By Tatsien Li, Yongji Tan, Zhijie Cai, Wei Chen, Jingnong Wang

Spontaneous capability (SP) well-logging is without doubt one of the commonest and helpful well-logging innovations in petroleum exploitation. This monograph is the 1st of its type at the mathematical version of spontaneous power well-logging and its numerical ideas. The mathematical version confirmed during this publication exhibits the need of introducing Sobolev areas with fractional energy, which heavily raises the trouble of proving the well-posedness and offering numerical resolution schemes. during this booklet, within the axisymmetric state of affairs the well-posedness of the corresponding mathematical version is proved and 3 effective schemes of numerical resolution are proposed, supported by means of a few numerical examples to fulfill useful computation needs.

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2. 1, there exists wε ∈ Wε such that as ε → 0, wε → w strongly in H∗1 (Ω). Denote vε = uε − wε |Ωε . 28), for any given φε ∈ Wε0 (Ωε ) we have aε (vε , φε ) = lε (φε ) − aε (wε , φε ). 33) Evidently, vε ∈ Wε0 (Ωε ). Taking φε = vε in the above formula yields aε (vε , vε ) = lε (vε ) − aε (wε , vε ). 34) Then we obtain ∇vε 2 L2∗ (Ωε ) ≤ C vε H∗1 (Ωε ) + ∇wε L2∗ (Ωε ) ∇vε L2∗ (Ωε ) . 35) Since as ε → 0, wε converges to w strongly in H∗1 (Ω), ∇wε L2∗ (Ωε ) is uniformly bounded with respect to ε. 2.

Since there are spontaneous potential differences on the interfaces γk (k = 1, . . , 5), we have to make a corresponding treatment. 12). For any given element e, if the node point q ∈ γk+ , we take it as a common node point to which no extra treatment is needed since u+ q (denoted by uq ) is kept as a − freedom. 12), we have u− q = uq − Ek (q), the contribution of which to the variational form is then σe ∇Np · ∇Nq uq − Ek (q) rdrdz e = σe ∇Np · ∇Nq rdrdz uq − e σe ∇Np · ∇Nq rdrdz Ek (q). 8) where Ae·q stands for the q-column of the element stiffness matrix Ae , and Ek (q) stands for the value of the spontaneous potential difference Ek at point q.

We fix the physical parameters and geometric parameters as in the case of Fig. 4 except E1 . Take E1 = 20 mV and E1 = 250 mV for the numerical simulation, respectively, and plot the vectors of electric field and the level curves of potential for these two cases, as shown in Figs. 6 and Figs. 8, respectively. From these figures we find two important facts. Firstly, the electric field varies seriously around point A but evenly around point B, since the compatible condition is satisfied at point B but violated at point A in both cases.

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