Fracture Mechanics Approach to the Growth Process of Fracture Network for Direct Heat Heat Extraction from Magma

在直接从岩浆中提取热量的系统中，工作流体（水）通过裂缝流动来提取热量，裂缝是由凝固的岩浆热收缩产生的。裂缝的作用是双重的，即流体路径和换热面。因此，这些骨折的特征是至关重要的。在本研究中，研究了断裂的扩展过程。据推测，与热交换系统相比，岩浆房要大得多。选取以下条件作为基本情况，即体系深度为7km，固相温度为800℃，杨氏模量乘以线性膨胀系数4.5X10^5（Pa/℃）。应力场由三个场组成，即首先是由作用于固化区外边界的过料压力引起的应力场，其次是由作用于固化区内边界的压力引起的应力场，最后是热应力场。分析是基于断裂力学进行的。结果表明，在基准情况下，前两条裂缝在463℃时扩展，随后依次扩展。第9条断口发育的温度为457℃。这两种温度的差别只有6℃。这意味着裂缝的生长和裂缝网络的形成几乎是同时发生的，因此我们无法控制。影响裂缝网络形成的因素似乎很多，如裂缝韧性、井筒半径、弹性常数等。其中，换热系统深度对断裂尺寸的影响最为显著。断裂韧性作用较小。

In the system of direct heat extraction from magma, working fluid (water) extracts heat by flowing through fractures, which are created by thermal contraction of solidified magma. Role of fractures is two fold, i.e., fluid path and heat exchange surfaces. Thus, the characteristics of these fractures are crucial. In the present research, the process of fracture growth is studied. It is supposed that the magma chamber is significantly large compared with the heat exchange system. The following condition is adopted as the base case, i.e., depth of the system is 7km, solidus temperature 800℃ and the Young's modulus times linear expansion coefficient 4.5X10^5(Pa/℃). The stress field consists of three fields, i.e., firstly the one due to over burden pressure acting on the outer boundary of the solidified region, secondly the one induced by pressure acting on the inner boundary of the solidified region and lastly the thermal stress field. The analysis is performed based on the Fracture Mechanics. It is found that growth of the first two fractures takes place at 463℃ in the base case and is followed by sequential fracture growth. The temperature at which the growth of ninth fracture takes place, is 457℃. The difference is only 6℃ between the two temperatures. This implies that the fracture growth and formation of fracture network would be almost simultaneous and thus be out of our control. There seem to be many factors which have effects on the formation of fracture network, such as fracture toughness, radius of wellbore, elastic constants and so on. Among these factors, depth of the heat exchange system has most significant effect, regarding the fracture size. Fracture toughness is less effective.