where the fluid pressure is based on the Fourier series. It is shown that the amplitudes with average wavelengths longer than \(2\pi \) times the thickness of the reservoir give expansion of the reservoir, but average wavelengths much shorter than this limit do not. Similarly, amplitudes with average wavelengths much longer than \(2\pi \) times the thickness of the overburden produce surface uplift, but wavelengths much shorter do not. The stress in the overburden, which is generated by the reservoir fluid pressure, is also analysed in terms of the wavelengths. A case is given where the analytical uplift is compared with the results of a numerical simulation and the agreement is excellent.

中文翻译：

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压力储层引起的储层膨胀，地表隆升和盖层应力的三维解析解

提出了具有三层的三维孔隙弹性模型的位移，应变和应力的解析解，其中三层是下伏岩层，具有给定流体压力的储层和上覆岩层。假设油藏中的流体压力绕z对称轴并以傅立叶余弦级数表示。孔隙弹性解表示为傅立叶级数中每个项的解的叠加。结果表明，储层中的整体应变与流体压力成正比，下伏和上覆岩层的总应变为零。利用整体应变的这些特性，可以得出三层模型的解决方案，其中流体流动和力学完全耦合。该模型的一个特殊目的是研究给定油藏压力下的地表隆起。根据压力的傅立叶表示中的波长研究了储层的膨胀和地表的隆升。结果表明，表面隆起可以用与储层的一维垂直扩展相似的形式书写，但流体压力基于傅立叶级数。结果表明，平均波长的振幅比\（2 \ pi \）倍于储层的厚度会使储层膨胀，但比该极限短得多的平均波长却不会。类似地，平均波长的振幅比上覆层厚度的\（2 \ pi \）长得多，但会产生表面隆起，而短得多的波长却不会。还根据波长分析了由储层流体压力产生的覆盖层应力。给出了将解析隆起与数值模拟结果进行比较的情况，并且一致性非常好。