Equations

Domain is a cylinder \(D = [rw, R] * [0, 2\pi] * [-L, L]\), which includes fracture \(D_f = [rw, R] * [0, 2\pi] * [-\delta, \delta]\).

Reservoir:

• pressure:

  \[\text{div}(u) = 0, \;\;\; u = -\sigma(\boldsymbol{x}, s) \cdot \triangledown p, \;\;\; \sigma = k(\boldsymbol{x}) / \mu(s).\]

here \(\boldsymbol{x}\) is a point, \(\mu(s) -\) fluid viscosity, \(u -\) flow rate, \(s -\) (water) saturation, \(k -\) - permeability, \(\sigma -\) transmissibility of fluid.

Functions \(\mu(s), f(s)\) defined from relative permeability degree functions

\[\begin{split}\dfrac{1}{\mu(s)} = \dfrac{\phi(s)}{\mu_w}, \;\;\; \phi(s) = k_w(s) + k_{\mu} k_o(s), \;\;\; k_{\mu} = \dfrac{\mu_w}{\mu_o}, \\ k_w(s) = s^N, \;\;\; k_o(s) = \left(1-s\right)^N.\end{split}\]

here \(k_w -\) relative permeability of water, \(k_o -\) relative permeability of oil, \(\mu_w -\) water viscosity, \(\mu_o -\) oil viscosity.

Note

Note that irreducible water and oil saturations are equal to 0.

• saturation:

  \[s = const.\]

Fracture:

Let us define average value over pressure variables

(1)\[\left<\cdot\right>_f = \dfrac{1}{2\delta} \int_{-\delta}^{\delta} \left(\cdot\right)dz.\]

• pressure:

  (2)\[\text{div}(u_f) = 0, \;\;\; u_f = -\sigma_f(s_f) \cdot \triangledown p_f, \;\;\; \sigma_f = k_f / \mu(s).\]

here \(u_f -\) flow rate in fracture, \(s -\) (water) saturation, \(k_f -\) constant fracture permeability.

Assume that

\[<u_f>_r \approx -\dfrac{k_f}{\mu}\triangledown <p_f>.\]

If we integrate equation (2) from \(-\delta\) to \(\delta\) using (1) and consider that we get (average signs \(<>\) are gone)

(3)\[\triangledown_r u_f + \dfrac{1}{2\delta} \left.u_z\right|_{\Gamma_f(-\delta)}^{\Gamma_f(\delta)} = 0.\]

The last member in (3) defines the flow from reservoir to fracture from bounds \(\Gamma_f(-\delta), \Gamma_f(\delta)\).

• saturation:

  (4)\[m_f \dfrac{\partial s_f}{\partial t} + \triangledown_r \left(f(s_f)u_f\right) + \dfrac{1}{2\delta} \left.\left(f(s)u_z\right)\right|_{\Gamma_f(-\delta)}^{\Gamma_f(\delta)} = 0, \;\;\; f(x) = \dfrac{k_w(x)}{\phi(x)},\]

  here \(f(x) -\) Backley-Leverette function.

Conditions:

• boundary:

  \[\begin{split}\Gamma_w^f \in [r_w, rw]*[0,2\pi]*[-\delta, \delta]: p = p_w, \\ \Gamma_T \in [r_w, R]*[0, 2\pi]*[\pm L,\pm L]: p = p_c, \\ \Gamma_w \in [r_w, rw]*[0,2\pi]*[-L, L]/[-\delta, \delta]: u_n = -\sigma \dfrac{\partial p}{\partial n} = 0, \\ \Gamma_B \in [R, R]*[0, 2\pi]*[-L, L]: u_n = 0, \\ \Gamma_f \in [R, R]*[0, 2\pi]*[\pm \delta, \pm \delta]: p = p_f, u = u_f, s = s_0.\end{split}\]

• initial:

  \[\begin{split}s = s_0 = const, \\ s_f = s_f^0.\end{split}\]

Well flow rate:

• from fract to well:

  \[q = \int_{\Gamma_w^f} u_f d\Gamma = 4\pi r_w \delta u_f.\]

• from reservoir to fract:

  \[q = \int_{\Gamma_T} u d\Gamma.\]