I have 3 implicit equations in 5 variables: $f_1(S, y_h, y_d, x_h, J)=0$, $f_2(S, y_h, y_d, x_h, J)=0$, and $f_3(S, y_h, y_d, x_h, J)=0$. These equations determine a 3D surface in $S-y_h-y_d$ coordinate system. The equations are given below. 
```
M = 1;
roo = 0.01;
xd = 3;

f1[S_, yh_, yd_, xh_, J_]:= 2 M (xh^2 - (yh + 1)^2)/(xh^2 + (yh + 1)^2)^2 + 
  2 S ((xh - xd)^2 - (yh + yd)^2)/((xh - xd)^2 + (yh + yd)^2)^2 + 
  J/(2 yh) == 0,

f2[S_, yh_, yd_, xh_, J_]:= M*xh (yh + 1)/(xh^2 + (yh + 1)^2)^2 + 
  S (xh - xd) (yh + yd)/((xh - xd)^2 + (yh + yd)^2)^2 == 0,

f3[S_, yh_, yd_, xh_, J_]:= J*Log[2 yh*J/roo] + 2 M (yh + 1)/(xh^2 + (yh + 1)^2) + 
  2 S (yh + yd)/((xh - xd)^2 + (yh + yd)^2) - (Log[2/roo] + 1) == 0,
```
where $S, y_h, y_d, x_h, J$ are `Reals`, $yh>0$, and $yd>0$.

 I am trying to obtain the surface in the $S-yh-yd$ coordinate system like the following image. How to plot it?
[![enter image description here][1]][1]

What I have tried: For 1 equation in 3 variables: $f(S, y_h, y_d) = 0$, I can use `ContourPlot3D` to get the surface; For equations expressed in an explicit form: $S = S(x_h, J)$, $y_h = y_h(x_h, J)$, $y_d = y_d(x_h, J)$, I can use `ParametricPlot3D`. But I failed when I faced this issue introduced above. Please help me. Thanks a lot for your support!

Background Information
---
This problem is from the paper [_Prominence eruptions and coronal mass ejections triggered by newly emerging flux_ by _J. Lin, T. G. Forbes, P. A. Isenberg_](https://doi.org/10.1029/2001JA000046).
 


  [1]: https://i.sstatic.net/jK0jf.png